Core

Core module. Provides the basic operations needed in diofant.

sympify

diofant.core.sympify.sympify(a, locals=None, convert_xor=True, strict=False, rational=False, evaluate=None)[source]

Converts an arbitrary expression to a type that can be used inside Diofant.

For example, it will convert Python ints into instance of diofant.Rational, floats into instances of diofant.Float, etc. It is also able to coerce symbolic expressions which inherit from Basic. This can be useful in cooperation with SAGE.

It currently accepts as arguments:

any object defined in diofant
standard numeric python types: int, long, float, Decimal
strings (like “0.09” or “2e-19”)
booleans, including None (will leave None unchanged)
lists, sets or tuples containing any of the above

If the argument is already a type that Diofant understands, it will do nothing but return that value. This can be used at the beginning of a function to ensure you are working with the correct type.

>>> sympify(2).is_integer
True
>>> sympify(2).is_real
True

>>> sympify(2.0).is_real
True
>>> sympify('2.0').is_real
True
>>> sympify('2e-45').is_real
True

If the expression could not be converted, a SympifyError is raised.

>>> sympify('x***2')
Traceback (most recent call last):
...
SympifyError: SympifyError: "could not parse u'x***2'"

Locals

The sympification happens with access to everything that is loaded by from diofant import *; anything used in a string that is not defined by that import will be converted to a symbol. In the following, the bitcount function is treated as a symbol and the O is interpreted as the Order object (used with series) and it raises an error when used improperly:

>>> s = 'bitcount(42)'
>>> sympify(s)
bitcount(42)
>>> sympify('O(x)')
O(x)
>>> sympify('O + 1')
Traceback (most recent call last):
...
TypeError: unbound method...

In order to have bitcount be recognized it can be imported into a namespace dictionary and passed as locals:

>>> ns = {}
>>> exec('from diofant.core.evalf import bitcount', ns)
>>> sympify(s, locals=ns)
6

In order to have the O interpreted as a Symbol, identify it as such in the namespace dictionary. This can be done in a variety of ways; all three of the following are possibilities:

>>> ns['O'] = Symbol('O')  # method 1
>>> exec('from diofant.abc import O', ns)  # method 2
>>> ns.update({O: Symbol('O')})  # method 3
>>> sympify('O + 1', locals=ns)
O + 1

If you want all single-letter and Greek-letter variables to be symbols then you can use the clashing-symbols dictionaries that have been defined there as private variables: _clash1 (single-letter variables), _clash2 (the multi-letter Greek names) or _clash (both single and multi-letter names that are defined in abc).

>>> from diofant.abc import _clash1
>>> _clash1
{'E': E, 'I': I, 'N': N, 'O': O, 'S': S}
>>> sympify('E & O', _clash1)
E & O

Strict

If the option strict is set to True, only the types for which an explicit conversion has been defined are converted. In the other cases, a SympifyError is raised.

>>> print(sympify(None))
None
>>> sympify(None, strict=True)
Traceback (most recent call last):
...
SympifyError: SympifyError: None

Evaluation

If the option evaluate is set to False, then arithmetic and operators will be converted into their Diofant equivalents and the evaluate=False option will be added. Nested Add or Mul will be denested first. This is done via an AST transformation that replaces operators with their Diofant equivalents, so if an operand redefines any of those operations, the redefined operators will not be used.

>>> sympify('2**2 / 3 + 5')
19/3
>>> sympify('2**2 / 3 + 5', evaluate=False)
2**2/3 + 5

Sometimes autosimplification during sympification results in expressions that are very different in structure than what was entered. Below you can see how an expression reduces to -1 by autosimplification, but does not do so when evaluate option is used.

>>> -2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1
-1
>>> s = '-2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1'
>>> sympify(s)
-1
>>> sympify(s, evaluate=False)
-2*((x - 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1

Extending

To extend sympify to convert custom objects (not derived from Basic), just define a _diofant_ method to your class. You can do that even to classes that you do not own by subclassing or adding the method at runtime.

>>> class MyList1:
...     def __iter__(self):
...         yield 1
...         yield 2
...         return
...
...     def __getitem__(self, i):
...         return list(self)[i]
...
...     def _diofant_(self):
...         return Matrix(self)
>>> sympify(MyList1())
Matrix([
[1],
[2]])

If you do not have control over the class definition you could also use the converter global dictionary. The key is the class and the value is a function that takes a single argument and returns the desired Diofant object, e.g. converter[MyList] = lambda x: Matrix(x).

>>> class MyList2:   # XXX Do not do this if you control the class!
...     def __iter__(self):  # Use _diofant_!
...         yield 1
...         yield 2
...         return
...
...     def __getitem__(self, i):
...         return list(self)[i]
>>> converter[MyList2] = lambda x: Matrix(x)
>>> sympify(MyList2())
Matrix([
[1],
[2]])

assumptions

This module contains the machinery handling assumptions.

All symbolic objects have assumption attributes that can be accessed via .is_<assumption name> attribute, i.e. is_integer. Full set of defined assumption names are accessible as Expr.default_assumptions.rules.defined_facts attribute.

