algorithms.statistics.formula.formulae

Module: algorithms.statistics.formula.formulae

Inheritance diagram for nipy.algorithms.statistics.formula.formulae:

Inheritance diagram of nipy.algorithms.statistics.formula.formulae

Formula objects

A formula is basically a sympy expression for the mean of something of the form:

mean = sum([Beta(e)*e for e in expr])

Or, a linear combination of sympy expressions, with each one multiplied by its own “Beta”. The elements of expr can be instances of Term (for a linear regression formula, they would all be instances of Term). But, in general, there might be some other parameters (i.e. sympy.Symbol instances) that are not Terms.

The design matrix is made up of columns that are the derivatives of mean with respect to everything that is not a Term, evaluted at a recarray that has field names given by [str(t) for t in self.terms].

For those familiar with R’s formula syntax, if we wanted a design matrix like the following:

> s.table = read.table("http://www-stat.stanford.edu/~jtaylo/courses/stats191/data/supervisor.table", header=T)
> d = model.matrix(lm(Y ~ X1*X3, s.table)
)
> d
   (Intercept) X1 X3 X1:X3
1            1 51 39  1989
2            1 64 54  3456
3            1 70 69  4830
4            1 63 47  2961
5            1 78 66  5148
6            1 55 44  2420
7            1 67 56  3752
8            1 75 55  4125
9            1 82 67  5494
10           1 61 47  2867
11           1 53 58  3074
12           1 60 39  2340
13           1 62 42  2604
14           1 83 45  3735
15           1 77 72  5544
16           1 90 72  6480
17           1 85 69  5865
18           1 60 75  4500
19           1 70 57  3990
20           1 58 54  3132
21           1 40 34  1360
22           1 61 62  3782
23           1 66 50  3300
24           1 37 58  2146
25           1 54 48  2592
26           1 77 63  4851
27           1 75 74  5550
28           1 57 45  2565
29           1 85 71  6035
30           1 82 59  4838
attr(,"assign")
[1] 0 1 2 3
>

With the Formula, it looks like this:

>>> r = np.rec.array([
...     (43, 51, 30, 39, 61, 92, 45), (63, 64, 51, 54, 63, 73, 47), 
...     (71, 70, 68, 69, 76, 86, 48), (61, 63, 45, 47, 54, 84, 35),
...     (81, 78, 56, 66, 71, 83, 47), (43, 55, 49, 44, 54, 49, 34),
...     (58, 67, 42, 56, 66, 68, 35), (71, 75, 50, 55, 70, 66, 41),
...     (72, 82, 72, 67, 71, 83, 31), (67, 61, 45, 47, 62, 80, 41),
...     (64, 53, 53, 58, 58, 67, 34), (67, 60, 47, 39, 59, 74, 41),
...     (69, 62, 57, 42, 55, 63, 25), (68, 83, 83, 45, 59, 77, 35),
...     (77, 77, 54, 72, 79, 77, 46), (81, 90, 50, 72, 60, 54, 36),
...     (74, 85, 64, 69, 79, 79, 63), (65, 60, 65, 75, 55, 80, 60),
...     (65, 70, 46, 57, 75, 85, 46), (50, 58, 68, 54, 64, 78, 52),
...     (50, 40, 33, 34, 43, 64, 33), (64, 61, 52, 62, 66, 80, 41),
...     (53, 66, 52, 50, 63, 80, 37), (40, 37, 42, 58, 50, 57, 49),
...     (63, 54, 42, 48, 66, 75, 33), (66, 77, 66, 63, 88, 76, 72),
...     (78, 75, 58, 74, 80, 78, 49), (48, 57, 44, 45, 51, 83, 38),
...     (85, 85, 71, 71, 77, 74, 55), (82, 82, 39, 59, 64, 78, 39)],
...              dtype=[('y', '<i8'), ('x1', '<i8'), ('x2', '<i8'),
...                     ('x3', '<i8'), ('x4', '<i8'), ('x5', '<i8'),
...                     ('x6', '<i8')])
>>> x1 = Term('x1'); x3 = Term('x3')
>>> f = Formula([x1, x3, x1*x3]) + I
>>> f.mean
_b0*x1 + _b1*x3 + _b2*x1*x3 + _b3

The I is the “intercept” term, I have explicity not used R’s default of adding it to everything.

>>> f.design(r)  
array([(51.0, 39.0, 1989.0, 1.0), (64.0, 54.0, 3456.0, 1.0),
       (70.0, 69.0, 4830.0, 1.0), (63.0, 47.0, 2961.0, 1.0),
       (78.0, 66.0, 5148.0, 1.0), (55.0, 44.0, 2420.0, 1.0),
       (67.0, 56.0, 3752.0, 1.0), (75.0, 55.0, 4125.0, 1.0),
       (82.0, 67.0, 5494.0, 1.0), (61.0, 47.0, 2867.0, 1.0),
       (53.0, 58.0, 3074.0, 1.0), (60.0, 39.0, 2340.0, 1.0),
       (62.0, 42.0, 2604.0, 1.0), (83.0, 45.0, 3735.0, 1.0),
       (77.0, 72.0, 5544.0, 1.0), (90.0, 72.0, 6480.0, 1.0),
       (85.0, 69.0, 5865.0, 1.0), (60.0, 75.0, 4500.0, 1.0),
       (70.0, 57.0, 3990.0, 1.0), (58.0, 54.0, 3132.0, 1.0),
       (40.0, 34.0, 1360.0, 1.0), (61.0, 62.0, 3782.0, 1.0),
       (66.0, 50.0, 3300.0, 1.0), (37.0, 58.0, 2146.0, 1.0),
       (54.0, 48.0, 2592.0, 1.0), (77.0, 63.0, 4851.0, 1.0),
       (75.0, 74.0, 5550.0, 1.0), (57.0, 45.0, 2565.0, 1.0),
       (85.0, 71.0, 6035.0, 1.0), (82.0, 59.0, 4838.0, 1.0)], 
      dtype=[('x1', '<f8'), ('x3', '<f8'), ('x1*x3', '<f8'), ('1', '<f8')])

Classes

Beta

class nipy.algorithms.statistics.formula.formulae.Beta

Bases: sympy.core.symbol.Dummy

A symbol tied to a Term term

Methods

__call__(*args)
adjoint()
apart([x]) See the apart function in sympy.polys
args_cnc([cset, warn, split_1]) Return [commutative factors, non-commutative factors] of self.
as_base_exp()
as_coeff_Add([rational]) Efficiently extract the coefficient of a summation.
as_coeff_Mul([rational]) Efficiently extract the coefficient of a product.
as_coeff_add(*deps) Return the tuple (c, args) where self is written as an Add, a.
as_coeff_exponent(x) c*x**e -> c,e where x can be any symbolic expression.
as_coeff_mul(*deps, **kwargs) Return the tuple (c, args) where self is written as a Mul, m.
as_coefficient(expr) Extracts symbolic coefficient at the given expression.
as_coefficients_dict() Return a dictionary mapping terms to their Rational coefficient.
as_content_primitive([radical, clear]) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive).
as_dummy() Return a Dummy having the same name and same assumptions as self.
as_expr(*gens) Convert a polynomial to a SymPy expression.
as_independent(*deps, **hint) A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps.
as_leading_term(*args, **kwargs) Returns the leading (nonzero) term of the series expansion of self.
as_numer_denom() expression -> a/b -> a, b
as_ordered_factors([order]) Return list of ordered factors (if Mul) else [self].
as_ordered_terms([order, data]) Transform an expression to an ordered list of terms.
as_poly(*gens, **args) Converts self to a polynomial or returns None.
as_powers_dict() Return self as a dictionary of factors with each factor being treated as a power.
as_real_imag([deep])
as_terms() Transform an expression to a list of terms.
atoms(*types) Returns the atoms that form the current object.
cancel(*gens, **args) See the cancel function in sympy.polys
class_key()
coeff(x[, n, right]) Returns the coefficient from the term(s) containing x**n.
collect(syms[, func, evaluate, exact, …]) See the collect function in sympy.simplify
combsimp() See the combsimp function in sympy.simplify
compare(other) Return -1, 0, 1 if the object is smaller, equal, or greater than other.
compute_leading_term(x[, logx]) as_leading_term is only allowed for results of .series()
conjugate()
copy()
could_extract_minus_sign() Canonical way to choose an element in the set {e, -e} where e is any expression.
count(query) Count the number of matching subexpressions.
count_ops([visual]) wrapper for count_ops that returns the operation count.
diff(*symbols, **assumptions)
doit(**hints)
dummy_eq(other[, symbol]) Compare two expressions and handle dummy symbols.
equals(other[, failing_expression]) Return True if self == other, False if it doesn’t, or None.
evalf([n, subs, maxn, chop, strict, quad, …]) Evaluate the given formula to an accuracy of n digits.
expand(*args, **kwargs) Expand an expression using hints.
extract_additively(c) Return self - c if it’s possible to subtract c from self and make all matching coefficients move towards zero, else return None.
extract_branch_factor([allow_half]) Try to write self as exp_polar(2*pi*I*n)*z in a nice way.
extract_multiplicatively(c) Return None if it’s not possible to make self in the form c * something in a nice way, i.e.
factor(*gens, **args) See the factor() function in sympy.polys.polytools
find(query[, group]) Find all subexpressions matching a query.
fourier_series([limits]) Compute fourier sine/cosine series of self.
fps([x, x0, dir, hyper, order, rational, full]) Compute formal power power series of self.
fromiter(args, **assumptions) Create a new object from an iterable.
getO() Returns the additive O(..) symbol if there is one, else None.
getn() Returns the order of the expression.
has(*args, **kwargs) Test whether any subexpression matches any of the patterns.
integrate(*args, **kwargs) See the integrate function in sympy.integrals
invert(g, *gens, **args) Return the multiplicative inverse of self mod g where self (and g) may be symbolic expressions).
is_algebraic_expr(*syms) This tests whether a given expression is algebraic or not, in the given symbols, syms.
is_constant(*wrt, **flags)
is_hypergeometric(k)
is_polynomial(*syms) Return True if self is a polynomial in syms and False otherwise.
is_rational_function(*syms) Test whether function is a ratio of two polynomials in the given symbols, syms.
leadterm(x) Returns the leading term a*x**b as a tuple (a, b).
limit(x, xlim[, dir]) Compute limit x->xlim.
lseries([x, x0, dir, logx]) Wrapper for series yielding an iterator of the terms of the series.
match(pattern[, old]) Pattern matching.
matches(expr[, repl_dict, old])
n([n, subs, maxn, chop, strict, quad, verbose]) Evaluate the given formula to an accuracy of n digits.
normal()
nseries([x, x0, n, dir, logx]) Wrapper to _eval_nseries if assumptions allow, else to series.
nsimplify([constants, tolerance, full]) See the nsimplify function in sympy.simplify
powsimp(*args, **kwargs) See the powsimp function in sympy.simplify
primitive() Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add).
radsimp(**kwargs) See the radsimp function in sympy.simplify
ratsimp() See the ratsimp function in sympy.simplify
rcall(*args) Apply on the argument recursively through the expression tree.
refine([assumption]) See the refine function in sympy.assumptions
removeO() Removes the additive O(..) symbol if there is one
replace(query, value[, map, simultaneous, exact]) Replace matching subexpressions of self with value.
rewrite(*args, **hints) Rewrite functions in terms of other functions.
round([p]) Return x rounded to the given decimal place.
separate([deep, force]) See the separate function in sympy.simplify
series([x, x0, n, dir, logx]) Series expansion of “self” around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.
simplify([ratio, measure]) See the simplify function in sympy.simplify
sort_key(*args, **kwargs)
subs(*args, **kwargs) Substitutes old for new in an expression after sympifying args.
taylor_term(n, x, *previous_terms) General method for the taylor term.
together(*args, **kwargs) See the together function in sympy.polys
transpose()
trigsimp(**args) See the trigsimp function in sympy.simplify
xreplace(rule[, hack2])
__init__()

x.__init__(…) initializes x; see help(type(x)) for signature

adjoint()
apart(x=None, **args)

See the apart function in sympy.polys

args

Returns a tuple of arguments of ‘self’.

Notes

Never use self._args, always use self.args. Only use _args in __new__ when creating a new function. Don’t override .args() from Basic (so that it’s easy to change the interface in the future if needed).

Examples

>>> from sympy import cot
>>> from sympy.abc import x, y
>>> cot(x).args
(x,)
>>> cot(x).args[0]
x
>>> (x*y).args
(x, y)
>>> (x*y).args[1]
y
args_cnc(cset=False, warn=True, split_1=True)

Return [commutative factors, non-commutative factors] of self.

self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly supressed by setting warn to False.

Note: -1 is always separated from a Number unless split_1 is False.

>>> from sympy import symbols, oo
>>> A, B = symbols('A B', commutative=0)
>>> x, y = symbols('x y')
>>> (-2*x*y).args_cnc()
[[-1, 2, x, y], []]
>>> (-2.5*x).args_cnc()
[[-1, 2.5, x], []]
>>> (-2*x*A*B*y).args_cnc()
[[-1, 2, x, y], [A, B]]
>>> (-2*x*A*B*y).args_cnc(split_1=False)
[[-2, x, y], [A, B]]
>>> (-2*x*y).args_cnc(cset=True)
[{-1, 2, x, y}, []]

The arg is always treated as a Mul:

>>> (-2 + x + A).args_cnc()
[[], [x - 2 + A]]
>>> (-oo).args_cnc() # -oo is a singleton
[[-1, oo], []]
as_base_exp()
as_coeff_Add(rational=False)

Efficiently extract the coefficient of a summation.

as_coeff_Mul(rational=False)

Efficiently extract the coefficient of a product.

as_coeff_add(*deps)

Return the tuple (c, args) where self is written as an Add, a.

c should be a Rational added to any terms of the Add that are independent of deps.

args should be a tuple of all other terms of a; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you don’t know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add.

  • if you know self is an Add and want only the head, use self.args[0];
  • if you don’t want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail.
  • if you want to split self into an independent and dependent parts use self.as_independent(*deps)
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_add()
(3, ())
>>> (3 + x).as_coeff_add()
(3, (x,))
>>> (3 + x + y).as_coeff_add(x)
(y + 3, (x,))
>>> (3 + y).as_coeff_add(x)
(y + 3, ())
as_coeff_exponent(x)

c*x**e -> c,e where x can be any symbolic expression.

as_coeff_mul(*deps, **kwargs)

Return the tuple (c, args) where self is written as a Mul, m.

c should be a Rational multiplied by any factors of the Mul that are independent of deps.

args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you don’t know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul.

  • if you know self is a Mul and want only the head, use self.args[0];
  • if you don’t want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail;
  • if you want to split self into an independent and dependent parts use self.as_independent(*deps)
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_mul()
(3, ())
>>> (3*x*y).as_coeff_mul()
(3, (x, y))
>>> (3*x*y).as_coeff_mul(x)
(3*y, (x,))
>>> (3*y).as_coeff_mul(x)
(3*y, ())
as_coefficient(expr)

Extracts symbolic coefficient at the given expression. In other words, this functions separates ‘self’ into the product of ‘expr’ and ‘expr’-free coefficient. If such separation is not possible it will return None.

See also

coeff
return sum of terms have a given factor
as_coeff_Add
separate the additive constant from an expression
as_coeff_Mul
separate the multiplicative constant from an expression
as_independent
separate x-dependent terms/factors from others
sympy.polys.polytools.coeff_monomial
efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.nth
like coeff_monomial but powers of monomial terms are used

Examples

>>> from sympy import E, pi, sin, I, Poly
>>> from sympy.abc import x
>>> E.as_coefficient(E)
1
>>> (2*E).as_coefficient(E)
2
>>> (2*sin(E)*E).as_coefficient(E)

Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.)

>>> (2*E + x*E).as_coefficient(E)
x + 2
>>> _.args[0]  # just want the exact match
2
>>> p = Poly(2*E + x*E); p
Poly(x*E + 2*E, x, E, domain='ZZ')
>>> p.coeff_monomial(E)
2
>>> p.nth(0, 1)
2

Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.)

>>> (2*E*x + x).as_coefficient(E)
>>> (2*E*x + x).coeff(E)
2*x
>>> (E*(x + 1) + x).as_coefficient(E)
>>> (2*pi*I).as_coefficient(pi*I)
2
>>> (2*I).as_coefficient(pi*I)
as_coefficients_dict()

Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term.

Examples

>>> from sympy.abc import a, x
>>> (3*x + a*x + 4).as_coefficients_dict()
{1: 4, x: 3, a*x: 1}
>>> _[a]
0
>>> (3*a*x).as_coefficients_dict()
{a*x: 3}
as_content_primitive(radical=False, clear=True)

This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo. The primitive need no be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).

Examples

>>> from sympy import sqrt
>>> from sympy.abc import x, y, z
>>> eq = 2 + 2*x + 2*y*(3 + 3*y)

The as_content_primitive function is recursive and retains structure:

>>> eq.as_content_primitive()
(2, x + 3*y*(y + 1) + 1)

Integer powers will have Rationals extracted from the base:

>>> ((2 + 6*x)**2).as_content_primitive()
(4, (3*x + 1)**2)
>>> ((2 + 6*x)**(2*y)).as_content_primitive()
(1, (2*(3*x + 1))**(2*y))

Terms may end up joining once their as_content_primitives are added:

>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(11, x*(y + 1))
>>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(9, x*(y + 1))
>>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive()
(1, 6.0*x*(y + 1) + 3*z*(y + 1))
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive()
(121, x**2*(y + 1)**2)
>>> ((5*(x*(1 + y)) + 2.0*x*(3 + 3*y))**2).as_content_primitive()
(1, 121.0*x**2*(y + 1)**2)

Radical content can also be factored out of the primitive:

>>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
(2, sqrt(2)*(1 + 2*sqrt(5)))

If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients.

>>> (x/2 + y).as_content_primitive()
(1/2, x + 2*y)
>>> (x/2 + y).as_content_primitive(clear=False)
(1, x/2 + y)
as_dummy()

Return a Dummy having the same name and same assumptions as self.

as_expr(*gens)

Convert a polynomial to a SymPy expression.

Examples

>>> from sympy import sin
>>> from sympy.abc import x, y
>>> f = (x**2 + x*y).as_poly(x, y)
>>> f.as_expr()
x**2 + x*y
>>> sin(x).as_expr()
sin(x)
as_independent(*deps, **hint)

A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.:

  • separatevars() to change Mul, Add and Pow (including exp) into Mul
  • .expand(mul=True) to change Add or Mul into Add
  • .expand(log=True) to change log expr into an Add

The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints.

For nonzero self, the returned tuple (i, d) has the following interpretation:

  • i will has no variable that appears in deps
  • d will be 1 or else have terms that contain variables that are in deps
  • if self is an Add then self = i + d
  • if self is a Mul then self = i*d
  • otherwise (self, S.One) or (S.One, self) is returned.

To force the expression to be treated as an Add, use the hint as_Add=True

See also

separatevars, expand, Add.as_two_terms, Mul.as_two_terms, as_coeff_add, as_coeff_mul

Examples

– self is an Add

>>> from sympy import sin, cos, exp
>>> from sympy.abc import x, y, z
>>> (x + x*y).as_independent(x)
(0, x*y + x)
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> (2*x*sin(x) + y + x + z).as_independent(x)
(y + z, 2*x*sin(x) + x)
>>> (2*x*sin(x) + y + x + z).as_independent(x, y)
(z, 2*x*sin(x) + x + y)

– self is a Mul

>>> (x*sin(x)*cos(y)).as_independent(x)
(cos(y), x*sin(x))

non-commutative terms cannot always be separated out when self is a Mul

>>> from sympy import symbols
>>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
>>> (n1 + n1*n2).as_independent(n2)
(n1, n1*n2)
>>> (n2*n1 + n1*n2).as_independent(n2)
(0, n1*n2 + n2*n1)
>>> (n1*n2*n3).as_independent(n1)
(1, n1*n2*n3)
>>> (n1*n2*n3).as_independent(n2)
(n1, n2*n3)
>>> ((x-n1)*(x-y)).as_independent(x)
(1, (x - y)*(x - n1))

– self is anything else:

>>> (sin(x)).as_independent(x)
(1, sin(x))
>>> (sin(x)).as_independent(y)
(sin(x), 1)
>>> exp(x+y).as_independent(x)
(1, exp(x + y))

– force self to be treated as an Add:

>>> (3*x).as_independent(x, as_Add=True)
(0, 3*x)

– force self to be treated as a Mul:

>>> (3+x).as_independent(x, as_Add=False)
(1, x + 3)
>>> (-3+x).as_independent(x, as_Add=False)
(1, x - 3)

Note how the below differs from the above in making the constant on the dep term positive.

>>> (y*(-3+x)).as_independent(x)
(y, x - 3)
– use .as_independent() for true independence testing instead
of .has(). The former considers only symbols in the free symbols while the latter considers all symbols
>>> from sympy import Integral
>>> I = Integral(x, (x, 1, 2))
>>> I.has(x)
True
>>> x in I.free_symbols
False
>>> I.as_independent(x) == (I, 1)
True
>>> (I + x).as_independent(x) == (I, x)
True

Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values

>>> from sympy import separatevars, log
>>> separatevars(exp(x+y)).as_independent(x)
(exp(y), exp(x))
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> separatevars(x + x*y).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).expand(mul=True).as_independent(y)
(x, x*y)
>>> a, b=symbols('a b', positive=True)
>>> (log(a*b).expand(log=True)).as_independent(b)
(log(a), log(b))
as_leading_term(*args, **kwargs)

Returns the leading (nonzero) term of the series expansion of self.

The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.

Examples

>>> from sympy.abc import x
>>> (1 + x + x**2).as_leading_term(x)
1
>>> (1/x**2 + x + x**2).as_leading_term(x)
x**(-2)
as_numer_denom()

expression -> a/b -> a, b

This is just a stub that should be defined by an object’s class methods to get anything else.

See also

normal
return a/b instead of a, b
as_ordered_factors(order=None)

Return list of ordered factors (if Mul) else [self].

as_ordered_terms(order=None, data=False)

Transform an expression to an ordered list of terms.

Examples

>>> from sympy import sin, cos
>>> from sympy.abc import x
>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]
as_poly(*gens, **args)

Converts self to a polynomial or returns None.

>>> from sympy import sin
>>> from sympy.abc import x, y
>>> print((x**2 + x*y).as_poly())
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + x*y).as_poly(x, y))
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + sin(y)).as_poly(x, y))
None
as_powers_dict()

Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.

as_real_imag(deep=True, **hints)
as_terms()

Transform an expression to a list of terms.

assumptions0
atoms(*types)

Returns the atoms that form the current object.

By default, only objects that are truly atomic and can’t be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.

Examples

>>> from sympy import Number, NumberSymbol, Symbol
>>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol)
{x, y}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number)
{1, 2}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)
{1, 2, pi}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)
{1, 2, I, pi}

Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class.

The type can be given implicitly, too:

>>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol
{x, y}

Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One, a special type of sympy atom, while type(S(2)) is type Integer and will find all integers in an expression:

>>> from sympy import S
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(1))
{1}
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(2))
{1, 2}

Finally, arguments to atoms() can select more than atomic atoms: any sympy type (loaded in core/__init__.py) can be listed as an argument and those types of “atoms” as found in scanning the arguments of the expression recursively:

>>> from sympy import Function, Mul
>>> from sympy.core.function import AppliedUndef
>>> f = Function('f')
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)
{f(x), sin(y + I*pi)}
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)
{f(x)}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
{I*pi, 2*sin(y + I*pi)}
cancel(*gens, **args)

See the cancel function in sympy.polys

canonical_variables

Return a dictionary mapping any variable defined in self.variables as underscore-suffixed numbers corresponding to their position in self.variables. Enough underscores are added to ensure that there will be no clash with existing free symbols.

Examples

>>> from sympy import Lambda
>>> from sympy.abc import x
>>> Lambda(x, 2*x).canonical_variables
{x: 0_}
class_key()
coeff(x, n=1, right=False)

Returns the coefficient from the term(s) containing x**n. If n is zero then all terms independent of x will be returned.

When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword ‘right’ is ignored when x is commutative.

See also

as_coefficient
separate the expression into a coefficient and factor
as_coeff_Add
separate the additive constant from an expression
as_coeff_Mul
separate the multiplicative constant from an expression
as_independent
separate x-dependent terms/factors from others
sympy.polys.polytools.coeff_monomial
efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.nth
like coeff_monomial but powers of monomial terms are used

Examples

>>> from sympy import symbols
>>> from sympy.abc import x, y, z

You can select terms that have an explicit negative in front of them:

>>> (-x + 2*y).coeff(-1)
x
>>> (x - 2*y).coeff(-1)
2*y

You can select terms with no Rational coefficient:

>>> (x + 2*y).coeff(1)
x
>>> (3 + 2*x + 4*x**2).coeff(1)
0

You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None):

>>> (3 + 2*x + 4*x**2).coeff(x, 0)
3
>>> eq = ((x + 1)**3).expand() + 1
>>> eq
x**3 + 3*x**2 + 3*x + 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 2]
>>> eq -= 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 0]

You can select terms that have a numerical term in front of them:

>>> (-x - 2*y).coeff(2)
-y
>>> from sympy import sqrt
>>> (x + sqrt(2)*x).coeff(sqrt(2))
x

The matching is exact:

>>> (3 + 2*x + 4*x**2).coeff(x)
2
>>> (3 + 2*x + 4*x**2).coeff(x**2)
4
>>> (3 + 2*x + 4*x**2).coeff(x**3)
0
>>> (z*(x + y)**2).coeff((x + y)**2)
z
>>> (z*(x + y)**2).coeff(x + y)
0

In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following:

>>> (x + z*(x + x*y)).coeff(x)
1

If such factoring is desired, factor_terms can be used first:

>>> from sympy import factor_terms
>>> factor_terms(x + z*(x + x*y)).coeff(x)
z*(y + 1) + 1
>>> n, m, o = symbols('n m o', commutative=False)
>>> n.coeff(n)
1
>>> (3*n).coeff(n)
3
>>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m
1 + m
>>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m
m

If there is more than one possible coefficient 0 is returned:

>>> (n*m + m*n).coeff(n)
0

If there is only one possible coefficient, it is returned:

>>> (n*m + x*m*n).coeff(m*n)
x
>>> (n*m + x*m*n).coeff(m*n, right=1)
1
collect(syms, func=None, evaluate=True, exact=False, distribute_order_term=True)

See the collect function in sympy.simplify

combsimp()

See the combsimp function in sympy.simplify

compare(other)

Return -1, 0, 1 if the object is smaller, equal, or greater than other.

Not in the mathematical sense. If the object is of a different type from the “other” then their classes are ordered according to the sorted_classes list.

Examples

>>> from sympy.abc import x, y
>>> x.compare(y)
-1
>>> x.compare(x)
0
>>> y.compare(x)
1
compute_leading_term(x, logx=None)

as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.

conjugate()
copy()
could_extract_minus_sign()

Canonical way to choose an element in the set {e, -e} where e is any expression. If the canonical element is e, we have e.could_extract_minus_sign() == True, else e.could_extract_minus_sign() == False.

For any expression, the set {e.could_extract_minus_sign(), (-e).could_extract_minus_sign()} must be {True, False}.

>>> from sympy.abc import x, y
>>> (x-y).could_extract_minus_sign() != (y-x).could_extract_minus_sign()
True
count(query)

Count the number of matching subexpressions.

count_ops(visual=None)

wrapper for count_ops that returns the operation count.

default_assumptions = {}
diff(*symbols, **assumptions)
doit(**hints)
dummy_eq(other, symbol=None)

Compare two expressions and handle dummy symbols.

Examples

>>> from sympy import Dummy
>>> from sympy.abc import x, y
>>> u = Dummy('u')
>>> (u**2 + 1).dummy_eq(x**2 + 1)
True
>>> (u**2 + 1) == (x**2 + 1)
False
>>> (u**2 + y).dummy_eq(x**2 + y, x)
True
>>> (u**2 + y).dummy_eq(x**2 + y, y)
False
dummy_index
equals(other, failing_expression=False)

Return True if self == other, False if it doesn’t, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.

If self is a Number (or complex number) that is not zero, then the result is False.

If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.

evalf(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)

Evaluate the given formula to an accuracy of n digits. Optional keyword arguments:

subs=<dict>
Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi}. The substitutions must be given as a dictionary.
maxn=<integer>
Allow a maximum temporary working precision of maxn digits (default=100)
chop=<bool>
Replace tiny real or imaginary parts in subresults by exact zeros (default=False)
strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec (default=False)
quad=<str>
Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad=’osc’.
verbose=<bool>
Print debug information (default=False)
expand(*args, **kwargs)

Expand an expression using hints.

See the docstring of the expand() function in sympy.core.function for more information.

extract_additively(c)

Return self - c if it’s possible to subtract c from self and make all matching coefficients move towards zero, else return None.

Examples

>>> from sympy.abc import x, y
>>> e = 2*x + 3
>>> e.extract_additively(x + 1)
x + 2
>>> e.extract_additively(3*x)
>>> e.extract_additively(4)
>>> (y*(x + 1)).extract_additively(x + 1)
>>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1)
(x + 1)*(x + 2*y) + 3

Sometimes auto-expansion will return a less simplified result than desired; gcd_terms might be used in such cases:

>>> from sympy import gcd_terms
>>> (4*x*(y + 1) + y).extract_additively(x)
4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y
>>> gcd_terms(_)
x*(4*y + 3) + y
extract_branch_factor(allow_half=False)

Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n).

>>> from sympy import exp_polar, I, pi
>>> from sympy.abc import x, y
>>> exp_polar(I*pi).extract_branch_factor()
(exp_polar(I*pi), 0)
>>> exp_polar(2*I*pi).extract_branch_factor()
(1, 1)
>>> exp_polar(-pi*I).extract_branch_factor()
(exp_polar(I*pi), -1)
>>> exp_polar(3*pi*I + x).extract_branch_factor()
(exp_polar(x + I*pi), 1)
>>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor()
(y*exp_polar(2*pi*x), -1)
>>> exp_polar(-I*pi/2).extract_branch_factor()
(exp_polar(-I*pi/2), 0)

If allow_half is True, also extract exp_polar(I*pi):

>>> exp_polar(I*pi).extract_branch_factor(allow_half=True)
(1, 1/2)
>>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True)
(1, 1)
>>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True)
(1, 3/2)
>>> exp_polar(-I*pi).extract_branch_factor(allow_half=True)
(1, -1/2)
extract_multiplicatively(c)

Return None if it’s not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.

>>> from sympy import symbols, Rational
>>> x, y = symbols('x,y', real=True)
>>> ((x*y)**3).extract_multiplicatively(x**2 * y)
x*y**2
>>> ((x*y)**3).extract_multiplicatively(x**4 * y)
>>> (2*x).extract_multiplicatively(2)
x
>>> (2*x).extract_multiplicatively(3)
>>> (Rational(1, 2)*x).extract_multiplicatively(3)
x/6
factor(*gens, **args)

See the factor() function in sympy.polys.polytools

find(query, group=False)

Find all subexpressions matching a query.

fourier_series(limits=None)

Compute fourier sine/cosine series of self.

See the docstring of the fourier_series() in sympy.series.fourier for more information.

fps(x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False)

Compute formal power power series of self.

See the docstring of the fps() function in sympy.series.formal for more information.

free_symbols
fromiter(args, **assumptions)

Create a new object from an iterable.

This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.

Examples

>>> from sympy import Tuple
>>> Tuple.fromiter(i for i in range(5))
(0, 1, 2, 3, 4)
func

The top-level function in an expression.

The following should hold for all objects:

>> x == x.func(*x.args)

Examples

>>> from sympy.abc import x
>>> a = 2*x
>>> a.func
<class 'sympy.core.mul.Mul'>
>>> a.args
(2, x)
>>> a.func(*a.args)
2*x
>>> a == a.func(*a.args)
True
getO()

Returns the additive O(..) symbol if there is one, else None.

getn()

Returns the order of the expression.

The order is determined either from the O(…) term. If there is no O(…) term, it returns None.

Examples

>>> from sympy import O
>>> from sympy.abc import x
>>> (1 + x + O(x**2)).getn()
2
>>> (1 + x).getn()
has(*args, **kwargs)

Test whether any subexpression matches any of the patterns.

Examples

>>> from sympy import sin
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True

Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval:

>>> from sympy.sets import Interval
>>> i = Interval.Lopen(0, 5); i
Interval.Lopen(0, 5)
>>> i.args
(0, 5, True, False)
>>> i.has(4)  # there is no "4" in the arguments
False
>>> i.has(0)  # there *is* a "0" in the arguments
True

Instead, use contains to determine whether a number is in the interval or not:

>>> i.contains(4)
True
>>> i.contains(0)
False

Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns). In particular, False is returned when the list of patterns is empty.

>>> x.has()
False
integrate(*args, **kwargs)

See the integrate function in sympy.integrals

invert(g, *gens, **args)

Return the multiplicative inverse of self mod g where self (and g) may be symbolic expressions).

See also

sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert

is_Add = False
is_AlgebraicNumber = False
is_Atom = True
is_Boolean = False
is_Derivative = False
is_Dummy = True
is_Equality = False
is_Float = False
is_Function = False
is_Indexed = False
is_Integer = False
is_Matrix = False
is_Mul = False
is_Not = False
is_Number = False
is_NumberSymbol = False
is_Order = False
is_Piecewise = False
is_Point = False
is_Poly = False
is_Pow = False
is_Rational = False
is_Relational = False
is_Symbol = True
is_Vector = False
is_Wild = False
is_algebraic
is_algebraic_expr(*syms)

This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form.

This function returns False for expressions that are “algebraic expressions” with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.

References

Examples

>>> from sympy import Symbol, sqrt
>>> x = Symbol('x', real=True)
>>> sqrt(1 + x).is_rational_function()
False
>>> sqrt(1 + x).is_algebraic_expr()
True

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one.

>>> from sympy import exp, factor
>>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1)
>>> a.is_algebraic_expr(x)
False
>>> factor(a).is_algebraic_expr()
True
is_antihermitian
is_commutative
is_comparable = False
is_complex
is_composite
is_constant(*wrt, **flags)
is_even
is_finite
is_hermitian
is_hypergeometric(k)
is_imaginary
is_infinite
is_integer
is_irrational
is_negative
is_noninteger
is_nonnegative
is_nonpositive
is_nonzero
is_number = False
is_odd
is_polar
is_polynomial(*syms)

Return True if self is a polynomial in syms and False otherwise.

This checks if self is an exact polynomial in syms. This function returns False for expressions that are “polynomials” with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used.

This is not part of the assumptions system. You cannot do Symbol(‘z’, polynomial=True).

Examples

>>> from sympy import Symbol
>>> x = Symbol('x')
>>> ((x**2 + 1)**4).is_polynomial(x)
True
>>> ((x**2 + 1)**4).is_polynomial()
True
>>> (2**x + 1).is_polynomial(x)
False
>>> n = Symbol('n', nonnegative=True, integer=True)
>>> (x**n + 1).is_polynomial(x)
False

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one.

>>> from sympy import sqrt, factor, cancel
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)
>>> a.is_polynomial(y)
False
>>> factor(a)
y + 1
>>> factor(a).is_polynomial(y)
True
>>> b = (y**2 + 2*y + 1)/(y + 1)
>>> b.is_polynomial(y)
False
>>> cancel(b)
y + 1
>>> cancel(b).is_polynomial(y)
True

See also .is_rational_function()

is_positive
is_prime
is_rational
is_rational_function(*syms)

Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form.

This function returns False for expressions that are “rational functions” with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True.

This is not part of the assumptions system. You cannot do Symbol(‘z’, rational_function=True).

Examples

>>> from sympy import Symbol, sin
>>> from sympy.abc import x, y
>>> (x/y).is_rational_function()
True
>>> (x**2).is_rational_function()
True
>>> (x/sin(y)).is_rational_function(y)
False
>>> n = Symbol('n', integer=True)
>>> (x**n + 1).is_rational_function(x)
False

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one.

>>> from sympy import sqrt, factor
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)/y
>>> a.is_rational_function(y)
False
>>> factor(a)
(y + 1)/y
>>> factor(a).is_rational_function(y)
True

See also is_algebraic_expr().

is_real
is_symbol = True
is_transcendental
is_zero
leadterm(x)

Returns the leading term a*x**b as a tuple (a, b).

Examples

>>> from sympy.abc import x
>>> (1+x+x**2).leadterm(x)
(1, 0)
>>> (1/x**2+x+x**2).leadterm(x)
(1, -2)
limit(x, xlim, dir='+')

Compute limit x->xlim.

lseries(x=None, x0=0, dir='+', logx=None)

Wrapper for series yielding an iterator of the terms of the series.

Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:

for term in sin(x).lseries(x):
    print term

The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you don’t know how many you should ask for in nseries() using the “n” parameter.

See also nseries().

match(pattern, old=False)

Pattern matching.

Wild symbols match all.

Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:

pattern.xreplace(self.match(pattern)) == self

Examples

>>> from sympy import Wild
>>> from sympy.abc import x, y
>>> p = Wild("p")
>>> q = Wild("q")
>>> r = Wild("r")
>>> e = (x+y)**(x+y)
>>> e.match(p**p)
{p_: x + y}
>>> e.match(p**q)
{p_: x + y, q_: x + y}
>>> e = (2*x)**2
>>> e.match(p*q**r)
{p_: 4, q_: x, r_: 2}
>>> (p*q**r).xreplace(e.match(p*q**r))
4*x**2

The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True:

>>> (x - 2).match(p - x, old=True)
{p_: 2*x - 2}
>>> (2/x).match(p*x, old=True)
{p_: 2/x**2}
matches(expr, repl_dict={}, old=False)
n(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)

Evaluate the given formula to an accuracy of n digits. Optional keyword arguments:

subs=<dict>
Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi}. The substitutions must be given as a dictionary.
maxn=<integer>
Allow a maximum temporary working precision of maxn digits (default=100)
chop=<bool>
Replace tiny real or imaginary parts in subresults by exact zeros (default=False)
strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec (default=False)
quad=<str>
Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad=’osc’.
verbose=<bool>
Print debug information (default=False)
name
normal()
nseries(x=None, x0=0, n=6, dir='+', logx=None)

Wrapper to _eval_nseries if assumptions allow, else to series.

If x is given, x0 is 0, dir=’+’, and self has x, then _eval_nseries is called. This calculates “n” terms in the innermost expressions and then builds up the final series just by “cross-multiplying” everything out.

The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided.