Assumptions determine certain properties of symbolic objects and can have 3 possible values: True, False, None. True is returned if the object has the property and False is returned if it doesn’t or can’t (i.e. doesn’t make sense):

>>> I.is_algebraic
True
>>> I.is_real
False
>>> I.is_prime
False

When the property cannot be determined (or when a method is not implemented) None will be returned, e.g. a generic symbol, x, may or may not be positive so a value of None is returned for x.is_positive.

By default, all symbolic values are in the largest set in the given context without specifying the property. For example, a symbol that has a property being integer, is also real, complex, etc.

Notes

Assumption values are stored in obj._assumptions dictionary or are returned by getter methods (with property decorators) or are attributes of objects/classes.

class diofant.core.assumptions.ManagedProperties(*args, **kws)[source]: Metaclass for classes with old-style assumptions.

class diofant.core.assumptions.StdFactKB(facts=None)[source]

A FactKB specialised for the built-in rules

This is the only kind of FactKB that Basic objects should use.

diofant.core.assumptions.as_property(fact)[source]: Convert a fact name to the name of the corresponding property.

diofant.core.assumptions.check_assumptions(expr, **assumptions)[source]

Checks if expression \(expr\) satisfies all assumptions.

Parameters:

expr (Expr) – Expression to test.
**assumptions (dict) – Keyword arguments to specify assumptions to test.

Returns:

True, False or None (if can’t conclude).

Examples

>>> check_assumptions(-5, integer=True)
True
>>> x = Symbol('x', positive=True)
>>> check_assumptions(2*x + 1, negative=True)
False
>>> check_assumptions(z, real=True) is None
True

diofant.core.assumptions.make_property(fact)[source]: Create the automagic property corresponding to a fact.

cache

cacheit

diofant.core.cache.cacheit(f, maxsize=None)[source]

Caching decorator.

The result of cached function must be immutable.

Examples

>>> @cacheit
... def f(a, b):
...     print(a, b)
...     return a + b
>>> f(x, y)
x y
x + y
>>> f(x, y)
x + y

basic

Base class for all the objects in Diofant.

class diofant.core.basic.Atom(*args)[source]

A parent class for atomic things.

An atom is an expression with no subexpressions, for example Symbol, Number, Rational or Integer, but not Add, Mul, Pow.

classmethod class_key()[source]: Nice order of classes.

doit(**hints)[source]: Evaluate objects that are not evaluated by default.

See also

Basic.doit

sort_key(order=None)[source]: Return a sort key.

class diofant.core.basic.Basic(*args)[source]

Base class for all objects in Diofant.

Always use args property, when accessing parameters of some instance.

property args

Returns a tuple of arguments of ‘self’.

Examples

>>> cot(x).args
(x,)
>>> (x*y).args
(x, y)

atoms(*types)[source]

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

>>> e = 1 + x + 2*sin(y + I*pi)
>>> e.atoms()
{1, 2, I, pi, x, y}

If one or more types are given, the results will contain only those types of atoms.

>>> e.atoms(Symbol)
{x, y}

>>> e.atoms(Number)
{1, 2}

>>> e.atoms(Number, NumberSymbol)
{1, 2, pi}

>>> e.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:

>>> e.atoms(x)
{x, y}

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

>>> e.atoms(Integer(1))
{1}

>>> e.atoms(Integer(2))
{1, 2}

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

>>> from diofant.core.function import AppliedUndef

>>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
{I*pi, 2*sin(y + I*pi)}

>>> f = Function('f')
>>> e = 1 + f(x) + 2*sin(y + I*pi)
>>> e.atoms(Function)
{f(x), sin(y + I*pi)}
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)
{f(x)}

classmethod class_key()[source]: Nice order of classes.

copy()[source]: Return swallow copy of self.

count(query)[source]: Count the number of matching subexpressions.

count_ops(visual=None)[source]: Wrapper for count_ops that returns the operation count.

doit(**hints)[source]

Evaluate objects that are not evaluated by default.

For example, limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

Examples

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)

find(query)[source]: Find all subexpressions matching a query.

property free_symbols

Return from the atoms of self those which are free symbols.

For most expressions, all symbols are free symbols. For some classes this is not true. e.g. Integrals use Symbols for the dummy variables which are bound variables, so Integral has a method to return all symbols except those. Derivative keeps track of symbols with respect to which it will perform a derivative; those are bound variables, too, so it has its own free_symbols method.

Any other method that uses bound variables should implement a free_symbols method.

property func

The top-level function in an expression.

The following should hold for all objects:

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

Examples

>>> a = 2*x
>>> a.func
<class 'diofant.core.mul.Mul'>
>>> a.args
(2, x)
>>> a.func(*a.args)
2*x
>>> a == a.func(*a.args)
True

has(*patterns)[source]

Test if any subexpression matches any of the patterns.

Parameters:: *patterns (tuple of Expr) – List of expressions to search for match.
Returns:: bool – False if there is no match or patterns list is empty, else True.

Examples

>>> e = x**2 + sin(x*y)
>>> e.has(z)
False
>>> e.has(x, y, z)
True
>>> x.has()
False

property is_evaluated: Test if an expession is evaluated.

match(pattern)[source]

Pattern matching.

Wild symbols match all.

Parameters:

pattern (Expr) – An expression that may contain Wild symbols.

Returns:

dict or None – If pattern match self, return a dictionary of replacement rules, such that:

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

Examples

>>> p = Wild('p')
>>> q = Wild('q')
>>> e = (x + y)**(x + y)
>>> e.match(p**p)
{p_: x + y}
>>> e.match(p**q)
{p_: x + y, q_: x + y}
>>> (p**q).xreplace(_)
(x + y)**(x + y)

See also

xreplace, diofant.core.symbol.Wild

rcall(*args)[source]

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 Diofant the the following will not work:

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

however you can use

>>> (x + Lambda(y, 2*y)).rcall(z)
x + 2*z

replace(query, value, exact=False)[source]

Replace matching subexpressions of self with value.

Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree in a simultaneous fashion so changes made are targeted only once. 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:

Examples

Initial setup

>>> 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))
>>> (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)

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

rewrite(*args, **hints)[source]

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

Unspecified pattern:

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

Pattern as a single function:

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

Pattern as a list of functions:

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

sort_key(order=None)[source]

Return a sort key.

Examples

>>> sorted([Rational(1, 2), I, -I], key=lambda x: x.sort_key())
[1/2, -I, I]

>>> [x, 1/x, 1/x**2, x**2, sqrt(x), root(x, 4), x**Rational(3, 2)]
[x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)]
>>> sorted(_, key=lambda x: x.sort_key())
[x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2]

subs(*args, **kwargs)[source]

Substitutes old for new in an expression after sympifying args.

\(args\) is either:

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.

Examples

>>> (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 diofant.abc import e

>>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)

>>> expr.subs({sqrt(sin(2*x)): a, sin(2*x): b,
...            cos(2*x): c, x: d, exp(x): 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(21, subs={x: 3.0}, strict=False)
0.333333333333333333333

rather than

>>> (1/x).subs({x: 3.0}).evalf(21, strict=False)
0.333333333333333

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

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
diofant.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

xreplace(rule)[source]

Replace occurrences of objects within the expression.

Parameters:: rule (dict-like) – Expresses a replacement rule
Returns:: xreplace (the result of the replacement)

Examples

>>> (1 + x*y).xreplace({x: pi})
pi*y + 1
>>> (1 + x*y).xreplace({x: pi, y: 2})
1 + 2*pi

Replacements occur only if an entire node in the expression tree is matched:

>>> (x*y + z).xreplace({x*y: pi})
z + pi
>>> (x*y*z).xreplace({x*y: pi})
x*y*z
>>> (2*x).xreplace({2*x: y, x: z})
y
>>> (2*2*x).xreplace({2*x: y, x: z})
4*z
>>> (x + y + 2).xreplace({x + y: 2})
x + y + 2
>>> (x + 2 + exp(x + 2)).xreplace({x + 2: y})
E**y + x + 2

xreplace doesn’t differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does:

>>> Integral(x, (x, 1, 2*x)).xreplace({x: y})
Integral(y, (y, 1, 2*y))

Trying to replace x with an expression raises an error:

>>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y})
Traceback (most recent call last):
...
ValueError: Invalid limits given: ((2*y, 1, 4*y),)

See also

replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements
subs: substitution of subexpressions as defined by the objects themselves.

class diofant.core.basic.preorder_traversal(node, keys=None)[source]

Do a pre-order traversal of a tree.

This iterator recursively yields nodes that it has visited in a pre-order fashion. That is, it yields the current node then descends through the tree breadth-first to yield all of a node’s children’s pre-order traversal.

For an expression, the order of the traversal depends on the order of .args, which in many cases can be arbitrary.

Parameters:

node (diofant expression) – The expression to traverse.
keys ((default None) sort key(s)) – The key(s) used to sort args of Basic objects. When None, args of Basic objects are processed in arbitrary order. If key is defined, it will be passed along to ordered() as the only key(s) to use to sort the arguments; if key is simply True then the default keys of ordered will be used.

Yields:

subtree (diofant expression) – All of the subtrees in the tree.

Examples

The nodes are returned in the order that they are encountered unless key is given; simply passing key=True will guarantee that the traversal is unique.

>>> list(preorder_traversal((x + y)*z, keys=True))
[z*(x + y), z, x + y, x, y]

skip()[source]

Skip yielding current node’s (last yielded node’s) subtrees.

Examples

>>> pt = preorder_traversal((x+y*z)*z)
>>> for i in pt:
...     print(i)
...     if i == x + y*z:
...         pt.skip()
z*(x + y*z)
z
x + y*z

core

singleton

Singleton mechanism

diofant.core.singleton.S: SingletonRegistry = S: Alias for instance of SingletonRegistry.

class diofant.core.singleton.Singleton(*args, **kwargs)[source]

Metaclass for singleton classes.

A singleton class has only one instance which is returned every time the class is instantiated. Additionally, this instance can be accessed through the global registry object S as S.<class_name>.

Examples

>>> class MySingleton(Basic, metaclass=Singleton):
...     pass
>>> Basic() is Basic()
False
>>> MySingleton() is MySingleton()
True
>>> S.MySingleton is MySingleton()
True

Notes

Instance creation is delayed until the first time the value is accessed.

This metaclass is a subclass of ManagedProperties because that is the metaclass of many classes that need to be Singletons (Python does not allow subclasses to have a different metaclass than the superclass, except the subclass may use a subclassed metaclass).

class diofant.core.singleton.SingletonRegistry[source]

The registry for the singleton classes.