Advantage – it’s fast, because we don’t have to determine how many terms we need to calculate in advance.

Disadvantage – you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct.

If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms.

See also lseries().

Examples

>>> from sympy import sin, log, Symbol
>>> from sympy.abc import x, y
>>> sin(x).nseries(x, 0, 6)
x - x**3/6 + x**5/120 + O(x**6)
>>> log(x+1).nseries(x, 0, 5)
x - x**2/2 + x**3/3 - x**4/4 + O(x**5)

Handling of the logx parameter — in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0):

>>> e = sin(log(x))
>>> e.nseries(x, 0, 6)
Traceback (most recent call last):
...
PoleError: ...
...
>>> logx = Symbol('logx')
>>> e.nseries(x, 0, 6, logx=logx)
sin(logx)

In the following example, the expansion works but gives only an Order term unless the logx parameter is used:

>>> e = x**y
>>> e.nseries(x, 0, 2)
O(log(x)**2)
>>> e.nseries(x, 0, 2, logx=logx)
exp(logx*y)
nsimplify(constants=[], tolerance=None, full=False)

See the nsimplify function in sympy.simplify

powsimp(*args, **kwargs)

See the powsimp function in sympy.simplify

primitive()

Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).

Examples

>>> from sympy.abc import x
>>> (3*(x + 1)**2).primitive()
(3, (x + 1)**2)
>>> a = (6*x + 2); a.primitive()
(2, 3*x + 1)
>>> b = (x/2 + 3); b.primitive()
(1/2, x + 6)
>>> (a*b).primitive() == (1, a*b)
True
radsimp(**kwargs)

See the radsimp function in sympy.simplify

ratsimp()

See the ratsimp function in sympy.simplify

rcall(*args)

Apply on the argument recursively through the expression tree.

This method is used to simulate a common abuse of notation for operators. For instance in SymPy the the following will not work:

(x+Lambda(y, 2*y))(z) == x+2*z,

however you can use

>>> from sympy import Lambda
>>> from sympy.abc import x, y, z
>>> (x + Lambda(y, 2*y)).rcall(z)
x + 2*z
refine(assumption=True)

See the refine function in sympy.assumptions

removeO()

Removes the additive O(..) symbol if there is one

replace(query, value, map=False, simultaneous=True, exact=False)

Replace matching subexpressions of self with value.

If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself doesn’t match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())).

Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is True, then the match will only succeed if non-zero values are received for each Wild that appears in the match pattern.

The list of possible combinations of queries and replacement values is listed below:

See also

subs
substitution of subexpressions as defined by the objects themselves.
xreplace
exact node replacement in expr tree; also capable of using matching rules

Examples

Initial setup

>>> from sympy import log, sin, cos, tan, Wild, Mul, Add
>>> from sympy.abc import x, y
>>> f = log(sin(x)) + tan(sin(x**2))
1.1. type -> type

obj.replace(type, newtype)

When object of type type is found, replace it with the result of passing its argument(s) to newtype.

>>> f.replace(sin, cos)
log(cos(x)) + tan(cos(x**2))
>>> sin(x).replace(sin, cos, map=True)
(cos(x), {sin(x): cos(x)})
>>> (x*y).replace(Mul, Add)
x + y
1.2. type -> func

obj.replace(type, func)

When object of type type is found, apply func to its argument(s). func must be written to handle the number of arguments of type.

>>> f.replace(sin, lambda arg: sin(2*arg))
log(sin(2*x)) + tan(sin(2*x**2))
>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
sin(2*x*y)
2.1. pattern -> expr

obj.replace(pattern(wild), expr(wild))

Replace subexpressions matching pattern with the expression written in terms of the Wild symbols in pattern.

>>> a = Wild('a')
>>> f.replace(sin(a), tan(a))
log(tan(x)) + tan(tan(x**2))
>>> f.replace(sin(a), tan(a/2))
log(tan(x/2)) + tan(tan(x**2/2))
>>> f.replace(sin(a), a)
log(x) + tan(x**2)
>>> (x*y).replace(a*x, a)
y

When the default value of False is used with patterns that have more than one Wild symbol, non-intuitive results may be obtained:

>>> b = Wild('b')
>>> (2*x).replace(a*x + b, b - a)
2/x

For this reason, the exact option can be used to make the replacement only when the match gives non-zero values for all Wild symbols:

>>> (2*x + y).replace(a*x + b, b - a, exact=True)
y - 2
>>> (2*x).replace(a*x + b, b - a, exact=True)
2*x
2.2. pattern -> func

obj.replace(pattern(wild), lambda wild: expr(wild))

All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression:

>>> f.replace(sin(a), lambda a: sin(2*a))
log(sin(2*x)) + tan(sin(2*x**2))
3.1. func -> func

obj.replace(filter, func)

Replace subexpression e with func(e) if filter(e) is True.

>>> g = 2*sin(x**3)
>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
4*sin(x**9)

The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice.

>>> e = x*(x*y + 1)
>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
2*x*(2*x*y + 1)
rewrite(*args, **hints)

Rewrite functions in terms of other functions.

Rewrites expression containing applications of functions of one kind in terms of functions of different kind. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function.

As a pattern this function accepts a list of functions to to rewrite (instances of DefinedFunction class). As rule you can use string or a destination function instance (in this case rewrite() will use the str() function).

There is also the possibility to pass hints on how to rewrite the given expressions. For now there is only one such hint defined called ‘deep’. When ‘deep’ is set to False it will forbid functions to rewrite their contents.

Examples

>>> from sympy import sin, exp
>>> from sympy.abc import x

Unspecified pattern:

>>> sin(x).rewrite(exp)
-I*(exp(I*x) - exp(-I*x))/2

Pattern as a single function:

>>> sin(x).rewrite(sin, exp)
-I*(exp(I*x) - exp(-I*x))/2

Pattern as a list of functions:

>>> sin(x).rewrite([sin, ], exp)
-I*(exp(I*x) - exp(-I*x))/2
round(p=0)

Return x rounded to the given decimal place.

If a complex number would results, apply round to the real and imaginary components of the number.

Notes

Do not confuse the Python builtin function, round, with the SymPy method of the same name. The former always returns a float (or raises an error if applied to a complex value) while the latter returns either a Number or a complex number:

>>> isinstance(round(S(123), -2), Number)
False
>>> isinstance(S(123).round(-2), Number)
True
>>> isinstance((3*I).round(), Mul)
True
>>> isinstance((1 + 3*I).round(), Add)
True

Examples

>>> from sympy import pi, E, I, S, Add, Mul, Number
>>> S(10.5).round()
11.
>>> pi.round()
3.
>>> pi.round(2)
3.14
>>> (2*pi + E*I).round()
6. + 3.*I

The round method has a chopping effect:

>>> (2*pi + I/10).round()
6.
>>> (pi/10 + 2*I).round()
2.*I
>>> (pi/10 + E*I).round(2)
0.31 + 2.72*I
separate(deep=False, force=False)

See the separate function in sympy.simplify

series(x=None, x0=0, n=6, dir='+', logx=None)

Series expansion of “self” around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.

Returns the series expansion of “self” around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.

>>> from sympy import cos, exp
>>> from sympy.abc import x, y
>>> cos(x).series()
1 - x**2/2 + x**4/24 + O(x**6)
>>> cos(x).series(n=4)
1 - x**2/2 + O(x**4)
>>> cos(x).series(x, x0=1, n=2)
cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1))
>>> e = cos(x + exp(y))
>>> e.series(y, n=2)
cos(x + 1) - y*sin(x + 1) + O(y**2)
>>> e.series(x, n=2)
cos(exp(y)) - x*sin(exp(y)) + O(x**2)

If n=None then a generator of the series terms will be returned.

>>> term=cos(x).series(n=None)
>>> [next(term) for i in range(2)]
[1, -x**2/2]

For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results.

>>> abs(x).series(dir="+")
x
>>> abs(x).series(dir="-")
-x
simplify(ratio=1.7, measure=None)

See the simplify function in sympy.simplify

sort_key(*args, **kwargs)
subs(*args, **kwargs)

Substitutes old for new in an expression after sympifying args.

args is either:
  • two arguments, e.g. foo.subs(old, new)
  • one iterable argument, e.g. foo.subs(iterable). The iterable may be
    o an iterable container with (old, new) pairs. In this case the
    replacements are processed in the order given with successive patterns possibly affecting replacements already made.
    o a dict or set whose key/value items correspond to old/new pairs.
    In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous).

If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.

See also

replace
replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements
xreplace
exact node replacement in expr tree; also capable of using matching rules
evalf
calculates the given formula to a desired level of precision

Examples

>>> from sympy import pi, exp, limit, oo
>>> from sympy.abc import x, y
>>> (1 + x*y).subs(x, pi)
pi*y + 1
>>> (1 + x*y).subs({x:pi, y:2})
1 + 2*pi
>>> (1 + x*y).subs([(x, pi), (y, 2)])
1 + 2*pi
>>> reps = [(y, x**2), (x, 2)]
>>> (x + y).subs(reps)
6
>>> (x + y).subs(reversed(reps))
x**2 + 2
>>> (x**2 + x**4).subs(x**2, y)
y**2 + y

To replace only the x**2 but not the x**4, use xreplace:

>>> (x**2 + x**4).xreplace({x**2: y})
x**4 + y

To delay evaluation until all substitutions have been made, set the keyword simultaneous to True:

>>> (x/y).subs([(x, 0), (y, 0)])
0
>>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True)
nan

This has the added feature of not allowing subsequent substitutions to affect those already made:

>>> ((x + y)/y).subs({x + y: y, y: x + y})
1
>>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)
y/(x + y)

In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted.

>>> from sympy import sqrt, sin, cos
>>> from sympy.abc import a, b, c, d, e
>>> A = (sqrt(sin(2*x)), a)
>>> B = (sin(2*x), b)
>>> C = (cos(2*x), c)
>>> D = (x, d)
>>> E = (exp(x), e)
>>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)
>>> expr.subs(dict([A, B, C, D, E]))
a*c*sin(d*e) + b

The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression:

>>> (x**3 - 3*x).subs({x: oo})
nan
>>> limit(x**3 - 3*x, x, oo)
oo

If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as

>>> (1/x).evalf(subs={x: 3.0}, n=21)
0.333333333333333333333

rather than

>>> (1/x).subs({x: 3.0}).evalf(21)
0.333333333333333314830

as the former will ensure that the desired level of precision is obtained.

taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

together(*args, **kwargs)

See the together function in sympy.polys

transpose()
trigsimp(**args)

See the trigsimp function in sympy.simplify

xreplace(rule, hack2=False)

Factor

class nipy.algorithms.statistics.formula.formulae.Factor(name, levels, char='b')

Bases: nipy.algorithms.statistics.formula.formulae.Formula

A qualitative variable in a regression model

A Factor is similar to R’s factor. The levels of the Factor can be either strings or ints.

Methods

design(input[, param, return_float, contrasts]) Construct the design matrix, and optional contrast matrices.
fromcol(col, name) Create a Factor from a column array.
fromrec(rec[, keep, drop]) Construct Formula from recarray
get_term(level) Retrieve a term of the Factor…
stratify(variable) Create a new variable, stratified by the levels of a Factor.
subs(old, new) Perform a sympy substitution on all terms in the Formula
__init__(name, levels, char='b')

Initialize Factor

Parameters:

name : str

levels : [str or int]

A sequence of strings or ints.

char : str, optional

prefix character for regression coefficients

coefs

Coefficients in the linear regression formula.

design(input, param=None, return_float=False, contrasts=None)

Construct the design matrix, and optional contrast matrices.

Parameters:

input : np.recarray

Recarray including fields needed to compute the Terms in getparams(self.design_expr).

param : None or np.recarray

Recarray including fields that are not Terms in getparams(self.design_expr)

return_float : bool, optional

If True, return a np.float array rather than a np.recarray

contrasts : None or dict, optional

Contrasts. The items in this dictionary should be (str, Formula) pairs where a contrast matrix is constructed for each Formula by evaluating its design at the same parameters as self.design. If not None, then the return_float is set to True.

Returns:

des : 2D array

design matrix

cmatrices : dict, optional

Dictionary with keys from contrasts input, and contrast matrices corresponding to des design matrix. Returned only if contrasts input is not None

design_expr
dtype

The dtype of the design matrix of the Formula.

static fromcol(col, name)

Create a Factor from a column array.

Parameters:

col : ndarray

an array with ndim==1

name : str

name of the Factor

Returns:

factor : Factor

Examples

>>> data = np.array([(3,'a'),(4,'a'),(5,'b'),(3,'b')], np.dtype([('x', np.float), ('y', 'S1')]))
>>> f1 = Factor.fromcol(data['y'], 'y')
>>> f2 = Factor.fromcol(data['x'], 'x')
>>> d = f1.design(data)
>>> print(d.dtype.descr)
[('y_a', '<f8'), ('y_b', '<f8')]
>>> d = f2.design(data)
>>> print(d.dtype.descr)
[('x_3', '<f8'), ('x_4', '<f8'), ('x_5', '<f8')]
fromrec(rec, keep=[], drop=[])

Construct Formula from recarray

For fields with a string-dtype, it is assumed that these are qualtiatitve regressors, i.e. Factors.

Parameters:

rec: recarray :

Recarray whose field names will be used to create a formula.

keep: [] :

Field names to explicitly keep, dropping all others.

drop: [] :

Field names to drop.

get_term(level)

Retrieve a term of the Factor…

main_effect
mean

Expression for the mean, expressed as a linear combination of terms, each with dummy variables in front.

params

The parameters in the Formula.

stratify(variable)

Create a new variable, stratified by the levels of a Factor.

Parameters:

variable : str or simple sympy expression

If sympy expression, then string representation must be all lower or upper case letters, i.e. it can be interpreted as a name.

Returns:

formula : Formula

Formula whose mean has one parameter named variable%d, for each level in self.levels

Examples

>>> f = Factor('a', ['x','y'])
>>> sf = f.stratify('theta')
>>> sf.mean
_theta0*a_x + _theta1*a_y
subs(old, new)

Perform a sympy substitution on all terms in the Formula

Returns a new instance of the same class

Parameters:

old : sympy.Basic

The expression to be changed

new : sympy.Basic

The value to change it to.

Returns:

newf : Formula

Examples

>>> s, t = [Term(l) for l in 'st']
>>> f, g = [sympy.Function(l) for l in 'fg']
>>> form = Formula([f(t),g(s)])
>>> newform = form.subs(g, sympy.Function('h'))
>>> newform.terms
array([f(t), h(s)], dtype=object)
>>> form.terms
array([f(t), g(s)], dtype=object)
terms

Terms in the linear regression formula.

FactorTerm

class nipy.algorithms.statistics.formula.formulae.FactorTerm

Bases: nipy.algorithms.statistics.formula.formulae.Term

Boolean Term derived from a Factor.

Its properties are the same as a Term except that its product with itself is itself.

Methods

__call__(*args)
adjoint()
apart([x]) See the apart function in sympy.polys
args_cnc([cset, warn, split_1]) Return [commutative factors, non-commutative factors] of self.
as_base_exp()
as_coeff_Add([rational]) Efficiently extract the coefficient of a summation.
as_coeff_Mul([rational]) Efficiently extract the coefficient of a product.
as_coeff_add(*deps) Return the tuple (c, args) where self is written as an Add, a.
as_coeff_exponent(x) c*x**e -> c,e where x can be any symbolic expression.
as_coeff_mul(*deps, **kwargs) Return the tuple (c, args) where self is written as a Mul, m.
as_coefficient(expr) Extracts symbolic coefficient at the given expression.
as_coefficients_dict() Return a dictionary mapping terms to their Rational coefficient.
as_content_primitive([radical, clear]) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive).
as_dummy() Return a Dummy having the same name and same assumptions as self.
as_expr(*gens) Convert a polynomial to a SymPy expression.
as_independent(*deps, **hint) A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps.
as_leading_term(*args, **kwargs) Returns the leading (nonzero) term of the series expansion of self.
as_numer_denom() expression -> a/b -> a, b
as_ordered_factors([order]) Return list of ordered factors (if Mul) else [self].
as_ordered_terms([order, data]) Transform an expression to an ordered list of terms.
as_poly(*gens, **args) Converts self to a polynomial or returns None.
as_powers_dict() Return self as a dictionary of factors with each factor being treated as a power.
as_real_imag([deep])
as_terms() Transform an expression to a list of terms.
atoms(*types) Returns the atoms that form the current object.
cancel(*gens, **args) See the cancel function in sympy.polys
class_key()
coeff(x[, n, right]) Returns the coefficient from the term(s) containing x**n.
collect(syms[, func, evaluate, exact, …]) See the collect function in sympy.simplify
combsimp() See the combsimp function in sympy.simplify
compare(other) Return -1, 0, 1 if the object is smaller, equal, or greater than other.
compute_leading_term(x[, logx]) as_leading_term is only allowed for results of .series()
conjugate()
copy()
could_extract_minus_sign() Canonical way to choose an element in the set {e, -e} where e is any expression.
count(query) Count the number of matching subexpressions.
count_ops([visual]) wrapper for count_ops that returns the operation count.
diff(*symbols, **assumptions)
doit(**hints)
dummy_eq(other[, symbol]) Compare two expressions and handle dummy symbols.
equals(other[, failing_expression]) Return True if self == other, False if it doesn’t, or None.
evalf([n, subs, maxn, chop, strict, quad, …]) Evaluate the given formula to an accuracy of n digits.
expand(*args, **kwargs) Expand an expression using hints.
extract_additively(c) Return self - c if it’s possible to subtract c from self and make all matching coefficients move towards zero, else return None.
extract_branch_factor([allow_half]) Try to write self as exp_polar(2*pi*I*n)*z in a nice way.
extract_multiplicatively(c) Return None if it’s not possible to make self in the form c * something in a nice way, i.e.
factor(*gens, **args) See the factor() function in sympy.polys.polytools
find(query[, group]) Find all subexpressions matching a query.
fourier_series([limits]) Compute fourier sine/cosine series of self.
fps([x, x0, dir, hyper, order, rational, full]) Compute formal power power series of self.
fromiter(args, **assumptions) Create a new object from an iterable.
getO() Returns the additive O(..) symbol if there is one, else None.
getn() Returns the order of the expression.
has(*args, **kwargs) Test whether any subexpression matches any of the patterns.
integrate(*args, **kwargs) See the integrate function in sympy.integrals
invert(g, *gens, **args) Return the multiplicative inverse of self mod g where self (and g) may be symbolic expressions).
is_algebraic_expr(*syms) This tests whether a given expression is algebraic or not, in the given symbols, syms.
is_constant(*wrt, **flags)
is_hypergeometric(k)
is_polynomial(*syms) Return True if self is a polynomial in syms and False otherwise.
is_rational_function(*syms) Test whether function is a ratio of two polynomials in the given symbols, syms.
leadterm(x) Returns the leading term a*x**b as a tuple (a, b).
limit(x, xlim[, dir]) Compute limit x->xlim.
lseries([x, x0, dir, logx]) Wrapper for series yielding an iterator of the terms of the series.
match(pattern[, old]) Pattern matching.
matches(expr[, repl_dict, old])
n([n, subs, maxn, chop, strict, quad, verbose]) Evaluate the given formula to an accuracy of n digits.
normal()
nseries([x, x0, n, dir, logx]) Wrapper to _eval_nseries if assumptions allow, else to series.
nsimplify([constants, tolerance, full]) See the nsimplify function in sympy.simplify
powsimp(*args, **kwargs) See the powsimp function in sympy.simplify
primitive() Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add).
radsimp(**kwargs) See the radsimp function in sympy.simplify
ratsimp() See the ratsimp function in sympy.simplify
rcall(*args) Apply on the argument recursively through the expression tree.
refine([assumption]) See the refine function in sympy.assumptions
removeO() Removes the additive O(..) symbol if there is one
replace(query, value[, map, simultaneous, exact]) Replace matching subexpressions of self with value.
rewrite(*args, **hints) Rewrite functions in terms of other functions.
round([p]) Return x rounded to the given decimal place.
separate([deep, force]) See the separate function in sympy.simplify
series([x, x0, n, dir, logx]) Series expansion of “self” around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.
simplify([ratio, measure]) See the simplify function in sympy.simplify
sort_key(*args, **kwargs)
subs(*args, **kwargs) Substitutes old for new in an expression after sympifying args.
taylor_term(n, x, *previous_terms) General method for the taylor term.
together(*args, **kwargs) See the together function in sympy.polys
transpose()
trigsimp(**args) See the trigsimp function in sympy.simplify
xreplace(rule[, hack2])
__init__()

x.__init__(…) initializes x; see help(type(x)) for signature

adjoint()
apart(x=None, **args)

See the apart function in sympy.polys

args

Returns a tuple of arguments of ‘self’.