Several classes in Diofant appear so often that they are singletonized, that is, using some metaprogramming they are made so that they can only be instantiated once (see the diofant.core.singleton.Singleton class for details). For instance, every time you create Integer(0), this will return the same instance, diofant.core.numbers.Zero.

>>> a = Integer(0)
>>> a is S.Zero
True

All singleton instances are attributes of the S object, so Integer(0) can also be accessed as S.Zero.

Notes

For the most part, the fact that certain objects are singletonized is an implementation detail that users shouldn’t need to worry about. In Diofant library code, is comparison is often used for performance purposes. The primary advantage of S for end users is the convenient access to certain instances that are otherwise difficult to type, like S.Half (instead of Rational(1, 2)).

When using is comparison, make sure the argument is a Basic instance. For example,

>>> int(0) is S.Zero
False

class diofant.core.singleton.SingletonWithManagedProperties(*args, **kwargs)[source]: Metaclass for singleton classes with managed properties.

evaluate

diofant.core.evaluate.evaluate(x)[source]

Control automatic evaluation.

This context managers controls whether or not all Diofant functions evaluate by default.

Note that much of Diofant expects evaluated expressions. This functionality is experimental and is unlikely to function as intended on large expressions.

Examples

>>> x + x
2*x
>>> with evaluate(False):
...     x + x
x + x

expr

Expr

class diofant.core.expr.Expr(*args)[source]

Base class for algebraic expressions.

Everything that requires arithmetic operations to be defined should subclass this class, instead of Basic (which should be used only for argument storage and expression manipulation, i.e. pattern matching, substitutions, etc).

See also

diofant.core.basic.Basic

adjoint()[source]: Compute conjugate transpose or Hermite conjugation.

See also

diofant.functions.elementary.complexes.adjoint

apart(x=None, **args)[source]: See the apart function in diofant.polys.

args_cnc(cset=False, warn=True, split_1=True)[source]

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 suppressed by setting warn to False.

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

>>> A, B = symbols('A B', commutative=0)
>>> (-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()[source]: Return base and exp of self.

See also

diofant.core.power.Pow.as_base_exp

as_coeff_Add(rational=False)[source]: Efficiently extract the coefficient of a summation.

as_coeff_Mul(rational=False)[source]: Efficiently extract the coefficient of a product.

as_coeff_add(*deps)[source]

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)

>>> (Integer(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)[source]: c*x**e -> c,e where x can be any symbolic expression.

as_coeff_mul(*deps, **kwargs)[source]

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

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

args should be a tuple of all other terms 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)

>>> (Integer(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)[source]

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.

Examples

>>> 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 = (2*E + x*E).as_poly()
>>> p
Poly(x*E + 2*E, x, E, domain='ZZ')
>>> p.coeff_monomial(E)
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)

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
diofant.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly

as_coefficients_dict()[source]

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

>>> (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)[source]

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

>>> 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)))

as_expr(*gens)[source]

Convert a polynomial to a Diofant expression.

Examples

>>> 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)[source]

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.

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
if self is anything else, either tuple (self, Integer(1)) or (Integer(1), self) is returned.

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

Examples

– self is an Add

>>> (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

>>> 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, E**(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

>>> 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

>>> separatevars(exp(x+y)).as_independent(x)
(E**y, E**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(x)[source]

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

>>> (1 + x + x**2).as_leading_term(x)
1
>>> (1/x**2 + x + x**2).as_leading_term(x)
x**(-2)

as_numer_denom()[source]

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

as_ordered_terms(order=None, data=False)[source]

Transform an expression to an ordered list of terms.

Examples

>>> (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)[source]

Converts self to a polynomial or returns None.

Examples

>>> (x**2 + x*y).as_poly()
Poly(x**2 + x*y, x, y, domain='ZZ')

>>> (x**2 + x*y).as_poly(x, y)
Poly(x**2 + x*y, x, y, domain='ZZ')

>>> (x**2 + sin(y)).as_poly(x, y) is None
True

as_powers_dict()[source]: 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)[source]

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method can’t be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> x, y = symbols('x y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> (z + t*I).as_real_imag()
(re(z) - im(t), re(t) + im(z))

as_terms()[source]: Transform an expression to a list of terms.

aseries(x, n=6, bound=0, hir=False)[source]

Returns asymptotic expansion for “self”.

This is equivalent to self.series(x, oo, n)

Use the hir parameter to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results.

If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalized representation of the mrv set and rewrites f using this normalized representation. Use the bound parameter to give limit on rewriting coefficients in its normalized form.

If the expansion contains an order term, it will be either O(x**(-n)) or O(w**(-n)) where w belongs to the most rapidly varying expression of self.

Examples

>>> e = sin(1/x + exp(-x)) - sin(1/x)
>>> e.series(x, oo)
E**(-x)*(1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), x, oo))
>>> e.aseries(x, n=3, hir=True)
-E**(-2*x)*sin(1/x)/2 + E**(-x)*cos(1/x) + O(E**(-3*x), x, oo)

>>> e = exp(exp(x)/(1 - 1/x))
>>> e.aseries(x, bound=3)
E**(E**x)*E**(E**x/x**2)*E**(E**x/x)*E**(-E**x + E**x/(1 - 1/x) - E**x/x - E**x/x**2)
>>> e.series(x, oo)
E**(E**x/(1 - 1/x))

Notes

This algorithm is directly induced from the limit computational algorithm provided by Gruntz [Gru96], p.90. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients.

References

https://en.wikipedia.org/wiki/Asymptotic_expansion

cancel(*gens, **args)[source]: See the cancel function in diofant.polys.

property 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

>>> Lambda(x, 2*x).canonical_variables
{x: 0_}

coeff(x, n=1, right=False)[source]

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.

Examples

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
>>> (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:

>>> 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)[source]: See the collect function in diofant.simplify.

combsimp()[source]: See the combsimp function in diofant.simplify.

compute_leading_term(x, logx=None)[source]: as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.

conjugate()[source]: Returns the complex conjugate of self.

See also

diofant.functions.elementary.complexes.conjugate

could_extract_minus_sign()[source]

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}.

>>> (x-y).could_extract_minus_sign() != (y-x).could_extract_minus_sign()
True

count_ops(visual=None)[source]: Wrapper for count_ops that returns the operation count.

diff(*args, **kwargs)[source]: Alias for diff().

equals(other, failing_expression=False)[source]

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.

expand(deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints)[source]: Expand an expression using hints.

See also

diofant.core.function.expand

extract_additively(c)[source]

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

Examples

>>> 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:

>>> (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)[source]

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

>>> 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)[source]

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.

>>> 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)[source]: See the factor() function in diofant.polys.polytools.

getO()[source]: Returns the additive O(..) symbol if there is one, else None.

getn()[source]

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

>>> (1 + x + O(x**2)).getn()
2
>>> (1 + x).getn()

integrate(*args, **kwargs)[source]: See the integrate function in diofant.integrals.

invert(other, *gens, **args)[source]: Return the multiplicative inverse of self mod other where self (and other) may be symbolic expressions).

See also

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

property is_algebraic

Test if self can have only values from the set of algebraic numbers.

References

https://en.wikipedia.org/wiki/Algebraic_number

is_algebraic_expr(*syms)[source]

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.

Examples

>>> 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.

>>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1)
>>> a.is_algebraic_expr(x)
False
>>> factor(a).is_algebraic_expr()
True

See also

is_rational_function

References

https://en.wikipedia.org/wiki/Algebraic_expression

property is_commutative: Test if self commutes with any other object wrt multiplication operation.

property is_comparable

Test if self can be computed to a real number with precision.

Examples

>>> (I*exp_polar(I*pi/2)).is_comparable
True
>>> (I*exp_polar(I*pi*2)).is_comparable
False

property is_complex: Test if self can have only values from the set of complex numbers.

See also

is_real

property is_composite

Test if self is a positive integer that has at least one positive divisor other than 1 or the number itself.

References

https://en.wikipedia.org/wiki/Composite_number

is_constant(*wrt, **flags)[source]

Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.

If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, two strategies are tried:

1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols.

2) differentiation with respect to variables in ‘wrt’ (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols.

If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True – in that case the numerical value will be returned.

If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.

Examples

>>> x.is_constant()
False
>>> Integer(2).is_constant()
True
>>> Sum(x, (x, 1, 10)).is_constant()
True
>>> Sum(x, (x, 1, n)).is_constant()
False
>>> Sum(x, (x, 1, n)).is_constant(y)
True
>>> Sum(x, (x, 1, n)).is_constant(n)
False
>>> Sum(x, (x, 1, n)).is_constant(x)
True
>>> eq = a*cos(x)**2 + a*sin(x)**2 - a
>>> eq.is_constant()
True
>>> eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
True

>>> (0**x).is_constant()
False
>>> x.is_constant()
False
>>> (x**x).is_constant()
False
>>> one = cos(x)**2 + sin(x)**2
>>> one.is_constant()
True
>>> ((one - 1)**(x + 1)).is_constant() in (True, False)  # could be 0 or 1
True

property is_even

Test if self can have only values from the set of even integers.

See also

is_odd

References

https://en.wikipedia.org/wiki/Parity_%28mathematics%29

property is_extended_real

Test if self can have only values on the extended real number line.

See also

is_real

References

https://en.wikipedia.org/wiki/Extended_real_number_line

property is_finite

Test if self absolute value is bounded.

References

https://en.wikipedia.org/wiki/Finite

is_hypergeometric(n)[source]

Test if self is a hypergeometric term in n.

Term \(a(n)\) is hypergeometric if it is annihilated by first order linear difference equations with polynomial coefficients or, in simpler words, if consecutive term ratio is a rational function.

See also

diofant.simplify.simplify.hypersimp

property is_imaginary

Test if self is an imaginary number.

I.e. that it can be written as a real number multiplied by the imaginary unit I.

References

https://en.wikipedia.org/wiki/Imaginary_number

property is_infinite

Test if self absolute value can be arbitrarily large.

References

property is_integer: Test if self can have only values from the set of integers.

property is_irrational

Test if self value cannot be represented exactly by Rational.

References

https://en.wikipedia.org/wiki/Irrational_number

property is_negative

Test if self can have only negative values.

References

https://en.wikipedia.org/wiki/Negative_number

property is_noninteger: Test if self can have only values from the subset of real numbers, that aren’t integers.

property is_nonnegative

Test if self can have only nonnegative values.