Notes

Never use self._args, always use self.args. Only use _args in __new__ when creating a new function. Don’t override .args() from Basic (so that it’s easy to change the interface in the future if needed).

Examples

>>> from sympy import cot
>>> from sympy.abc import x, y
>>> cot(x).args
(x,)
>>> cot(x).args[0]
x
>>> (x*y).args
(x, y)
>>> (x*y).args[1]
y
args_cnc(cset=False, warn=True, split_1=True)

Return [commutative factors, non-commutative factors] of self.

self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly supressed by setting warn to False.

Note: -1 is always separated from a Number unless split_1 is False.

>>> from sympy import symbols, oo
>>> A, B = symbols('A B', commutative=0)
>>> x, y = symbols('x y')
>>> (-2*x*y).args_cnc()
[[-1, 2, x, y], []]
>>> (-2.5*x).args_cnc()
[[-1, 2.5, x], []]
>>> (-2*x*A*B*y).args_cnc()
[[-1, 2, x, y], [A, B]]
>>> (-2*x*A*B*y).args_cnc(split_1=False)
[[-2, x, y], [A, B]]
>>> (-2*x*y).args_cnc(cset=True)
[{-1, 2, x, y}, []]

The arg is always treated as a Mul:

>>> (-2 + x + A).args_cnc()
[[], [x - 2 + A]]
>>> (-oo).args_cnc() # -oo is a singleton
[[-1, oo], []]
as_base_exp()
as_coeff_Add(rational=False)

Efficiently extract the coefficient of a summation.

as_coeff_Mul(rational=False)

Efficiently extract the coefficient of a product.

as_coeff_add(*deps)

Return the tuple (c, args) where self is written as an Add, a.

c should be a Rational added to any terms of the Add that are independent of deps.

args should be a tuple of all other terms of a; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you don’t know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add.

  • if you know self is an Add and want only the head, use self.args[0];
  • if you don’t want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail.
  • if you want to split self into an independent and dependent parts use self.as_independent(*deps)
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_add()
(3, ())
>>> (3 + x).as_coeff_add()
(3, (x,))
>>> (3 + x + y).as_coeff_add(x)
(y + 3, (x,))
>>> (3 + y).as_coeff_add(x)
(y + 3, ())
as_coeff_exponent(x)

c*x**e -> c,e where x can be any symbolic expression.

as_coeff_mul(*deps, **kwargs)

Return the tuple (c, args) where self is written as a Mul, m.

c should be a Rational multiplied by any factors of the Mul that are independent of deps.

args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you don’t know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul.

  • if you know self is a Mul and want only the head, use self.args[0];
  • if you don’t want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail;
  • if you want to split self into an independent and dependent parts use self.as_independent(*deps)
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_mul()
(3, ())
>>> (3*x*y).as_coeff_mul()
(3, (x, y))
>>> (3*x*y).as_coeff_mul(x)
(3*y, (x,))
>>> (3*y).as_coeff_mul(x)
(3*y, ())
as_coefficient(expr)

Extracts symbolic coefficient at the given expression. In other words, this functions separates ‘self’ into the product of ‘expr’ and ‘expr’-free coefficient. If such separation is not possible it will return None.

See also

coeff
return sum of terms have a given factor
as_coeff_Add
separate the additive constant from an expression
as_coeff_Mul
separate the multiplicative constant from an expression
as_independent
separate x-dependent terms/factors from others
sympy.polys.polytools.coeff_monomial
efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.nth
like coeff_monomial but powers of monomial terms are used

Examples

>>> from sympy import E, pi, sin, I, Poly
>>> from sympy.abc import x
>>> E.as_coefficient(E)
1
>>> (2*E).as_coefficient(E)
2
>>> (2*sin(E)*E).as_coefficient(E)

Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.)

>>> (2*E + x*E).as_coefficient(E)
x + 2
>>> _.args[0]  # just want the exact match
2
>>> p = Poly(2*E + x*E); p
Poly(x*E + 2*E, x, E, domain='ZZ')
>>> p.coeff_monomial(E)
2
>>> p.nth(0, 1)
2

Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.)

>>> (2*E*x + x).as_coefficient(E)
>>> (2*E*x + x).coeff(E)
2*x
>>> (E*(x + 1) + x).as_coefficient(E)
>>> (2*pi*I).as_coefficient(pi*I)
2
>>> (2*I).as_coefficient(pi*I)
as_coefficients_dict()

Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term.

Examples

>>> from sympy.abc import a, x
>>> (3*x + a*x + 4).as_coefficients_dict()
{1: 4, x: 3, a*x: 1}
>>> _[a]
0
>>> (3*a*x).as_coefficients_dict()
{a*x: 3}
as_content_primitive(radical=False, clear=True)

This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo. The primitive need no be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).

Examples

>>> from sympy import sqrt
>>> from sympy.abc import x, y, z
>>> eq = 2 + 2*x + 2*y*(3 + 3*y)

The as_content_primitive function is recursive and retains structure:

>>> eq.as_content_primitive()
(2, x + 3*y*(y + 1) + 1)

Integer powers will have Rationals extracted from the base:

>>> ((2 + 6*x)**2).as_content_primitive()
(4, (3*x + 1)**2)
>>> ((2 + 6*x)**(2*y)).as_content_primitive()
(1, (2*(3*x + 1))**(2*y))

Terms may end up joining once their as_content_primitives are added:

>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(11, x*(y + 1))
>>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(9, x*(y + 1))
>>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive()
(1, 6.0*x*(y + 1) + 3*z*(y + 1))
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive()
(121, x**2*(y + 1)**2)
>>> ((5*(x*(1 + y)) + 2.0*x*(3 + 3*y))**2).as_content_primitive()
(1, 121.0*x**2*(y + 1)**2)

Radical content can also be factored out of the primitive:

>>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
(2, sqrt(2)*(1 + 2*sqrt(5)))

If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients.

>>> (x/2 + y).as_content_primitive()
(1/2, x + 2*y)
>>> (x/2 + y).as_content_primitive(clear=False)
(1, x/2 + y)
as_dummy()

Return a Dummy having the same name and same assumptions as self.

as_expr(*gens)

Convert a polynomial to a SymPy expression.

Examples

>>> from sympy import sin
>>> from sympy.abc import x, y
>>> f = (x**2 + x*y).as_poly(x, y)
>>> f.as_expr()
x**2 + x*y
>>> sin(x).as_expr()
sin(x)
as_independent(*deps, **hint)

A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.:

  • separatevars() to change Mul, Add and Pow (including exp) into Mul
  • .expand(mul=True) to change Add or Mul into Add
  • .expand(log=True) to change log expr into an Add

The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints.

For nonzero self, the returned tuple (i, d) has the following interpretation:

  • i will has no variable that appears in deps
  • d will be 1 or else have terms that contain variables that are in deps
  • if self is an Add then self = i + d
  • if self is a Mul then self = i*d
  • otherwise (self, S.One) or (S.One, self) is returned.

To force the expression to be treated as an Add, use the hint as_Add=True

See also

separatevars, expand, Add.as_two_terms, Mul.as_two_terms, as_coeff_add, as_coeff_mul

Examples

– self is an Add

>>> from sympy import sin, cos, exp
>>> from sympy.abc import x, y, z
>>> (x + x*y).as_independent(x)
(0, x*y + x)
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> (2*x*sin(x) + y + x + z).as_independent(x)
(y + z, 2*x*sin(x) + x)
>>> (2*x*sin(x) + y + x + z).as_independent(x, y)
(z, 2*x*sin(x) + x + y)

– self is a Mul

>>> (x*sin(x)*cos(y)).as_independent(x)
(cos(y), x*sin(x))

non-commutative terms cannot always be separated out when self is a Mul

>>> from sympy import symbols
>>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
>>> (n1 + n1*n2).as_independent(n2)
(n1, n1*n2)
>>> (n2*n1 + n1*n2).as_independent(n2)
(0, n1*n2 + n2*n1)
>>> (n1*n2*n3).as_independent(n1)
(1, n1*n2*n3)
>>> (n1*n2*n3).as_independent(n2)
(n1, n2*n3)
>>> ((x-n1)*(x-y)).as_independent(x)
(1, (x - y)*(x - n1))

– self is anything else:

>>> (sin(x)).as_independent(x)
(1, sin(x))
>>> (sin(x)).as_independent(y)
(sin(x), 1)
>>> exp(x+y).as_independent(x)
(1, exp(x + y))

– force self to be treated as an Add:

>>> (3*x).as_independent(x, as_Add=True)
(0, 3*x)

– force self to be treated as a Mul:

>>> (3+x).as_independent(x, as_Add=False)
(1, x + 3)
>>> (-3+x).as_independent(x, as_Add=False)
(1, x - 3)

Note how the below differs from the above in making the constant on the dep term positive.

>>> (y*(-3+x)).as_independent(x)
(y, x - 3)
– use .as_independent() for true independence testing instead
of .has(). The former considers only symbols in the free symbols while the latter considers all symbols
>>> from sympy import Integral
>>> I = Integral(x, (x, 1, 2))
>>> I.has(x)
True
>>> x in I.free_symbols
False
>>> I.as_independent(x) == (I, 1)
True
>>> (I + x).as_independent(x) == (I, x)
True

Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values

>>> from sympy import separatevars, log
>>> separatevars(exp(x+y)).as_independent(x)
(exp(y), exp(x))
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> separatevars(x + x*y).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).expand(mul=True).as_independent(y)
(x, x*y)
>>> a, b=symbols('a b', positive=True)
>>> (log(a*b).expand(log=True)).as_independent(b)
(log(a), log(b))
as_leading_term(*args, **kwargs)

Returns the leading (nonzero) term of the series expansion of self.

The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.

Examples

>>> from sympy.abc import x
>>> (1 + x + x**2).as_leading_term(x)
1
>>> (1/x**2 + x + x**2).as_leading_term(x)
x**(-2)
as_numer_denom()

expression -> a/b -> a, b

This is just a stub that should be defined by an object’s class methods to get anything else.

See also

normal
return a/b instead of a, b
as_ordered_factors(order=None)

Return list of ordered factors (if Mul) else [self].

as_ordered_terms(order=None, data=False)

Transform an expression to an ordered list of terms.

Examples

>>> from sympy import sin, cos
>>> from sympy.abc import x
>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]
as_poly(*gens, **args)

Converts self to a polynomial or returns None.

>>> from sympy import sin
>>> from sympy.abc import x, y
>>> print((x**2 + x*y).as_poly())
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + x*y).as_poly(x, y))
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + sin(y)).as_poly(x, y))
None
as_powers_dict()

Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.

as_real_imag(deep=True, **hints)
as_terms()

Transform an expression to a list of terms.

assumptions0
atoms(*types)

Returns the atoms that form the current object.

By default, only objects that are truly atomic and can’t be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.

Examples

>>> from sympy import Number, NumberSymbol, Symbol
>>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol)
{x, y}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number)
{1, 2}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)
{1, 2, pi}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)
{1, 2, I, pi}

Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class.

The type can be given implicitly, too:

>>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol
{x, y}

Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One, a special type of sympy atom, while type(S(2)) is type Integer and will find all integers in an expression:

>>> from sympy import S
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(1))
{1}
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(2))
{1, 2}

Finally, arguments to atoms() can select more than atomic atoms: any sympy type (loaded in core/__init__.py) can be listed as an argument and those types of “atoms” as found in scanning the arguments of the expression recursively:

>>> from sympy import Function, Mul
>>> from sympy.core.function import AppliedUndef
>>> f = Function('f')
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)
{f(x), sin(y + I*pi)}
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)
{f(x)}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
{I*pi, 2*sin(y + I*pi)}
cancel(*gens, **args)

See the cancel function in sympy.polys

canonical_variables

Return a dictionary mapping any variable defined in self.variables as underscore-suffixed numbers corresponding to their position in self.variables. Enough underscores are added to ensure that there will be no clash with existing free symbols.

Examples

>>> from sympy import Lambda
>>> from sympy.abc import x
>>> Lambda(x, 2*x).canonical_variables
{x: 0_}
class_key()
coeff(x, n=1, right=False)

Returns the coefficient from the term(s) containing x**n. If n is zero then all terms independent of x will be returned.

When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword ‘right’ is ignored when x is commutative.

See also

as_coefficient
separate the expression into a coefficient and factor
as_coeff_Add
separate the additive constant from an expression
as_coeff_Mul
separate the multiplicative constant from an expression
as_independent
separate x-dependent terms/factors from others
sympy.polys.polytools.coeff_monomial
efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.nth
like coeff_monomial but powers of monomial terms are used

Examples

>>> from sympy import symbols
>>> from sympy.abc import x, y, z

You can select terms that have an explicit negative in front of them:

>>> (-x + 2*y).coeff(-1)
x
>>> (x - 2*y).coeff(-1)
2*y

You can select terms with no Rational coefficient:

>>> (x + 2*y).coeff(1)
x
>>> (3 + 2*x + 4*x**2).coeff(1)
0

You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None):

>>> (3 + 2*x + 4*x**2).coeff(x, 0)
3
>>> eq = ((x + 1)**3).expand() + 1
>>> eq
x**3 + 3*x**2 + 3*x + 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 2]
>>> eq -= 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 0]

You can select terms that have a numerical term in front of them:

>>> (-x - 2*y).coeff(2)
-y
>>> from sympy import sqrt
>>> (x + sqrt(2)*x).coeff(sqrt(2))
x

The matching is exact:

>>> (3 + 2*x + 4*x**2).coeff(x)
2
>>> (3 + 2*x + 4*x**2).coeff(x**2)
4
>>> (3 + 2*x + 4*x**2).coeff(x**3)
0
>>> (z*(x + y)**2).coeff((x + y)**2)
z
>>> (z*(x + y)**2).coeff(x + y)
0

In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following:

>>> (x + z*(x + x*y)).coeff(x)
1

If such factoring is desired, factor_terms can be used first:

>>> from sympy import factor_terms
>>> factor_terms(x + z*(x + x*y)).coeff(x)
z*(y + 1) + 1
>>> n, m, o = symbols('n m o', commutative=False)
>>> n.coeff(n)
1
>>> (3*n).coeff(n)
3
>>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m
1 + m
>>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m
m

If there is more than one possible coefficient 0 is returned:

>>> (n*m + m*n).coeff(n)
0

If there is only one possible coefficient, it is returned:

>>> (n*m + x*m*n).coeff(m*n)
x
>>> (n*m + x*m*n).coeff(m*n, right=1)
1
collect(syms, func=None, evaluate=True, exact=False, distribute_order_term=True)

See the collect function in sympy.simplify

combsimp()

See the combsimp function in sympy.simplify

compare(other)

Return -1, 0, 1 if the object is smaller, equal, or greater than other.

Not in the mathematical sense. If the object is of a different type from the “other” then their classes are ordered according to the sorted_classes list.

Examples

>>> from sympy.abc import x, y
>>> x.compare(y)
-1
>>> x.compare(x)
0
>>> y.compare(x)
1
compute_leading_term(x, logx=None)

as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.

conjugate()
copy()
could_extract_minus_sign()

Canonical way to choose an element in the set {e, -e} where e is any expression. If the canonical element is e, we have e.could_extract_minus_sign() == True, else e.could_extract_minus_sign() == False.

For any expression, the set {e.could_extract_minus_sign(), (-e).could_extract_minus_sign()} must be {True, False}.

>>> from sympy.abc import x, y
>>> (x-y).could_extract_minus_sign() != (y-x).could_extract_minus_sign()
True
count(query)

Count the number of matching subexpressions.

count_ops(visual=None)

wrapper for count_ops that returns the operation count.

default_assumptions = {}
diff(*symbols, **assumptions)
doit(**hints)
dummy_eq(other, symbol=None)

Compare two expressions and handle dummy symbols.

Examples

>>> from sympy import Dummy
>>> from sympy.abc import x, y
>>> u = Dummy('u')
>>> (u**2 + 1).dummy_eq(x**2 + 1)
True
>>> (u**2 + 1) == (x**2 + 1)
False
>>> (u**2 + y).dummy_eq(x**2 + y, x)
True
>>> (u**2 + y).dummy_eq(x**2 + y, y)
False
equals(other, failing_expression=False)

Return True if self == other, False if it doesn’t, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.

If self is a Number (or complex number) that is not zero, then the result is False.

If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.

evalf(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)

Evaluate the given formula to an accuracy of n digits. Optional keyword arguments:

subs=<dict>
Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi}. The substitutions must be given as a dictionary.
maxn=<integer>
Allow a maximum temporary working precision of maxn digits (default=100)
chop=<bool>
Replace tiny real or imaginary parts in subresults by exact zeros (default=False)
strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec (default=False)
quad=<str>
Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad=’osc’.
verbose=<bool>
Print debug information (default=False)
expand(*args, **kwargs)

Expand an expression using hints.

See the docstring of the expand() function in sympy.core.function for more information.

extract_additively(c)

Return self - c if it’s possible to subtract c from self and make all matching coefficients move towards zero, else return None.

Examples

>>> from sympy.abc import x, y
>>> e = 2*x + 3
>>> e.extract_additively(x + 1)
x + 2
>>> e.extract_additively(3*x)
>>> e.extract_additively(4)
>>> (y*(x + 1)).extract_additively(x + 1)
>>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1)
(x + 1)*(x + 2*y) + 3

Sometimes auto-expansion will return a less simplified result than desired; gcd_terms might be used in such cases:

>>> from sympy import gcd_terms
>>> (4*x*(y + 1) + y).extract_additively(x)
4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y
>>> gcd_terms(_)
x*(4*y + 3) + y
extract_branch_factor(allow_half=False)

Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n).

>>> from sympy import exp_polar, I, pi
>>> from sympy.abc import x, y
>>> exp_polar(I*pi).extract_branch_factor()
(exp_polar(I*pi), 0)
>>> exp_polar(2*I*pi).extract_branch_factor()
(1, 1)
>>> exp_polar(-pi*I).extract_branch_factor()
(exp_polar(I*pi), -1)
>>> exp_polar(3*pi*I + x).extract_branch_factor()
(exp_polar(x + I*pi), 1)
>>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor()
(y*exp_polar(2*pi*x), -1)
>>> exp_polar(-I*pi/2).extract_branch_factor()
(exp_polar(-I*pi/2), 0)

If allow_half is True, also extract exp_polar(I*pi):

>>> exp_polar(I*pi).extract_branch_factor(allow_half=True)
(1, 1/2)
>>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True)
(1, 1)
>>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True)
(1, 3/2)
>>> exp_polar(-I*pi).extract_branch_factor(allow_half=True)
(1, -1/2)
extract_multiplicatively(c)

Return None if it’s not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.

>>> from sympy import symbols, Rational
>>> x, y = symbols('x,y', real=True)
>>> ((x*y)**3).extract_multiplicatively(x**2 * y)
x*y**2
>>> ((x*y)**3).extract_multiplicatively(x**4 * y)
>>> (2*x).extract_multiplicatively(2)
x
>>> (2*x).extract_multiplicatively(3)
>>> (Rational(1, 2)*x).extract_multiplicatively(3)
x/6
factor(*gens, **args)

See the factor() function in sympy.polys.polytools

find(query, group=False)

Find all subexpressions matching a query.

formula

Return a Formula with only terms=[self].

fourier_series(limits=None)

Compute fourier sine/cosine series of self.

See the docstring of the fourier_series() in sympy.series.fourier for more information.

fps(x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False)

Compute formal power power series of self.

See the docstring of the fps() function in sympy.series.formal for more information.

free_symbols
fromiter(args, **assumptions)

Create a new object from an iterable.

This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.

Examples

>>> from sympy import Tuple
>>> Tuple.fromiter(i for i in range(5))
(0, 1, 2, 3, 4)
func

The top-level function in an expression.

The following should hold for all objects:

>> x == x.func(*x.args)

Examples

>>> from sympy.abc import x
>>> a = 2*x
>>> a.func
<class 'sympy.core.mul.Mul'>
>>> a.args
(2, x)
>>> a.func(*a.args)
2*x
>>> a == a.func(*a.args)
True
getO()

Returns the additive O(..) symbol if there is one, else None.

getn()

Returns the order of the expression.

The order is determined either from the O(…) term. If there is no O(…) term, it returns None.

Examples

>>> from sympy import O
>>> from sympy.abc import x
>>> (1 + x + O(x**2)).getn()
2
>>> (1 + x).getn()
has(*args, **kwargs)

Test whether any subexpression matches any of the patterns.

Examples

>>> from sympy import sin
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True

Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval:

>>> from sympy.sets import Interval
>>> i = Interval.Lopen(0, 5); i
Interval.Lopen(0, 5)
>>> i.args
(0, 5, True, False)
>>> i.has(4)  # there is no "4" in the arguments
False
>>> i.has(0)  # there *is* a "0" in the arguments
True

Instead, use contains to determine whether a number is in the interval or not:

>>> i.contains(4)
True
>>> i.contains(0)
False

Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns). In particular, False is returned when the list of patterns is empty.

>>> x.has()
False
integrate(*args, **kwargs)

See the integrate function in sympy.integrals

invert(g, *gens, **args)

Return the multiplicative inverse of self mod g where self (and g) may be symbolic expressions).

See also

sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert

is_Add = False
is_AlgebraicNumber = False
is_Atom = True
is_Boolean = False
is_Derivative = False
is_Dummy = False
is_Equality = False
is_Float = False
is_Function = False
is_Indexed = False
is_Integer = False
is_Matrix = False
is_Mul = False
is_Not = False
is_Number = False
is_NumberSymbol = False
is_Order = False
is_Piecewise = False
is_Point = False
is_Poly = False
is_Pow = False
is_Rational = False
is_Relational = False
is_Symbol = True
is_Vector = False
is_Wild = False
is_algebraic
is_algebraic_expr(*syms)

This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form.

This function returns False for expressions that are “algebraic expressions” with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.

References

Examples

>>> from sympy import Symbol, sqrt
>>> x = Symbol('x', real=True)
>>> sqrt(1 + x).is_rational_function()
False
>>> sqrt(1 + x).is_algebraic_expr()
True

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one.

>>> from sympy import exp, factor
>>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1)
>>> a.is_algebraic_expr(x)
False
>>> factor(a).is_algebraic_expr()
True
is_antihermitian
is_commutative
is_comparable = False
is_complex
is_composite
is_constant(*wrt, **flags)
is_even
is_finite
is_hermitian
is_hypergeometric(k)
is_imaginary
is_infinite
is_integer
is_irrational
is_negative
is_noninteger
is_nonnegative
is_nonpositive
is_nonzero
is_number = False
is_odd
is_polar
is_polynomial(*syms)

Return True if self is a polynomial in syms and False otherwise.

This checks if self is an exact polynomial in syms. This function returns False for expressions that are “polynomials” with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used.

This is not part of the assumptions system. You cannot do Symbol(‘z’, polynomial=True).

Examples

>>> from sympy import Symbol
>>> x = Symbol('x')
>>> ((x**2 + 1)**4).is_polynomial(x)
True
>>> ((x**2 + 1)**4).is_polynomial()
True
>>> (2**x + 1).is_polynomial(x)
False
>>> n = Symbol('n', nonnegative=True, integer=True)
>>> (x**n + 1).is_polynomial(x)
False

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one.

>>> from sympy import sqrt, factor, cancel
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)
>>> a.is_polynomial(y)
False
>>> factor(a)
y + 1
>>> factor(a).is_polynomial(y)
True
>>> b = (y**2 + 2*y + 1)/(y + 1)
>>> b.is_polynomial(y)
False
>>> cancel(b)
y + 1
>>> cancel(b).is_polynomial(y)
True

See also .is_rational_function()

is_positive
is_prime
is_rational
is_rational_function(*syms)

Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form.

This function returns False for expressions that are “rational functions” with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True.

This is not part of the assumptions system. You cannot do Symbol(‘z’, rational_function=True).

Examples

>>> from sympy import Symbol, sin
>>> from sympy.abc import x, y
>>> (x/y).is_rational_function()
True
>>> (x**2).is_rational_function()
True
>>> (x/sin(y)).is_rational_function(y)
False
>>> n = Symbol('n', integer=True)
>>> (x**n + 1).is_rational_function(x)
False

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one.

>>> from sympy import sqrt, factor
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)/y
>>> a.is_rational_function(y)
False
>>> factor(a)
(y + 1)/y
>>> factor(a).is_rational_function(y)
True

See also is_algebraic_expr().

is_real
is_symbol = True
is_transcendental
is_zero
leadterm(x)

Returns the leading term a*x**b as a tuple (a, b).

Examples

>>> from sympy.abc import x
>>> (1+x+x**2).leadterm(x)
(1, 0)
>>> (1/x**2+x+x**2).leadterm(x)
(1, -2)
limit(x, xlim, dir='+')

Compute limit x->xlim.

lseries(x=None, x0=0, dir='+', logx=None)

Wrapper for series yielding an iterator of the terms of the series.

Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:

for term in sin(x).lseries(x):
    print term

The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you don’t know how many you should ask for in nseries() using the “n” parameter.

See also nseries().

match(pattern, old=False)

Pattern matching.

Wild symbols match all.

Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:

pattern.xreplace(self.match(pattern)) == self

Examples

>>> from sympy import Wild
>>> from sympy.abc import x, y
>>> p = Wild("p")
>>> q = Wild("q")
>>> r = Wild("r")
>>> e = (x+y)**(x+y)
>>> e.match(p**p)
{p_: x + y}
>>> e.match(p**q)
{p_: x + y, q_: x + y}
>>> e = (2*x)**2
>>> e.match(p*q**r)
{p_: 4, q_: x, r_: 2}
>>> (p*q**r).xreplace(e.match(p*q**r))
4*x**2

The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True:

>>> (x - 2).match(p - x, old=True)
{p_: 2*x - 2}
>>> (2/x).match(p*x, old=True)
{p_: 2/x**2}
matches(expr, repl_dict={}, old=False)
n(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)

Evaluate the given formula to an accuracy of n digits. Optional keyword arguments:

subs=<dict>
Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi}. The substitutions must be given as a dictionary.
maxn=<integer>
Allow a maximum temporary working precision of maxn digits (default=100)
chop=<bool>
Replace tiny real or imaginary parts in subresults by exact zeros (default=False)
strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec (default=False)
quad=<str>
Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad=’osc’.
verbose=<bool>
Print debug information (default=False)
name
normal()
nseries(x=None, x0=0, n=6, dir='+', logx=None)

Wrapper to _eval_nseries if assumptions allow, else to series.

If x is given, x0 is 0, dir=’+’, and self has x, then _eval_nseries is called. This calculates “n” terms in the innermost expressions and then builds up the final series just by “cross-multiplying” everything out.

The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided.

Advantage – it’s fast, because we don’t have to determine how many terms we need to calculate in advance.

Disadvantage – you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct.

If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms.

See also lseries().

Examples

>>> from sympy import sin, log, Symbol
>>> from sympy.abc import x, y
>>> sin(x).nseries(x, 0, 6)
x - x**3/6 + x**5/120 + O(x**6)
>>> log(x+1).nseries(x, 0, 5)
x - x**2/2 + x**3/3 - x**4/4 + O(x**5)

Handling of the logx parameter — in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0):

>>> e = sin(log(x))
>>> e.nseries(x, 0, 6)
Traceback (most recent call last):
...
PoleError: ...
...
>>> logx = Symbol('logx')
>>> e.nseries(x, 0, 6, logx=logx)
sin(logx)

In the following example, the expansion works but gives only an Order term unless the logx parameter is used:

>>> e = x**y
>>> e.nseries(x, 0, 2)
O(log(x)**2)
>>> e.nseries(x, 0, 2, logx=logx)
exp(logx*y)
nsimplify(constants=[], tolerance=None, full=False)

See the nsimplify function in sympy.simplify

powsimp(*args, **kwargs)

See the powsimp function in sympy.simplify

primitive()

Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).

Examples

>>> from sympy.abc import x
>>> (3*(x + 1)**2).primitive()
(3, (x + 1)**2)
>>> a = (6*x + 2); a.primitive()
(2, 3*x + 1)
>>> b = (x/2 + 3); b.primitive()
(1/2, x + 6)
>>> (a*b).primitive() == (1, a*b)
True
radsimp(**kwargs)

See the radsimp function in sympy.simplify

ratsimp()

See the ratsimp function in sympy.simplify

rcall(*args)

Apply on the argument recursively through the expression tree.

This method is used to simulate a common abuse of notation for operators. For instance in SymPy the the following will not work:

(x+Lambda(y, 2*y))(z) == x+2*z,

however you can use

>>> from sympy import Lambda
>>> from sympy.abc import x, y, z
>>> (x + Lambda(y, 2*y)).rcall(z)
x + 2*z
refine(assumption=True)

See the refine function in sympy.assumptions

removeO()

Removes the additive O(..) symbol if there is one

replace(query, value, map=False, simultaneous=True, exact=False)

Replace matching subexpressions of self with value.

If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself doesn’t match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())).

Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is True, then the match will only succeed if non-zero values are received for each Wild that appears in the match pattern.

The list of possible combinations of queries and replacement values is listed below:

See also

subs
substitution of subexpressions as defined by the objects themselves.
xreplace
exact node replacement in expr tree; also capable of using matching rules

Examples

Initial setup

>>> from sympy import log, sin, cos, tan, Wild, Mul, Add
>>> from sympy.abc import x, y
>>> f = log(sin(x)) + tan(sin(x**2))
1.1. type -> type

obj.replace(type, newtype)

When object of type type is found, replace it with the result of passing its argument(s) to newtype.

>>> f.replace(sin, cos)
log(cos(x)) + tan(cos(x**2))
>>> sin(x).replace(sin, cos, map=True)
(cos(x), {sin(x): cos(x)})
>>> (x*y).replace(Mul, Add)
x + y
1.2. type -> func

obj.replace(type, func)

When object of type type is found, apply func to its argument(s). func must be written to handle the number of arguments of type.

>>> f.replace(sin, lambda arg: sin(2*arg))
log(sin(2*x)) + tan(sin(2*x**2))
>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
sin(2*x*y)
2.1. pattern -> expr

obj.replace(pattern(wild), expr(wild))

Replace subexpressions matching pattern with the expression written in terms of the Wild symbols in pattern.

>>> a = Wild('a')
>>> f.replace(sin(a), tan(a))
log(tan(x)) + tan(tan(x**2))
>>> f.replace(sin(a), tan(a/2))
log(tan(x/2)) + tan(tan(x**2/2))
>>> f.replace(sin(a), a)
log(x) + tan(x**2)
>>> (x*y).replace(a*x, a)
y

When the default value of False is used with patterns that have more than one Wild symbol, non-intuitive results may be obtained:

>>> b = Wild('b')
>>> (2*x).replace(a*x + b, b - a)
2/x

For this reason, the exact option can be used to make the replacement only when the match gives non-zero values for all Wild symbols:

>>> (2*x + y).replace(a*x + b, b - a, exact=True)
y - 2
>>> (2*x).replace(a*x + b, b - a, exact=True)
2*x
2.2. pattern -> func

obj.replace(pattern(wild), lambda wild: expr(wild))

All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression:

>>> f.replace(sin(a), lambda a: sin(2*a))
log(sin(2*x)) + tan(sin(2*x**2))
3.1. func -> func

obj.replace(filter, func)

Replace subexpression e with func(e) if filter(e) is True.

>>> g = 2*sin(x**3)
>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
4*sin(x**9)

The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice.

>>> e = x*(x*y + 1)
>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
2*x*(2*x*y + 1)
rewrite(*args, **hints)

Rewrite functions in terms of other functions.

Rewrites expression containing applications of functions of one kind in terms of functions of different kind. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function.

As a pattern this function accepts a list of functions to to rewrite (instances of DefinedFunction class). As rule you can use string or a destination function instance (in this case rewrite() will use the str() function).

There is also the possibility to pass hints on how to rewrite the given expressions. For now there is only one such hint defined called ‘deep’. When ‘deep’ is set to False it will forbid functions to rewrite their contents.

Examples

>>> from sympy import sin, exp
>>> from sympy.abc import x

Unspecified pattern:

>>> sin(x).rewrite(exp)
-I*(exp(I*x) - exp(-I*x))/2

Pattern as a single function:

>>> sin(x).rewrite(sin, exp)
-I*(exp(I*x) - exp(-I*x))/2

Pattern as a list of functions:

>>> sin(x).rewrite([sin, ], exp)
-I*(exp(I*x) - exp(-I*x))/2
round(p=0)

Return x rounded to the given decimal place.

If a complex number would results, apply round to the real and imaginary components of the number.

Notes

Do not confuse the Python builtin function, round, with the SymPy method of the same name. The former always returns a float (or raises an error if applied to a complex value) while the latter returns either a Number or a complex number:

>>> isinstance(round(S(123), -2), Number)
False
>>> isinstance(S(123).round(-2), Number)
True
>>> isinstance((3*I).round(), Mul)
True
>>> isinstance((1 + 3*I).round(), Add)
True

Examples

>>> from sympy import pi, E, I, S, Add, Mul, Number
>>> S(10.5).round()
11.
>>> pi.round()
3.
>>> pi.round(2)
3.14
>>> (2*pi + E*I).round()
6. + 3.*I

The round method has a chopping effect:

>>> (2*pi + I/10).round()
6.
>>> (pi/10 + 2*I).round()
2.*I
>>> (pi/10 + E*I).round(2)
0.31 + 2.72*I
separate(deep=False, force=False)

See the separate function in sympy.simplify

series(x=None, x0=0, n=6, dir='+', logx=None)

Series expansion of “self” around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.

Returns the series expansion of “self” around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.

>>> from sympy import cos, exp
>>> from sympy.abc import x, y
>>> cos(x).series()
1 - x**2/2 + x**4/24 + O(x**6)
>>> cos(x).series(n=4)
1 - x**2/2 + O(x**4)
>>> cos(x).series(x, x0=1, n=2)
cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1))
>>> e = cos(x + exp(y))
>>> e.series(y, n=2)
cos(x + 1) - y*sin(x + 1) + O(y**2)
>>> e.series(x, n=2)
cos(exp(y)) - x*sin(exp(y)) + O(x**2)

If n=None then a generator of the series terms will be returned.

>>> term=cos(x).series(n=None)
>>> [next(term) for i in range(2)]
[1, -x**2/2]

For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results.

>>> abs(x).series(dir="+")
x
>>> abs(x).series(dir="-")
-x
simplify(ratio=1.7, measure=None)

See the simplify function in sympy.simplify

sort_key(*args, **kwargs)
subs(*args, **kwargs)

Substitutes old for new in an expression after sympifying args.

args is either:
  • two arguments, e.g. foo.subs(old, new)
  • one iterable argument, e.g. foo.subs(iterable). The iterable may be
    o an iterable container with (old, new) pairs. In this case the
    replacements are processed in the order given with successive patterns possibly affecting replacements already made.
    o a dict or set whose key/value items correspond to old/new pairs.
    In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous).

If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.

See also

replace
replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements
xreplace
exact node replacement in expr tree; also capable of using matching rules
evalf
calculates the given formula to a desired level of precision

Examples

>>> from sympy import pi, exp, limit, oo
>>> from sympy.abc import x, y
>>> (1 + x*y).subs(x, pi)
pi*y + 1
>>> (1 + x*y).subs({x:pi, y:2})
1 + 2*pi
>>> (1 + x*y).subs([(x, pi), (y, 2)])
1 + 2*pi
>>> reps = [(y, x**2), (x, 2)]
>>> (x + y).subs(reps)
6
>>> (x + y).subs(reversed(reps))
x**2 + 2
>>> (x**2 + x**4).subs(x**2, y)
y**2 + y

To replace only the x**2 but not the x**4, use xreplace:

>>> (x**2 + x**4).xreplace({x**2: y})
x**4 + y

To delay evaluation until all substitutions have been made, set the keyword simultaneous to True:

>>> (x/y).subs([(x, 0), (y, 0)])
0
>>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True)
nan

This has the added feature of not allowing subsequent substitutions to affect those already made:

>>> ((x + y)/y).subs({x + y: y, y: x + y})
1
>>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)
y/(x + y)

In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted.

>>> from sympy import sqrt, sin, cos
>>> from sympy.abc import a, b, c, d, e
>>> A = (sqrt(sin(2*x)), a)
>>> B = (sin(2*x), b)
>>> C = (cos(2*x), c)
>>> D = (x, d)
>>> E = (exp(x), e)
>>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)
>>> expr.subs(dict([A, B, C, D, E]))
a*c*sin(d*e) + b

The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression:

>>> (x**3 - 3*x).subs({x: oo})
nan
>>> limit(x**3 - 3*x, x, oo)
oo

If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as

>>> (1/x).evalf(subs={x: 3.0}, n=21)
0.333333333333333333333

rather than

>>> (1/x).subs({x: 3.0}).evalf(21)
0.333333333333333314830

as the former will ensure that the desired level of precision is obtained.

taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

together(*args, **kwargs)

See the together function in sympy.polys

transpose()
trigsimp(**args)

See the trigsimp function in sympy.simplify

xreplace(rule, hack2=False)

Formula

class nipy.algorithms.statistics.formula.formulae.Formula(seq, char='b')

Bases: object

A Formula is a model for a mean in a regression model.