See also

is_negative

References

https://en.wikipedia.org/wiki/Negative_number

property is_nonpositive: Test if self can have only nonpositive values.

property is_nonzero: Test if self is nonzero.

See also

is_zero

property is_number

Returns True if ‘self’ has no free symbols.

It will be faster than if not self.free_symbols, however, since is_number will fail as soon as it hits a free symbol.

Examples

>>> x.is_number
False
>>> (2*x).is_number
False
>>> (2 + log(2)).is_number
True
>>> (2 + Integral(2, x)).is_number
False
>>> (2 + Integral(2, (x, 1, 2))).is_number
True

property is_odd

Test if self can have only values from the set of odd integers.

See also

is_even

References

https://en.wikipedia.org/wiki/Parity_%28mathematics%29

property is_polar: Test if self can have values from the Riemann surface of the logarithm.

See also

diofant.functions.elementary.complexes.polar_lift, diofant.functions.elementary.complexes.principal_branch, diofant.functions.elementary.exponential.exp_polar

is_polynomial(*syms)[source]

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

>>> ((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.

>>> 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

property is_positive: Test if self can have only positive values.

property is_prime

Test if self is a natural number greater than 1 that has no positive divisors other than 1 and itself.

References

https://en.wikipedia.org/wiki/Prime_number

property is_rational: Test if self can have only values from the set of rationals.

is_rational_function(*syms)[source]

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

>>> (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.

>>> 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

property is_real

Test if self can have only values from the set of real numbers.

See also

is_complex

References

https://en.wikipedia.org/wiki/Real_number

property is_transcendental

Test if self can have only values from the set of transcendental numbers.

References

https://en.wikipedia.org/wiki/Transcendental_number

property is_zero: Test if self is zero.

See also

is_nonzero

limit(x, xlim, dir=-1)[source]: Compute limit x->xlim.

normal()[source]: Canonicalize ratio, i.e. return numerator if denominator is 1.

nseries(x, n=6, logx=None)[source]

Calculate “n” terms of series in x around 0

This calculates n terms of series in the innermost expressions and then builds up the final series just by “cross-multiplying” everything out.

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.

Parameters:

x (Symbol) – variable for series expansion (positive and finite symbol)
n (Integer, optional) – number of terms to calculate. Default is 6.
logx (Symbol, optional) – This can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0.

See also

series

Examples

>>> sin(x).nseries(x)
x - x**3/6 + x**5/120 + O(x**7)
>>> log(x + 1).nseries(x, 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)
Traceback (most recent call last):
...
PoleError: ...
>>> logx = Symbol('logx')
>>> e.nseries(x, logx=logx)
sin(logx)

Notes

This method call the helper method _eval_nseries. Such methods should be implemented in subclasses.

The series expansion code is an important part of the gruntz algorithm for determining limits. _eval_nseries has to return a generalized power series with coefficients in C(log(x), log):

c_0*x**e_0 + ... (finitely many terms)

where e_i are numbers (not necessarily integers) and c_i involve only numbers, the function log, and log(x). (This also means it must not contain log(x(1 + p)), this has to be expanded to log(x) + log(1 + p) if p.is_positive.)

nsimplify(constants=[], tolerance=None, full=False)[source]: See the nsimplify function in diofant.simplify.

powsimp(**args)[source]: See the powsimp function in diofant.simplify.

primitive()[source]

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

>>> (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, (x/2 + 3)*(6*x + 2))

radsimp(**kwargs)[source]: See the radsimp function in diofant.simplify.

ratsimp()[source]: See the ratsimp function in diofant.simplify.

removeO()[source]: Removes the additive O(..) symbol if there is one.

round(p=0)[source]

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.

Examples

>>> Float(10.5).round()
11.
>>> pi.round()
3.
>>> pi.round(2)
3.14
>>> (2*pi + E*I).round()
6.0 + 3.0*I

The round method has a chopping effect:

>>> (2*pi + I/10).round()
6.
>>> (pi/10 + 2*I).round()
2.0*I
>>> (pi/10 + E*I).round(2)
0.31 + 2.72*I

Notes

Do not confuse the Python builtin function, round, with the Diofant 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(Integer(123), -2), Number)
False
>>> isinstance(Integer(123).round(-2), Number)
True
>>> isinstance((3*I).round(), Mul)
True
>>> isinstance((1 + 3*I).round(), Add)
True

series(x=None, x0=0, n=6, dir=None, logx=None)[source]

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.

>>> 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(E**y) - x*sin(E**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=-1 (default) the series is calculated from the right and for dir=+1 the series from the left. For infinite x0 (oo or -oo), the dir argument is determined from the direction of the infinity (i.e. dir=+1 for oo). For smooth functions this flag will not alter the results.

>>> abs(x).series(dir=-1)
x
>>> abs(x).series(dir=+1)
-x

For rational expressions this method may return original expression.

>>> (1/x).series(x, n=8)
1/x

simplify(ratio=1.7, measure=None)[source]: See the simplify function in diofant.simplify.

sort_key(order=None)[source]: Return a sort key.

taylor_term(n, x, *previous_terms)[source]

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)[source]: See the together function in diofant.polys.

transpose()[source]: Transpose self.

See also

diofant.functions.elementary.complexes.transpose

trigsimp(**args)[source]: See the trigsimp function in diofant.simplify.