It is often given by a sequence of sympy expressions, with the mean model being the sum of each term multiplied by a linear regression coefficient.

The expressions may depend on additional Symbol instances, giving a non-linear regression model.

Methods

design(input[, param, return_float, contrasts]) Construct the design matrix, and optional contrast matrices.
fromrec(rec[, keep, drop]) Construct Formula from recarray
subs(old, new) Perform a sympy substitution on all terms in the Formula
__init__(seq, char='b')
Parameters:

seq : sequence of sympy.Basic

char : str, optional

character for regression coefficient

coefs

Coefficients in the linear regression formula.

design(input, param=None, return_float=False, contrasts=None)

Construct the design matrix, and optional contrast matrices.

Parameters:

input : np.recarray

Recarray including fields needed to compute the Terms in getparams(self.design_expr).

param : None or np.recarray

Recarray including fields that are not Terms in getparams(self.design_expr)

return_float : bool, optional

If True, return a np.float array rather than a np.recarray

contrasts : None or dict, optional

Contrasts. The items in this dictionary should be (str, Formula) pairs where a contrast matrix is constructed for each Formula by evaluating its design at the same parameters as self.design. If not None, then the return_float is set to True.

Returns:

des : 2D array

design matrix

cmatrices : dict, optional

Dictionary with keys from contrasts input, and contrast matrices corresponding to des design matrix. Returned only if contrasts input is not None

design_expr
dtype

The dtype of the design matrix of the Formula.

static fromrec(rec, keep=[], drop=[])

Construct Formula from recarray

For fields with a string-dtype, it is assumed that these are qualtiatitve regressors, i.e. Factors.

Parameters:

rec: recarray :

Recarray whose field names will be used to create a formula.

keep: [] :

Field names to explicitly keep, dropping all others.

drop: [] :

Field names to drop.

mean

Expression for the mean, expressed as a linear combination of terms, each with dummy variables in front.

params

The parameters in the Formula.

subs(old, new)

Perform a sympy substitution on all terms in the Formula

Returns a new instance of the same class

Parameters:

old : sympy.Basic

The expression to be changed

new : sympy.Basic

The value to change it to.

Returns:

newf : Formula

Examples

>>> s, t = [Term(l) for l in 'st']
>>> f, g = [sympy.Function(l) for l in 'fg']
>>> form = Formula([f(t),g(s)])
>>> newform = form.subs(g, sympy.Function('h'))
>>> newform.terms
array([f(t), h(s)], dtype=object)
>>> form.terms
array([f(t), g(s)], dtype=object)
terms

Terms in the linear regression formula.

RandomEffects

class nipy.algorithms.statistics.formula.formulae.RandomEffects(seq, sigma=None, char='e')

Bases: nipy.algorithms.statistics.formula.formulae.Formula

Covariance matrices for common random effects analyses.

Examples

Two subjects (here named 2 and 3):

>>> subj = make_recarray([2,2,2,3,3], 's')
>>> subj_factor = Factor('s', [2,3])

By default the covariance matrix is symbolic. The display differs a little between sympy versions (hence we don’t check it in the doctests):

>>> c = RandomEffects(subj_factor.terms)
>>> c.cov(subj) 
array([[_s2_0, _s2_0, _s2_0, 0, 0],
       [_s2_0, _s2_0, _s2_0, 0, 0],
       [_s2_0, _s2_0, _s2_0, 0, 0],
       [0, 0, 0, _s2_1, _s2_1],
       [0, 0, 0, _s2_1, _s2_1]], dtype=object)

With a numeric sigma, you get a numeric array:

>>> c = RandomEffects(subj_factor.terms, sigma=np.array([[4,1],[1,6]]))
>>> c.cov(subj)
array([[ 4.,  4.,  4.,  1.,  1.],
       [ 4.,  4.,  4.,  1.,  1.],
       [ 4.,  4.,  4.,  1.,  1.],
       [ 1.,  1.,  1.,  6.,  6.],
       [ 1.,  1.,  1.,  6.,  6.]])

Methods

cov(term[, param]) Compute the covariance matrix for some given data.
design(input[, param, return_float, contrasts]) Construct the design matrix, and optional contrast matrices.
fromrec(rec[, keep, drop]) Construct Formula from recarray
subs(old, new) Perform a sympy substitution on all terms in the Formula
__init__(seq, sigma=None, char='e')

Initialize random effects instance

Parameters:

seq : [sympy.Basic]

sigma : ndarray

Covariance of the random effects. Defaults to a diagonal with entries for each random effect.

char : character for regression coefficient

coefs

Coefficients in the linear regression formula.

cov(term, param=None)

Compute the covariance matrix for some given data.

Parameters:

term : np.recarray

Recarray including fields corresponding to the Terms in getparams(self.design_expr).

param : np.recarray

Recarray including fields that are not Terms in getparams(self.design_expr)

Returns:

C : ndarray

Covariance matrix implied by design and self.sigma.

design(input, param=None, return_float=False, contrasts=None)

Construct the design matrix, and optional contrast matrices.

Parameters:

input : np.recarray

Recarray including fields needed to compute the Terms in getparams(self.design_expr).

param : None or np.recarray

Recarray including fields that are not Terms in getparams(self.design_expr)

return_float : bool, optional

If True, return a np.float array rather than a np.recarray

contrasts : None or dict, optional

Contrasts. The items in this dictionary should be (str, Formula) pairs where a contrast matrix is constructed for each Formula by evaluating its design at the same parameters as self.design. If not None, then the return_float is set to True.

Returns:

des : 2D array

design matrix

cmatrices : dict, optional

Dictionary with keys from contrasts input, and contrast matrices corresponding to des design matrix. Returned only if contrasts input is not None

design_expr
dtype

The dtype of the design matrix of the Formula.

fromrec(rec, keep=[], drop=[])

Construct Formula from recarray

For fields with a string-dtype, it is assumed that these are qualtiatitve regressors, i.e. Factors.

Parameters:

rec: recarray :

Recarray whose field names will be used to create a formula.

keep: [] :

Field names to explicitly keep, dropping all others.

drop: [] :

Field names to drop.

mean

Expression for the mean, expressed as a linear combination of terms, each with dummy variables in front.

params

The parameters in the Formula.

subs(old, new)

Perform a sympy substitution on all terms in the Formula

Returns a new instance of the same class

Parameters:

old : sympy.Basic

The expression to be changed

new : sympy.Basic

The value to change it to.

Returns:

newf : Formula

Examples

>>> s, t = [Term(l) for l in 'st']
>>> f, g = [sympy.Function(l) for l in 'fg']
>>> form = Formula([f(t),g(s)])
>>> newform = form.subs(g, sympy.Function('h'))
>>> newform.terms
array([f(t), h(s)], dtype=object)
>>> form.terms
array([f(t), g(s)], dtype=object)
terms

Terms in the linear regression formula.

Term

class nipy.algorithms.statistics.formula.formulae.Term

Bases: sympy.core.symbol.Symbol

A sympy.Symbol type to represent a term an a regression model

Terms can be added to other sympy expressions with the single convention that a term plus itself returns itself.

It is meant to emulate something on the right hand side of a formula in R. In particular, its name can be the name of a field in a recarray used to create a design matrix.

>>> t = Term('x')
>>> xval = np.array([(3,),(4,),(5,)], np.dtype([('x', np.float)]))
>>> f = t.formula
>>> d = f.design(xval)
>>> print(d.dtype.descr)
[('x', '<f8')]
>>> f.design(xval, return_float=True)
array([ 3.,  4.,  5.])

Methods

__call__(*args)
adjoint()
apart([x]) See the apart function in sympy.polys
args_cnc([cset, warn, split_1]) Return [commutative factors, non-commutative factors] of self.
as_base_exp()
as_coeff_Add([rational]) Efficiently extract the coefficient of a summation.
as_coeff_Mul([rational]) Efficiently extract the coefficient of a product.
as_coeff_add(*deps) Return the tuple (c, args) where self is written as an Add, a.
as_coeff_exponent(x) c*x**e -> c,e where x can be any symbolic expression.
as_coeff_mul(*deps, **kwargs) Return the tuple (c, args) where self is written as a Mul, m.
as_coefficient(expr) Extracts symbolic coefficient at the given expression.
as_coefficients_dict() Return a dictionary mapping terms to their Rational coefficient.
as_content_primitive([radical, clear]) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive).
as_dummy() Return a Dummy having the same name and same assumptions as self.
as_expr(*gens) Convert a polynomial to a SymPy expression.
as_independent(*deps, **hint) A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps.
as_leading_term(*args, **kwargs) Returns the leading (nonzero) term of the series expansion of self.
as_numer_denom() expression -> a/b -> a, b
as_ordered_factors([order]) Return list of ordered factors (if Mul) else [self].
as_ordered_terms([order, data]) Transform an expression to an ordered list of terms.
as_poly(*gens, **args) Converts self to a polynomial or returns None.
as_powers_dict() Return self as a dictionary of factors with each factor being treated as a power.
as_real_imag([deep])
as_terms() Transform an expression to a list of terms.
atoms(*types) Returns the atoms that form the current object.
cancel(*gens, **args) See the cancel function in sympy.polys
class_key()
coeff(x[, n, right]) Returns the coefficient from the term(s) containing x**n.
collect(syms[, func, evaluate, exact, …]) See the collect function in sympy.simplify
combsimp() See the combsimp function in sympy.simplify
compare(other) Return -1, 0, 1 if the object is smaller, equal, or greater than other.
compute_leading_term(x[, logx]) as_leading_term is only allowed for results of .series()
conjugate()
copy()
could_extract_minus_sign() Canonical way to choose an element in the set {e, -e} where e is any expression.
count(query) Count the number of matching subexpressions.
count_ops([visual]) wrapper for count_ops that returns the operation count.
diff(*symbols, **assumptions)
doit(**hints)
dummy_eq(other[, symbol]) Compare two expressions and handle dummy symbols.
equals(other[, failing_expression]) Return True if self == other, False if it doesn’t, or None.
evalf([n, subs, maxn, chop, strict, quad, …]) Evaluate the given formula to an accuracy of n digits.
expand(*args, **kwargs) Expand an expression using hints.
extract_additively(c) Return self - c if it’s possible to subtract c from self and make all matching coefficients move towards zero, else return None.
extract_branch_factor([allow_half]) Try to write self as exp_polar(2*pi*I*n)*z in a nice way.
extract_multiplicatively(c) Return None if it’s not possible to make self in the form c * something in a nice way, i.e.
factor(*gens, **args) See the factor() function in sympy.polys.polytools
find(query[, group]) Find all subexpressions matching a query.
fourier_series([limits]) Compute fourier sine/cosine series of self.
fps([x, x0, dir, hyper, order, rational, full]) Compute formal power power series of self.
fromiter(args, **assumptions) Create a new object from an iterable.
getO() Returns the additive O(..) symbol if there is one, else None.
getn() Returns the order of the expression.
has(*args, **kwargs) Test whether any subexpression matches any of the patterns.
integrate(*args, **kwargs) See the integrate function in sympy.integrals
invert(g, *gens, **args) Return the multiplicative inverse of self mod g where self (and g) may be symbolic expressions).
is_algebraic_expr(*syms) This tests whether a given expression is algebraic or not, in the given symbols, syms.
is_constant(*wrt, **flags)
is_hypergeometric(k)
is_polynomial(*syms) Return True if self is a polynomial in syms and False otherwise.
is_rational_function(*syms) Test whether function is a ratio of two polynomials in the given symbols, syms.
leadterm(x) Returns the leading term a*x**b as a tuple (a, b).
limit(x, xlim[, dir]) Compute limit x->xlim.
lseries([x, x0, dir, logx]) Wrapper for series yielding an iterator of the terms of the series.
match(pattern[, old]) Pattern matching.
matches(expr[, repl_dict, old])
n([n, subs, maxn, chop, strict, quad, verbose]) Evaluate the given formula to an accuracy of n digits.
normal()
nseries([x, x0, n, dir, logx]) Wrapper to _eval_nseries if assumptions allow, else to series.
nsimplify([constants, tolerance, full]) See the nsimplify function in sympy.simplify
powsimp(*args, **kwargs) See the powsimp function in sympy.simplify
primitive() Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add).
radsimp(**kwargs) See the radsimp function in sympy.simplify
ratsimp() See the ratsimp function in sympy.simplify
rcall(*args) Apply on the argument recursively through the expression tree.
refine([assumption]) See the refine function in sympy.assumptions
removeO() Removes the additive O(..) symbol if there is one
replace(query, value[, map, simultaneous, exact]) Replace matching subexpressions of self with value.
rewrite(*args, **hints) Rewrite functions in terms of other functions.
round([p]) Return x rounded to the given decimal place.
separate([deep, force]) See the separate function in sympy.simplify
series([x, x0, n, dir, logx]) Series expansion of “self” around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.
simplify([ratio, measure]) See the simplify function in sympy.simplify
sort_key(*args, **kwargs)
subs(*args, **kwargs) Substitutes old for new in an expression after sympifying args.
taylor_term(n, x, *previous_terms) General method for the taylor term.
together(*args, **kwargs) See the together function in sympy.polys
transpose()
trigsimp(**args) See the trigsimp function in sympy.simplify
xreplace(rule[, hack2])
__init__()

x.__init__(…) initializes x; see help(type(x)) for signature

adjoint()
apart(x=None, **args)

See the apart function in sympy.polys

args

Returns a tuple of arguments of ‘self’.

Notes

Never use self._args, always use self.args. Only use _args in __new__ when creating a new function. Don’t override .args() from Basic (so that it’s easy to change the interface in the future if needed).

Examples

>>> from sympy import cot
>>> from sympy.abc import x, y
>>> cot(x).args
(x,)
>>> cot(x).args[0]
x
>>> (x*y).args
(x, y)
>>> (x*y).args[1]
y
args_cnc(cset=False, warn=True, split_1=True)

Return [commutative factors, non-commutative factors] of self.

self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly supressed by setting warn to False.

Note: -1 is always separated from a Number unless split_1 is False.

>>> from sympy import symbols, oo
>>> A, B = symbols('A B', commutative=0)
>>> x, y = symbols('x y')
>>> (-2*x*y).args_cnc()
[[-1, 2, x, y], []]
>>> (-2.5*x).args_cnc()
[[-1, 2.5, x], []]
>>> (-2*x*A*B*y).args_cnc()
[[-1, 2, x, y], [A, B]]
>>> (-2*x*A*B*y).args_cnc(split_1=False)
[[-2, x, y], [A, B]]
>>> (-2*x*y).args_cnc(cset=True)
[{-1, 2, x, y}, []]

The arg is always treated as a Mul:

>>> (-2 + x + A).args_cnc()
[[], [x - 2 + A]]
>>> (-oo).args_cnc() # -oo is a singleton
[[-1, oo], []]
as_base_exp()
as_coeff_Add(rational=False)

Efficiently extract the coefficient of a summation.

as_coeff_Mul(rational=False)

Efficiently extract the coefficient of a product.

as_coeff_add(*deps)

Return the tuple (c, args) where self is written as an Add, a.

c should be a Rational added to any terms of the Add that are independent of deps.

args should be a tuple of all other terms of a; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you don’t know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add.

  • if you know self is an Add and want only the head, use self.args[0];
  • if you don’t want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail.
  • if you want to split self into an independent and dependent parts use self.as_independent(*deps)
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_add()
(3, ())
>>> (3 + x).as_coeff_add()
(3, (x,))
>>> (3 + x + y).as_coeff_add(x)
(y + 3, (x,))
>>> (3 + y).as_coeff_add(x)
(y + 3, ())
as_coeff_exponent(x)

c*x**e -> c,e where x can be any symbolic expression.

as_coeff_mul(*deps, **kwargs)

Return the tuple (c, args) where self is written as a Mul, m.

c should be a Rational multiplied by any factors of the Mul that are independent of deps.

args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you don’t know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul.

  • if you know self is a Mul and want only the head, use self.args[0];
  • if you don’t want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail;
  • if you want to split self into an independent and dependent parts use self.as_independent(*deps)
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_mul()
(3, ())
>>> (3*x*y).as_coeff_mul()
(3, (x, y))
>>> (3*x*y).as_coeff_mul(x)
(3*y, (x,))
>>> (3*y).as_coeff_mul(x)
(3*y, ())
as_coefficient(expr)

Extracts symbolic coefficient at the given expression. In other words, this functions separates ‘self’ into the product of ‘expr’ and ‘expr’-free coefficient. If such separation is not possible it will return None.

See also

coeff
return sum of terms have a given factor
as_coeff_Add
separate the additive constant from an expression
as_coeff_Mul
separate the multiplicative constant from an expression
as_independent
separate x-dependent terms/factors from others
sympy.polys.polytools.coeff_monomial
efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.nth
like coeff_monomial but powers of monomial terms are used

Examples

>>> from sympy import E, pi, sin, I, Poly
>>> from sympy.abc import x
>>> E.as_coefficient(E)
1
>>> (2*E).as_coefficient(E)
2
>>> (2*sin(E)*E).as_coefficient(E)

Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.)

>>> (2*E + x*E).as_coefficient(E)
x + 2
>>> _.args[0]  # just want the exact match
2
>>> p = Poly(2*E + x*E); p
Poly(x*E + 2*E, x, E, domain='ZZ')
>>> p.coeff_monomial(E)
2
>>> p.nth(0, 1)
2

Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.)

>>> (2*E*x + x).as_coefficient(E)
>>> (2*E*x + x).coeff(E)
2*x
>>> (E*(x + 1) + x).as_coefficient(E)
>>> (2*pi*I).as_coefficient(pi*I)
2
>>> (2*I).as_coefficient(pi*I)
as_coefficients_dict()

Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term.

Examples

>>> from sympy.abc import a, x
>>> (3*x + a*x + 4).as_coefficients_dict()
{1: 4, x: 3, a*x: 1}
>>> _[a]
0
>>> (3*a*x).as_coefficients_dict()
{a*x: 3}
as_content_primitive(radical=False, clear=True)

This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo. The primitive need no be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).

Examples

>>> from sympy import sqrt
>>> from sympy.abc import x, y, z
>>> eq = 2 + 2*x + 2*y*(3 + 3*y)

The as_content_primitive function is recursive and retains structure:

>>> eq.as_content_primitive()
(2, x + 3*y*(y + 1) + 1)

Integer powers will have Rationals extracted from the base:

>>> ((2 + 6*x)**2).as_content_primitive()
(4, (3*x + 1)**2)
>>> ((2 + 6*x)**(2*y)).as_content_primitive()
(1, (2*(3*x + 1))**(2*y))

Terms may end up joining once their as_content_primitives are added:

>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(11, x*(y + 1))
>>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(9, x*(y + 1))
>>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive()
(1, 6.0*x*(y + 1) + 3*z*(y + 1))
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive()
(121, x**2*(y + 1)**2)
>>> ((5*(x*(1 + y)) + 2.0*x*(3 + 3*y))**2).as_content_primitive()
(1, 121.0*x**2*(y + 1)**2)

Radical content can also be factored out of the primitive:

>>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
(2, sqrt(2)*(1 + 2*sqrt(5)))

If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients.

>>> (x/2 + y).as_content_primitive()
(1/2, x + 2*y)
>>> (x/2 + y).as_content_primitive(clear=False)
(1, x/2 + y)
as_dummy()

Return a Dummy having the same name and same assumptions as self.

as_expr(*gens)

Convert a polynomial to a SymPy expression.