AtomicExpr

class diofant.core.expr.AtomicExpr(*args)[source]

A parent class for object which are both atoms and Exprs.

For example: Symbol, Number, Rational, Integer, … But not: Add, Mul, Pow, …

symbol

Symbol

class diofant.core.symbol.Symbol(name, **assumptions)[source]

Symbol is a placeholder for atomic symbolic expression.

It has a name and a set of assumptions.

Parameters:

name (str) – The name for Symbol.
**assumptions (dict) – Keyword arguments to specify assumptions for Symbol. Default assumption is commutative=True.

Examples

>>> a, b = symbols('a b')
>>> bool(a*b == b*a)
True

You can override default assumptions:

>>> A, B = symbols('A B', commutative=False)
>>> bool(A*B != B*A)
True
>>> bool(A*B*2 == 2*A*B) is True  # multiplication by scalars is commutative
True

Wild

class diofant.core.symbol.Wild(name, exclude=(), properties=(), **assumptions)[source]

A Wild symbol matches anything, whatever is not explicitly excluded.

Examples

>>> a = Wild('a')
>>> x.match(a)
{a_: x}
>>> pi.match(a)
{a_: pi}
>>> (3*x**2).match(a*x)
{a_: 3*x}
>>> cos(x).match(a)
{a_: cos(x)}
>>> b = Wild('b', exclude=[x])
>>> (3*x**2).match(b*x)
>>> b.match(a)
{a_: b_}
>>> A = WildFunction('A')
>>> A.match(a)
{a_: A_}

Notes

When using Wild, be sure to use the exclude keyword to make the pattern more precise. Without the exclude pattern, you may get matches that are technically correct, but not what you wanted. For example, using the above without exclude:

>>> a, b = symbols('a b', cls=Wild)
>>> (2 + 3*y).match(a*x + b*y)
{a_: 2/x, b_: 3}

This is technically correct, because (2/x)*x + 3*y == 2 + 3*y, but you probably wanted it to not match at all. The issue is that you really didn’t want a and b to include x and y, and the exclude parameter lets you specify exactly this. With the exclude parameter, the pattern will not match.

>>> a = Wild('a', exclude=[x, y])
>>> b = Wild('b', exclude=[x, y])
>>> (2 + 3*y).match(a*x + b*y)

Exclude also helps remove ambiguity from matches.

>>> E = 2*x**3*y*z
>>> a, b = symbols('a b', cls=Wild)
>>> E.match(a*b)
{a_: 2*y*z, b_: x**3}
>>> a = Wild('a', exclude=[x, y])
>>> E.match(a*b)
{a_: z, b_: 2*x**3*y}
>>> a = Wild('a', exclude=[x, y, z])
>>> E.match(a*b)
{a_: 2, b_: x**3*y*z}

See also

Symbol

Dummy

class diofant.core.symbol.Dummy(name=None, **assumptions)[source]

Dummy symbols are each unique, identified by an internal count index:

>>> bool(Dummy('x') == Dummy('x')) is True
False

If a name is not supplied then a string value of the count index will be used. This is useful when a temporary variable is needed and the name of the variable used in the expression is not important.

See also

Symbol

classmethod class_key()[source]: Nice order of classes.

sort_key(order=None)[source]: Return a sort key.

symbols

diofant.core.symbol.symbols(names, **args)[source]

Transform strings into instances of Symbol class.

symbols() function returns a sequence of symbols with names taken from names argument, which can be a comma or whitespace delimited string, or a sequence of strings:

>>> a, b, c = symbols('a b c')

The type of output is dependent on the properties of input arguments:

>>> symbols('x')
x
>>> symbols('x,')
(x,)
>>> symbols('x,y')
(x, y)
>>> symbols(('a', 'b', 'c'))
(a, b, c)
>>> symbols(['a', 'b', 'c'])
[a, b, c]
>>> symbols({'a', 'b', 'c'})
{a, b, c}

If an iterable container is needed for a single symbol, set the seq argument to True or terminate the symbol name with a comma:

>>> symbols('x', seq=True)
(x,)

To reduce typing, range syntax is supported to create indexed symbols. Ranges are indicated by a colon and the type of range is determined by the character to the right of the colon. If the character is a digit then all contiguous digits to the left are taken as the nonnegative starting value (or 0 if there is no digit left of the colon) and all contiguous digits to the right are taken as 1 greater than the ending value:

>>> symbols('x:10')
(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)

>>> symbols('x5:10')
(x5, x6, x7, x8, x9)
>>> symbols('x5(:2)')
(x50, x51)

>>> symbols('x5:10 y:5')
(x5, x6, x7, x8, x9, y0, y1, y2, y3, y4)

>>> symbols(('x5:10', 'y:5'))
((x5, x6, x7, x8, x9), (y0, y1, y2, y3, y4))

If the character to the right of the colon is a letter, then the single letter to the left (or ‘a’ if there is none) is taken as the start and all characters in the lexicographic range through the letter to the right are used as the range:

>>> symbols('x:z')
(x, y, z)
>>> symbols('x:c')  # null range
()
>>> symbols('x(:c)')
(xa, xb, xc)

>>> symbols(':c')
(a, b, c)

>>> symbols('a:d, x:z')
(a, b, c, d, x, y, z)

>>> symbols(('a:d', 'x:z'))
((a, b, c, d), (x, y, z))