Examples

>>> from sympy import sin
>>> from sympy.abc import x, y
>>> f = (x**2 + x*y).as_poly(x, y)
>>> f.as_expr()
x**2 + x*y
>>> sin(x).as_expr()
sin(x)
as_independent(*deps, **hint)

A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.:

  • separatevars() to change Mul, Add and Pow (including exp) into Mul
  • .expand(mul=True) to change Add or Mul into Add
  • .expand(log=True) to change log expr into an Add

The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints.

For nonzero self, the returned tuple (i, d) has the following interpretation:

  • i will has no variable that appears in deps
  • d will be 1 or else have terms that contain variables that are in deps
  • if self is an Add then self = i + d
  • if self is a Mul then self = i*d
  • otherwise (self, S.One) or (S.One, self) is returned.

To force the expression to be treated as an Add, use the hint as_Add=True

See also

separatevars, expand, Add.as_two_terms, Mul.as_two_terms, as_coeff_add, as_coeff_mul

Examples

– self is an Add

>>> from sympy import sin, cos, exp
>>> from sympy.abc import x, y, z
>>> (x + x*y).as_independent(x)
(0, x*y + x)
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> (2*x*sin(x) + y + x + z).as_independent(x)
(y + z, 2*x*sin(x) + x)
>>> (2*x*sin(x) + y + x + z).as_independent(x, y)
(z, 2*x*sin(x) + x + y)

– self is a Mul

>>> (x*sin(x)*cos(y)).as_independent(x)
(cos(y), x*sin(x))

non-commutative terms cannot always be separated out when self is a Mul

>>> from sympy import symbols
>>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
>>> (n1 + n1*n2).as_independent(n2)
(n1, n1*n2)
>>> (n2*n1 + n1*n2).as_independent(n2)
(0, n1*n2 + n2*n1)
>>> (n1*n2*n3).as_independent(n1)
(1, n1*n2*n3)
>>> (n1*n2*n3).as_independent(n2)
(n1, n2*n3)
>>> ((x-n1)*(x-y)).as_independent(x)
(1, (x - y)*(x - n1))

– self is anything else:

>>> (sin(x)).as_independent(x)
(1, sin(x))
>>> (sin(x)).as_independent(y)
(sin(x), 1)
>>> exp(x+y).as_independent(x)
(1, exp(x + y))

– force self to be treated as an Add:

>>> (3*x).as_independent(x, as_Add=True)
(0, 3*x)

– force self to be treated as a Mul:

>>> (3+x).as_independent(x, as_Add=False)
(1, x + 3)
>>> (-3+x).as_independent(x, as_Add=False)
(1, x - 3)

Note how the below differs from the above in making the constant on the dep term positive.

>>> (y*(-3+x)).as_independent(x)
(y, x - 3)
– use .as_independent() for true independence testing instead
of .has(). The former considers only symbols in the free symbols while the latter considers all symbols
>>> from sympy import Integral
>>> I = Integral(x, (x, 1, 2))
>>> I.has(x)
True
>>> x in I.free_symbols
False
>>> I.as_independent(x) == (I, 1)
True
>>> (I + x).as_independent(x) == (I, x)
True

Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values

>>> from sympy import separatevars, log
>>> separatevars(exp(x+y)).as_independent(x)
(exp(y), exp(x))
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> separatevars(x + x*y).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).expand(mul=True).as_independent(y)
(x, x*y)
>>> a, b=symbols('a b', positive=True)
>>> (log(a*b).expand(log=True)).as_independent(b)
(log(a), log(b))
as_leading_term(*args, **kwargs)

Returns the leading (nonzero) term of the series expansion of self.

The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.

Examples

>>> from sympy.abc import x
>>> (1 + x + x**2).as_leading_term(x)
1
>>> (1/x**2 + x + x**2).as_leading_term(x)
x**(-2)
as_numer_denom()

expression -> a/b -> a, b

This is just a stub that should be defined by an object’s class methods to get anything else.

See also

normal
return a/b instead of a, b
as_ordered_factors(order=None)

Return list of ordered factors (if Mul) else [self].

as_ordered_terms(order=None, data=False)

Transform an expression to an ordered list of terms.

Examples

>>> from sympy import sin, cos
>>> from sympy.abc import x
>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]
as_poly(*gens, **args)

Converts self to a polynomial or returns None.

>>> from sympy import sin
>>> from sympy.abc import x, y
>>> print((x**2 + x*y).as_poly())
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + x*y).as_poly(x, y))
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + sin(y)).as_poly(x, y))
None
as_powers_dict()

Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.

as_real_imag(deep=True, **hints)
as_terms()

Transform an expression to a list of terms.

assumptions0
atoms(*types)

Returns the atoms that form the current object.

By default, only objects that are truly atomic and can’t be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.

Examples

>>> from sympy import Number, NumberSymbol, Symbol
>>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol)
{x, y}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number)
{1, 2}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)
{1, 2, pi}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)
{1, 2, I, pi}

Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class.

The type can be given implicitly, too:

>>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol
{x, y}

Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One, a special type of sympy atom, while type(S(2)) is type Integer and will find all integers in an expression:

>>> from sympy import S
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(1))
{1}
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(2))
{1, 2}

Finally, arguments to atoms() can select more than atomic atoms: any sympy type (loaded in core/__init__.py) can be listed as an argument and those types of “atoms” as found in scanning the arguments of the expression recursively:

>>> from sympy import Function, Mul
>>> from sympy.core.function import AppliedUndef
>>> f = Function('f')
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)
{f(x), sin(y + I*pi)}
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)
{f(x)}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
{I*pi, 2*sin(y + I*pi)}
cancel(*gens, **args)

See the cancel function in sympy.polys

canonical_variables

Return a dictionary mapping any variable defined in self.variables as underscore-suffixed numbers corresponding to their position in self.variables. Enough underscores are added to ensure that there will be no clash with existing free symbols.

Examples

>>> from sympy import Lambda
>>> from sympy.abc import x
>>> Lambda(x, 2*x).canonical_variables
{x: 0_}
class_key()
coeff(x, n=1, right=False)

Returns the coefficient from the term(s) containing x**n. If n is zero then all terms independent of x will be returned.

When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword ‘right’ is ignored when x is commutative.

See also

as_coefficient
separate the expression into a coefficient and factor
as_coeff_Add
separate the additive constant from an expression
as_coeff_Mul
separate the multiplicative constant from an expression
as_independent
separate x-dependent terms/factors from others
sympy.polys.polytools.coeff_monomial
efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.nth
like coeff_monomial but powers of monomial terms are used

Examples

>>> from sympy import symbols
>>> from sympy.abc import x, y, z

You can select terms that have an explicit negative in front of them:

>>> (-x + 2*y).coeff(-1)
x
>>> (x - 2*y).coeff(-1)
2*y

You can select terms with no Rational coefficient:

>>> (x + 2*y).coeff(1)
x
>>> (3 + 2*x + 4*x**2).coeff(1)
0

You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None):

>>> (3 + 2*x + 4*x**2).coeff(x, 0)
3
>>> eq = ((x + 1)**3).expand() + 1
>>> eq
x**3 + 3*x**2 + 3*x + 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 2]
>>> eq -= 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 0]

You can select terms that have a numerical term in front of them:

>>> (-x - 2*y).coeff(2)
-y
>>> from sympy import sqrt
>>> (x + sqrt(2)*x).coeff(sqrt(2))
x

The matching is exact:

>>> (3 + 2*x + 4*x**2).coeff(x)
2
>>> (3 + 2*x + 4*x**2).coeff(x**2)
4
>>> (3 + 2*x + 4*x**2).coeff(x**3)
0
>>> (z*(x + y)**2).coeff((x + y)**2)
z
>>> (z*(x + y)**2).coeff(x + y)
0

In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following:

>>> (x + z*(x + x*y)).coeff(x)
1

If such factoring is desired, factor_terms can be used first:

>>> from sympy import factor_terms
>>> factor_terms(x + z*(x + x*y)).coeff(x)
z*(y + 1) + 1
>>> n, m, o = symbols('n m o', commutative=False)
>>> n.coeff(n)
1
>>> (3*n).coeff(n)
3
>>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m
1 + m
>>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m
m

If there is more than one possible coefficient 0 is returned:

>>> (n*m + m*n).coeff(n)
0

If there is only one possible coefficient, it is returned:

>>> (n*m + x*m*n).coeff(m*n)
x
>>> (n*m + x*m*n).coeff(m*n, right=1)
1
collect(syms, func=None, evaluate=True, exact=False, distribute_order_term=True)

See the collect function in sympy.simplify

combsimp()

See the combsimp function in sympy.simplify

compare(other)

Return -1, 0, 1 if the object is smaller, equal, or greater than other.

Not in the mathematical sense. If the object is of a different type from the “other” then their classes are ordered according to the sorted_classes list.

Examples

>>> from sympy.abc import x, y
>>> x.compare(y)
-1
>>> x.compare(x)
0
>>> y.compare(x)
1
compute_leading_term(x, logx=None)

as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.

conjugate()
copy()
could_extract_minus_sign()

Canonical way to choose an element in the set {e, -e} where e is any expression. If the canonical element is e, we have e.could_extract_minus_sign() == True, else e.could_extract_minus_sign() == False.

For any expression, the set {e.could_extract_minus_sign(), (-e).could_extract_minus_sign()} must be {True, False}.

>>> from sympy.abc import x, y
>>> (x-y).could_extract_minus_sign() != (y-x).could_extract_minus_sign()
True
count(query)

Count the number of matching subexpressions.

count_ops(visual=None)

wrapper for count_ops that returns the operation count.

default_assumptions = {}
diff(*symbols, **assumptions)
doit(**hints)
dummy_eq(other, symbol=None)

Compare two expressions and handle dummy symbols.

Examples

>>> from sympy import Dummy
>>> from sympy.abc import x, y
>>> u = Dummy('u')
>>> (u**2 + 1).dummy_eq(x**2 + 1)
True
>>> (u**2 + 1) == (x**2 + 1)
False
>>> (u**2 + y).dummy_eq(x**2 + y, x)
True
>>> (u**2 + y).dummy_eq(x**2 + y, y)
False
equals(other, failing_expression=False)

Return True if self == other, False if it doesn’t, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.

If self is a Number (or complex number) that is not zero, then the result is False.

If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.

evalf(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)

Evaluate the given formula to an accuracy of n digits. Optional keyword arguments:

subs=<dict>
Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi}. The substitutions must be given as a dictionary.
maxn=<integer>
Allow a maximum temporary working precision of maxn digits (default=100)
chop=<bool>
Replace tiny real or imaginary parts in subresults by exact zeros (default=False)
strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec (default=False)
quad=<str>
Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad=’osc’.
verbose=<bool>
Print debug information (default=False)
expand(*args, **kwargs)

Expand an expression using hints.

See the docstring of the expand() function in sympy.core.function for more information.

extract_additively(c)

Return self - c if it’s possible to subtract c from self and make all matching coefficients move towards zero, else return None.

Examples

>>> from sympy.abc import x, y
>>> e = 2*x + 3
>>> e.extract_additively(x + 1)
x + 2
>>> e.extract_additively(3*x)
>>> e.extract_additively(4)
>>> (y*(x + 1)).extract_additively(x + 1)
>>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1)
(x + 1)*(x + 2*y) + 3

Sometimes auto-expansion will return a less simplified result than desired; gcd_terms might be used in such cases:

>>> from sympy import gcd_terms
>>> (4*x*(y + 1) + y).extract_additively(x)
4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y
>>> gcd_terms(_)
x*(4*y + 3) + y
extract_branch_factor(allow_half=False)

Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n).

>>> from sympy import exp_polar, I, pi
>>> from sympy.abc import x, y
>>> exp_polar(I*pi).extract_branch_factor()
(exp_polar(I*pi), 0)
>>> exp_polar(2*I*pi).extract_branch_factor()
(1, 1)
>>> exp_polar(-pi*I).extract_branch_factor()
(exp_polar(I*pi), -1)
>>> exp_polar(3*pi*I + x).extract_branch_factor()
(exp_polar(x + I*pi), 1)
>>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor()
(y*exp_polar(2*pi*x), -1)
>>> exp_polar(-I*pi/2).extract_branch_factor()
(exp_polar(-I*pi/2), 0)

If allow_half is True, also extract exp_polar(I*pi):

>>> exp_polar(I*pi).extract_branch_factor(allow_half=True)
(1, 1/2)
>>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True)
(1, 1)
>>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True)
(1, 3/2)
>>> exp_polar(-I*pi).extract_branch_factor(allow_half=True)
(1, -1/2)
extract_multiplicatively(c)

Return None if it’s not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.

>>> from sympy import symbols, Rational
>>> x, y = symbols('x,y', real=True)
>>> ((x*y)**3).extract_multiplicatively(x**2 * y)
x*y**2
>>> ((x*y)**3).extract_multiplicatively(x**4 * y)
>>> (2*x).extract_multiplicatively(2)
x
>>> (2*x).extract_multiplicatively(3)
>>> (Rational(1, 2)*x).extract_multiplicatively(3)
x/6
factor(*gens, **args)

See the factor() function in sympy.polys.polytools

find(query, group=False)

Find all subexpressions matching a query.

formula

Return a Formula with only terms=[self].

fourier_series(limits=None)

Compute fourier sine/cosine series of self.

See the docstring of the fourier_series() in sympy.series.fourier for more information.

fps(x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False)

Compute formal power power series of self.

See the docstring of the fps() function in sympy.series.formal for more information.

free_symbols
fromiter(args, **assumptions)

Create a new object from an iterable.

This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.

Examples

>>> from sympy import Tuple
>>> Tuple.fromiter(i for i in range(5))
(0, 1, 2, 3, 4)
func

The top-level function in an expression.

The following should hold for all objects:

>> x == x.func(*x.args)

Examples

>>> from sympy.abc import x
>>> a = 2*x
>>> a.func
<class 'sympy.core.mul.Mul'>
>>> a.args
(2, x)
>>> a.func(*a.args)
2*x
>>> a == a.func(*a.args)
True
getO()

Returns the additive O(..) symbol if there is one, else None.

getn()

Returns the order of the expression.

The order is determined either from the O(…) term. If there is no O(…) term, it returns None.

Examples

>>> from sympy import O
>>> from sympy.abc import x
>>> (1 + x + O(x**2)).getn()
2
>>> (1 + x).getn()
has(*args, **kwargs)

Test whether any subexpression matches any of the patterns.

Examples

>>> from sympy import sin
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True

Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval:

>>> from sympy.sets import Interval
>>> i = Interval.Lopen(0, 5); i
Interval.Lopen(0, 5)
>>> i.args
(0, 5, True, False)
>>> i.has(4)  # there is no "4" in the arguments
False
>>> i.has(0)  # there *is* a "0" in the arguments
True

Instead, use contains to determine whether a number is in the interval or not:

>>> i.contains(4)
True
>>> i.contains(0)
False

Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns). In particular, False is returned when the list of patterns is empty.

>>> x.has()
False
integrate(*args, **kwargs)

See the integrate function in sympy.integrals

invert(g, *gens, **args)

Return the multiplicative inverse of self mod g where self (and g) may be symbolic expressions).

See also

sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert

is_Add = False
is_AlgebraicNumber = False
is_Atom = True
is_Boolean = False
is_Derivative = False
is_Dummy = False
is_Equality = False
is_Float = False
is_Function = False
is_Indexed = False
is_Integer = False
is_Matrix = False
is_Mul = False
is_Not = False
is_Number = False
is_NumberSymbol = False
is_Order = False
is_Piecewise = False
is_Point = False
is_Poly = False
is_Pow = False
is_Rational = False
is_Relational = False
is_Symbol = True
is_Vector = False
is_Wild = False
is_algebraic
is_algebraic_expr(*syms)

This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form.

This function returns False for expressions that are “algebraic expressions” with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.

References

Examples

>>> from sympy import Symbol, sqrt
>>> x = Symbol('x', real=True)
>>> sqrt(1 + x).is_rational_function()
False
>>> sqrt(1 + x).is_algebraic_expr()
True

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one.

>>> from sympy import exp, factor
>>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1)
>>> a.is_algebraic_expr(x)
False
>>> factor(a).is_algebraic_expr()
True
is_antihermitian
is_commutative
is_comparable = False
is_complex
is_composite
is_constant(*wrt, **flags)
is_even
is_finite
is_hermitian
is_hypergeometric(k)
is_imaginary
is_infinite
is_integer
is_irrational
is_negative
is_noninteger
is_nonnegative
is_nonpositive
is_nonzero
is_number = False
is_odd
is_polar
is_polynomial(*syms)

Return True if self is a polynomial in syms and False otherwise.

This checks if self is an exact polynomial in syms. This function returns False for expressions that are “polynomials” with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used.

This is not part of the assumptions system. You cannot do Symbol(‘z’, polynomial=True).

Examples

>>> from sympy import Symbol
>>> x = Symbol('x')
>>> ((x**2 + 1)**4).is_polynomial(x)
True
>>> ((x**2 + 1)**4).is_polynomial()
True
>>> (2**x + 1).is_polynomial(x)
False
>>> n = Symbol('n', nonnegative=True, integer=True)
>>> (x**n + 1).is_polynomial(x)
False

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one.

>>> from sympy import sqrt, factor, cancel
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)
>>> a.is_polynomial(y)
False
>>> factor(a)
y + 1
>>> factor(a).is_polynomial(y)
True
>>> b = (y**2 + 2*y + 1)/(y + 1)
>>> b.is_polynomial(y)
False
>>> cancel(b)
y + 1
>>> cancel(b).is_polynomial(y)
True

See also .is_rational_function()

is_positive
is_prime
is_rational
is_rational_function(*syms)

Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form.

This function returns False for expressions that are “rational functions” with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True.

This is not part of the assumptions system. You cannot do Symbol(‘z’, rational_function=True).

Examples

>>> from sympy import Symbol, sin
>>> from sympy.abc import x, y
>>> (x/y).is_rational_function()
True
>>> (x**2).is_rational_function()
True
>>> (x/sin(y)).is_rational_function(y)
False
>>> n = Symbol('n', integer=True)
>>> (x**n + 1).is_rational_function(x)
False

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one.

>>> from sympy import sqrt, factor
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)/y
>>> a.is_rational_function(y)
False
>>> factor(a)
(y + 1)/y
>>> factor(a).is_rational_function(y)
True

See also is_algebraic_expr().

is_real
is_symbol = True
is_transcendental
is_zero
leadterm(x)

Returns the leading term a*x**b as a tuple (a, b).

Examples

>>> from sympy.abc import x
>>> (1+x+x**2).leadterm(x)
(1, 0)
>>> (1/x**2+x+x**2).leadterm(x)
(1, -2)
limit(x, xlim, dir='+')

Compute limit x->xlim.

lseries(x=None, x0=0, dir='+', logx=None)

Wrapper for series yielding an iterator of the terms of the series.

Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:

for term in sin(x).lseries(x):
    print term

The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you don’t know how many you should ask for in nseries() using the “n” parameter.

See also nseries().

match(pattern, old=False)

Pattern matching.

Wild symbols match all.

Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:

pattern.xreplace(self.match(pattern)) == self

Examples

>>> from sympy import Wild
>>> from sympy.abc import x, y
>>> p = Wild("p")
>>> q = Wild("q")
>>> r = Wild("r")
>>> e = (x+y)**(x+y)
>>> e.match(p**p)
{p_: x + y}
>>> e.match(p**q)
{p_: x + y, q_: x + y}
>>> e = (2*x)**2
>>> e.match(p*q**r)
{p_: 4, q_: x, r_: 2}
>>> (p*q**r).xreplace(e.match(p*q**r))
4*x**2

The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True:

>>> (x - 2).match(p - x, old=True)
{p_: 2*x - 2}
>>> (2/x).match(p*x, old=True)
{p_: 2/x**2}
matches(expr, repl_dict={}, old=False)
n(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)

Evaluate the given formula to an accuracy of n digits. Optional keyword arguments:

subs=<dict>
Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi}. The substitutions must be given as a dictionary.
maxn=<integer>
Allow a maximum temporary working precision of maxn digits (default=100)
chop=<bool>
Replace tiny real or imaginary parts in subresults by exact zeros (default=False)
strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec (default=False)
quad=<str>
Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad=’osc’.
verbose=<bool>
Print debug information (default=False)
name
normal()
nseries(x=None, x0=0, n=6, dir='+', logx=None)

Wrapper to _eval_nseries if assumptions allow, else to series.

If x is given, x0 is 0, dir=’+’, and self has x, then _eval_nseries is called. This calculates “n” terms in the innermost expressions and then builds up the final series just by “cross-multiplying” everything out.

The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided.

Advantage – it’s fast, because we don’t have to determine how many terms we need to calculate in advance.

Disadvantage – you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct.

If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms.

See also lseries().

Examples

>>> from sympy import sin, log, Symbol
>>> from sympy.abc import x, y
>>> sin(x).nseries(x, 0, 6)
x - x**3/6 + x**5/120 + O(x**6)
>>> log(x+1).nseries(x, 0, 5)
x - x**2/2 + x**3/3 - x**4/4 + O(x**5)

Handling of the logx parameter — in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0):

>>> e = sin(log(x))
>>> e.nseries(x, 0, 6)
Traceback (most recent call last):
...
PoleError: ...
...
>>> logx = Symbol('logx')
>>> e.nseries(x, 0, 6, logx=logx)
sin(logx)

In the following example, the expansion works but gives only an Order term unless the logx parameter is used:

>>> e = x**y
>>> e.nseries(x, 0, 2)
O(log(x)**2)
>>> e.nseries(x, 0, 2, logx=logx)
exp(logx*y)
nsimplify(constants=[], tolerance=None, full=False)

See the nsimplify function in sympy.simplify

powsimp(*args, **kwargs)

See the powsimp function in sympy.simplify

primitive()

Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).

Examples

>>> from sympy.abc import x
>>> (3*(x + 1)**2).primitive()
(3, (x + 1)**2)
>>> a = (6*x + 2); a.primitive()
(2, 3*x + 1)
>>> b = (x/2 + 3); b.primitive()
(1/2, x + 6)
>>> (a*b).primitive() == (1, a*b)
True
radsimp(**kwargs)

See the radsimp function in sympy.simplify

ratsimp()

See the ratsimp function in sympy.simplify

rcall(*args)

Apply on the argument recursively through the expression tree.

This method is used to simulate a common abuse of notation for operators. For instance in SymPy the the following will not work:

(x+Lambda(y, 2*y))(z) == x+2*z,

however you can use

>>> from sympy import Lambda
>>> from sympy.abc import x, y, z
>>> (x + Lambda(y, 2*y)).rcall(z)
x + 2*z
refine(assumption=True)

See the refine function in sympy.assumptions

removeO()

Removes the additive O(..) symbol if there is one

replace(query, value, map=False, simultaneous=True, exact=False)

Replace matching subexpressions of self with value.

If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself doesn’t match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())).

Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is True, then the match will only succeed if non-zero values are received for each Wild that appears in the match pattern.

The list of possible combinations of queries and replacement values is listed below:

See also

subs
substitution of subexpressions as defined by the objects themselves.
xreplace
exact node replacement in expr tree; also capable of using matching rules

Examples

Initial setup

>>> from sympy import log, sin, cos, tan, Wild, Mul, Add
>>> from sympy.abc import x, y
>>> f = log(sin(x)) + tan(sin(x**2))
1.1. type -> type

obj.replace(type, newtype)

When object of type type is found, replace it with the result of passing its argument(s) to newtype.

>>> f.replace(sin, cos)
log(cos(x)) + tan(cos(x**2))
>>> sin(x).replace(sin, cos, map=True)
(cos(x), {sin(x): cos(x)})
>>> (x*y).replace(Mul, Add)
x + y
1.2. type -> func

obj.replace(type, func)

When object of type type is found, apply func to its argument(s). func must be written to handle the number of arguments of type.

>>> f.replace(sin, lambda arg: sin(2*arg))
log(sin(2*x)) + tan(sin(2*x**2))
>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
sin(2*x*y)
2.1. pattern -> expr

obj.replace(pattern(wild), expr(wild))

Replace subexpressions matching pattern with the expression written in terms of the Wild symbols in pattern.

>>> a = Wild('a')
>>> f.replace(sin(a), tan(a))
log(tan(x)) + tan(tan(x**2))
>>> f.replace(sin(a), tan(a/2))
log(tan(x/2)) + tan(tan(x**2/2))
>>> f.replace(sin(a), a)
log(x) + tan(x**2)
>>> (x*y).replace(a*x, a)
y

When the default value of False is used with patterns that have more than one Wild symbol, non-intuitive results may be obtained:

>>> b = Wild('b')
>>> (2*x).replace(a*x + b, b - a)
2/x

For this reason, the exact option can be used to make the replacement only when the match gives non-zero values for all Wild symbols:

>>> (2*x + y).replace(a*x + b, b - a, exact=True)
y - 2
>>> (2*x).replace(a*x + b, b - a, exact=True)
2*x
2.2. pattern -> func

obj.replace(pattern(wild), lambda wild: expr(wild))

All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression:

>>> f.replace(sin(a), lambda a: sin(2*a))
log(sin(2*x)) + tan(sin(2*x**2))
3.1. func -> func

obj.replace(filter, func)

Replace subexpression e with func(e) if filter(e) is True.

>>> g = 2*sin(x**3)
>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
4*sin(x**9)

The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice.

>>> e = x*(x*y + 1)
>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
2*x*(2*x*y + 1)
rewrite(*args, **hints)

Rewrite functions in terms of other functions.

Rewrites expression containing applications of functions of one kind in terms of functions of different kind. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function.

As a pattern this function accepts a list of functions to to rewrite (instances of DefinedFunction class). As rule you can use string or a destination function instance (in this case rewrite() will use the str() function).

There is also the possibility to pass hints on how to rewrite the given expressions. For now there is only one such hint defined called ‘deep’. When ‘deep’ is set to False it will forbid functions to rewrite their contents.

Examples

>>> from sympy import sin, exp
>>> from sympy.abc import x

Unspecified pattern:

>>> sin(x).rewrite(exp)
-I*(exp(I*x) - exp(-I*x))/2

Pattern as a single function:

>>> sin(x).rewrite(sin, exp)
-I*(exp(I*x) - exp(-I*x))/2

Pattern as a list of functions:

>>> sin(x).rewrite([sin, ], exp)
-I*(exp(I*x) - exp(-I*x))/2
round(p=0)

Return x rounded to the given decimal place.

If a complex number would results, apply round to the real and imaginary components of the number.

Notes

Do not confuse the Python builtin function, round, with the SymPy method of the same name. The former always returns a float (or raises an error if applied to a complex value) while the latter returns either a Number or a complex number:

>>> isinstance(round(S(123), -2), Number)
False
>>> isinstance(S(123).round(-2), Number)
True
>>> isinstance((3*I).round(), Mul)
True
>>> isinstance((1 + 3*I).round(), Add)
True

Examples

>>> from sympy import pi, E, I, S, Add, Mul, Number
>>> S(10.5).round()
11.
>>> pi.round()
3.
>>> pi.round(2)
3.14
>>> (2*pi + E*I).round()
6. + 3.*I

The round method has a chopping effect:

>>> (2*pi + I/10).round()
6.
>>> (pi/10 + 2*I).round()
2.*I
>>> (pi/10 + E*I).round(2)
0.31 + 2.72*I
separate(deep=False, force=False)

See the separate function in sympy.simplify

series(x=None, x0=0, n=6, dir='+', logx=None)

Series expansion of “self” around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.

Returns the series expansion of “self” around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.

>>> from sympy import cos, exp
>>> from sympy.abc import x, y
>>> cos(x).series()
1 - x**2/2 + x**4/24 + O(x**6)
>>> cos(x).series(n=4)
1 - x**2/2 + O(x**4)
>>> cos(x).series(x, x0=1, n=2)
cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1))
>>> e = cos(x + exp(y))
>>> e.series(y, n=2)
cos(x + 1) - y*sin(x + 1) + O(y**2)
>>> e.series(x, n=2)
cos(exp(y)) - x*sin(exp(y)) + O(x**2)

If n=None then a generator of the series terms will be returned.

>>> term=cos(x).series(n=None)
>>> [next(term) for i in range(2)]
[1, -x**2/2]

For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results.

>>> abs(x).series(dir="+")
x
>>> abs(x).series(dir="-")
-x
simplify(ratio=1.7, measure=None)

See the simplify function in sympy.simplify

sort_key(*args, **kwargs)
subs(*args, **kwargs)

Substitutes old for new in an expression after sympifying args.

args is either:
  • two arguments, e.g. foo.subs(old, new)
  • one iterable argument, e.g. foo.subs(iterable). The iterable may be
    o an iterable container with (old, new) pairs. In this case the
    replacements are processed in the order given with successive patterns possibly affecting replacements already made.
    o a dict or set whose key/value items correspond to old/new pairs.
    In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous).

If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.

See also

replace
replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements
xreplace
exact node replacement in expr tree; also capable of using matching rules
evalf
calculates the given formula to a desired level of precision

Examples

>>> from sympy import pi, exp, limit, oo
>>> from sympy.abc import x, y
>>> (1 + x*y).subs(x, pi)
pi*y + 1
>>> (1 + x*y).subs({x:pi, y:2})
1 + 2*pi
>>> (1 + x*y).subs([(x, pi), (y, 2)])
1 + 2*pi
>>> reps = [(y, x**2), (x, 2)]
>>> (x + y).subs(reps)
6
>>> (x + y).subs(reversed(reps))
x**2 + 2
>>> (x**2 + x**4).subs(x**2, y)
y**2 + y

To replace only the x**2 but not the x**4, use xreplace:

>>> (x**2 + x**4).xreplace({x**2: y})
x**4 + y

To delay evaluation until all substitutions have been made, set the keyword simultaneous to True:

>>> (x/y).subs([(x, 0), (y, 0)])
0
>>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True)
nan

This has the added feature of not allowing subsequent substitutions to affect those already made:

>>> ((x + y)/y).subs({x + y: y, y: x + y})
1
>>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)
y/(x + y)

In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted.

>>> from sympy import sqrt, sin, cos
>>> from sympy.abc import a, b, c, d, e
>>> A = (sqrt(sin(2*x)), a)
>>> B = (sin(2*x), b)
>>> C = (cos(2*x), c)
>>> D = (x, d)
>>> E = (exp(x), e)
>>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)
>>> expr.subs(dict([A, B, C, D, E]))
a*c*sin(d*e) + b

The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression:

>>> (x**3 - 3*x).subs({x: oo})
nan
>>> limit(x**3 - 3*x, x, oo)
oo

If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as

>>> (1/x).evalf(subs={x: 3.0}, n=21)
0.333333333333333333333

rather than

>>> (1/x).subs({x: 3.0}).evalf(21)
0.333333333333333314830

as the former will ensure that the desired level of precision is obtained.

taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

together(*args, **kwargs)

See the together function in sympy.polys

transpose()
trigsimp(**args)

See the trigsimp function in sympy.simplify

xreplace(rule, hack2=False)

Functions

nipy.algorithms.statistics.formula.formulae.contrast_from_cols_or_rows(L, D, pseudo=None)

Construct a contrast matrix from a design matrix D

(possibly with its pseudo inverse already computed) and a matrix L that either specifies something in the column space of D or the row space of D.

Parameters:

L : ndarray

Matrix used to try and construct a contrast.

D : ndarray

Design matrix used to create the contrast.

pseudo : None or array-like, optional

If not None, gives pseudo-inverse of D. Allows you to pass this if it is already calculated.

Returns:

C : ndarray

Matrix with C.shape[1] == D.shape[1] representing an estimable contrast.

Notes

From an n x p design matrix D and a matrix L, tries to determine a p x q contrast matrix C which determines a contrast of full rank, i.e. the n x q matrix

dot(transpose(C), pinv(D))

is full rank.

L must satisfy either L.shape[0] == n or L.shape[1] == p.

If L.shape[0] == n, then L is thought of as representing columns in the column space of D.

If L.shape[1] == p, then L is thought of as what is known as a contrast matrix. In this case, this function returns an estimable contrast corresponding to the dot(D, L.T)

This always produces a meaningful contrast, not always with the intended properties because q is always non-zero unless L is identically 0. That is, it produces a contrast that spans the column space of L (after projection onto the column space of D).

nipy.algorithms.statistics.formula.formulae.define(*args, **kwargs)
nipy.algorithms.statistics.formula.formulae.getparams(expression)

Return the parameters of an expression that are not Term instances but are instances of sympy.Symbol.

Examples

>>> x, y, z = [Term(l) for l in 'xyz']
>>> f = Formula([x,y,z])
>>> getparams(f)
[]
>>> f.mean
_b0*x + _b1*y + _b2*z
>>> getparams(f.mean)
[_b0, _b1, _b2]
>>> th = sympy.Symbol('theta')
>>> f.mean*sympy.exp(th)
(_b0*x + _b1*y + _b2*z)*exp(theta)
>>> getparams(f.mean*sympy.exp(th))
[_b0, _b1, _b2, theta]
nipy.algorithms.statistics.formula.formulae.getterms(expression)

Return the all instances of Term in an expression.

Examples

>>> x, y, z = [Term(l) for l in 'xyz']
>>> f = Formula([x,y,z])
>>> getterms(f)
[x, y, z]
>>> getterms(f.mean)
[x, y, z]
nipy.algorithms.statistics.formula.formulae.is_factor(obj)

Is obj a Factor?

nipy.algorithms.statistics.formula.formulae.is_factor_term(obj)

Is obj a FactorTerm?

nipy.algorithms.statistics.formula.formulae.is_formula(obj)

Is obj a Formula?

nipy.algorithms.statistics.formula.formulae.is_term(obj)

Is obj a Term?

nipy.algorithms.statistics.formula.formulae.make_dummy(*args, **kwds)

make_dummy is deprecated! Please use sympy.Dummy instead of this function

Make dummy variable of given name
Parameters:

name : str

name of dummy variable

Returns:

dum : Dummy instance

Notes

The interface to Dummy changed between 0.6.7 and 0.7.0, and we used this function to keep compatibility. Now we depend on sympy 0.7.0 and this function is obsolete.

nipy.algorithms.statistics.formula.formulae.make_recarray(rows, names, dtypes=None, drop_name_dim=<class nipy.utils._NoValue>)

Create recarray from rows with field names names

Create a recarray with named columns from a list or ndarray of rows and sequence of names for the columns. If rows is an ndarray, dtypes must be None, otherwise we raise a ValueError. Otherwise, if dtypes is None, we cast the data in all columns in rows as np.float. If dtypes is not None, the routine uses dtypes as a dtype specifier for the output structured array.

Parameters:

rows: list or array :

Rows that will be turned into an recarray.

names: sequence :

Sequence of strings - names for the columns.

dtypes: None or sequence of str or sequence of np.dtype, optional :

Used to create a np.dtype, can be sequence of np.dtype or string.

drop_name_dim : {_NoValue, False, True}, optional

Flag for compatibility with future default behavior. Current default is False. If True, drops the length 1 dimension corresponding to the axis transformed into fields when converting into a recarray. If _NoValue specified, gives default. Default will change to True in the next version of Nipy.

Returns:

v : np.ndarray

Structured array with field names given by names.

Raises:

ValueError :

dtypes not None when rows is array.

Examples

The following tests depend on machine byte order for their exact output.

>>> arr = np.array([[3, 4], [4, 6], [6, 8]])
>>> make_recarray(arr, ['x', 'y'],
...               drop_name_dim=True) 
array([(3, 4), (4, 6), (6, 8)],
      dtype=[('x', '<i8'), ('y', '<i8')])
>>> make_recarray(arr, ['x', 'y'],
...               drop_name_dim=False) 
array([[(3, 4)],
       [(4, 6)],
       [(6, 8)]],
      dtype=[('x', '<i8'), ('y', '<i8')])
>>> r = make_recarray(arr, ['w', 'u'], drop_name_dim=True)
>>> make_recarray(r, ['x', 'y'],
...               drop_name_dim=True) 
array([(3, 4), (4, 6), (6, 8)],
      dtype=[('x', '<i8'), ('y', '<i8')])
>>> make_recarray([[3, 4], [4, 6], [7, 9]], 'wv',
...               [np.float, np.int])  
array([(3.0, 4), (4.0, 6), (7.0, 9)],
      dtype=[('w', '<f8'), ('v', '<i8')])
nipy.algorithms.statistics.formula.formulae.natural_spline(t, knots=None, order=3, intercept=False)

Return a Formula containing a natural spline

Spline for a Term with specified knots and order.

Parameters:

t : Term

knots : None or sequence, optional

Sequence of float. Default None (same as empty list)

order : int, optional

Order of the spline. Defaults to a cubic (==3)

intercept : bool, optional

If True, include a constant function in the natural spline. Default is False

Returns:

formula : Formula

A Formula with (len(knots) + order) Terms (if intercept=False, otherwise includes one more Term), made up of the natural spline functions.

Examples

>>> x = Term('x')
>>> n = natural_spline(x, knots=[1,3,4], order=3)
>>> xval = np.array([3,5,7.]).view(np.dtype([('x', np.float)]))
>>> n.design(xval, return_float=True)
array([[   3.,    9.,   27.,    8.,    0.,   -0.],
       [   5.,   25.,  125.,   64.,    8.,    1.],
       [   7.,   49.,  343.,  216.,   64.,   27.]])
>>> d = n.design(xval)
>>> print(d.dtype.descr)
[('ns_1(x)', '<f8'), ('ns_2(x)', '<f8'), ('ns_3(x)', '<f8'), ('ns_4(x)', '<f8'), ('ns_5(x)', '<f8'), ('ns_6(x)', '<f8')]
nipy.algorithms.statistics.formula.formulae.terms(names, **kwargs)

Return list of terms with names given by names

This is just a convenience in defining a set of terms, and is the equivalent of sympy.symbols for defining symbols in sympy.

We enforce the sympy 0.7.0 behavior of returning symbol “abc” from input “abc”, rthan than 3 symbols “a”, “b”, “c”.

Parameters:

names : str or sequence of str

If a single str, can specify multiple ``Term``s with string containing space or ‘,’ as separator.

**kwargs : keyword arguments

keyword arguments as for sympy.symbols

Returns:

ts : Term or tuple

Term instance or list of Term instance objects named from names

Examples

>>> terms(('a', 'b', 'c'))
(a, b, c)
>>> terms('a, b, c')
(a, b, c)
>>> terms('abc')
abc