Multiple ranges are supported; contiguous numerical ranges should be separated by parentheses to disambiguate the ending number of one range from the starting number of the next:

>>> symbols('x:2(1:3)')
(x01, x02, x11, x12)
>>> symbols(':3:2')  # parsing is from left to right
(00, 01, 10, 11, 20, 21)

Only one pair of parentheses surrounding ranges are removed, so to include parentheses around ranges, double them. And to include spaces, commas, or colons, escape them with a backslash:

>>> symbols('x((a:b))')
(x(a), x(b))
>>> symbols(r'x(:1\,:2)')  # or 'x((:1)\,(:2))'
(x(0,0), x(0,1))

All newly created symbols have assumptions set according to args:

>>> a = symbols('a', integer=True)
>>> a.is_integer
True

>>> x, y, z = symbols('x y z', real=True)
>>> x.is_real and y.is_real and z.is_real
True

Despite its name, symbols() can create symbol-like objects like instances of Function or Wild classes. To achieve this, set cls keyword argument to the desired type:

>>> symbols('f g h', cls=Function)
(f, g, h)

>>> type(_[0])
<class 'diofant.core.function.UndefinedFunction'>

var

diofant.core.symbol.var(names, **args)[source]

Create symbols and inject them into the global namespace.

This calls symbols() with the same arguments and puts the results into the global namespace. It’s recommended not to use var() in library code, where symbols() has to be used.

Examples

>>> var('x')
x
>>> x
x

>>> var('a ab abc')
(a, ab, abc)
>>> abc
abc

>>> var('x y', real=True)
(x, y)
>>> x.is_real and y.is_real
True

See also

symbols

numbers

Number

class diofant.core.numbers.Number(*obj)[source]

Represents any kind of number in diofant.

Floating point numbers are represented by the Float class. Integer numbers (of any size), together with rational numbers (again, there is no limit on their size) are represented by the Rational class.

as_coeff_Add(rational=False)[source]: Efficiently extract the coefficient of a summation.

as_coeff_Mul(rational=False)[source]: Efficiently extract the coefficient of a product.

as_coeff_add(*deps)[source]: Return the tuple (c, args) where self is written as an Add.

See also

diofant.core.expr.Expr.as_coeff_add

as_coeff_mul(*deps, **kwargs)[source]: Return the tuple (c, args) where self is written as a Mul.

See also

diofant.core.expr.Expr.as_coeff_mul

classmethod class_key()[source]: Nice order of classes.

cofactors(other)[source]: Compute GCD and cofactors of \(self\) and \(other\).

gcd(other)[source]: Compute GCD of \(self\) and \(other\).

is_constant(*wrt, **flags)[source]: Return True if self is constant.

See also

diofant.core.expr.Expr.is_constant

lcm(other)[source]: Compute LCM of \(self\) and \(other\).

sort_key(order=None)[source]: Return a sort key.

Float

class diofant.core.numbers.Float(num, dps=None)[source]

Represent a floating-point number of arbitrary precision.

Examples

>>> Float(3.5)
3.50000000000000
>>> Float(3)
3.

Creating Floats from strings (and Python int type) will give a minimum precision of 15 digits, but the precision will automatically increase to capture all digits entered.

>>> Float(1)
1.
>>> Float(10**20)
100000000000000000000.
>>> Float('1e20')
1.e+20

However, floating-point numbers (Python float types) retain only 15 digits of precision:

>>> Float(1e20)
1.00000000000000e+20
>>> Float(1.23456789123456789)
1.23456789123457

It may be preferable to enter high-precision decimal numbers as strings:

>>> Float('1.23456789123456789')
1.23456789123456789

The desired number of digits can also be specified:

>>> Float('1e-3', 3)
0.00100
>>> Float(100, 4)
100.0

Float can automatically count significant figures if decimal precision argument is omitted. (Auto-counting is only allowed for strings and ints).

>>> Float('12e-3')
0.012
>>> Float(3)
3.
>>> Float('60.e2')  # 2 digits significant
6.0e+3
>>> Float('6000.')  # 4 digits significant
6000.
>>> Float('600e-2')  # 3 digits significant
6.00

Notes

Floats are inexact by their nature unless their value is a binary-exact value.

>>> approx, exact = Float(.1, 1), Float(.125, 1)

For calculation purposes, you can change the precision of Float, but this will not increase the accuracy of the inexact value. The following is the most accurate 5-digit approximation of a value of 0.1 that had only 1 digit of precision:

>>> Float(approx, 5)
0.099609

Please note that you can’t increase precision with evalf:

>>> approx.evalf(5)
Traceback (most recent call last):
...
PrecisionExhausted: ...

By contrast, 0.125 is exact in binary (as it is in base 10) and so it can be passed to Float constructor to obtain an arbitrary precision with matching accuracy:

>>> Float(exact, 5)
0.12500
>>> Float(exact, 20)
0.12500000000000000000

Trying to make a high-precision Float from a float is not disallowed, but one must keep in mind that the underlying float (not the apparent decimal value) is being obtained with high precision. For example, 0.3 does not have a finite binary representation. The closest rational is the fraction 5404319552844595/2**54. So if you try to obtain a Float of 0.3 to 20 digits of precision you will not see the same thing as 0.3 followed by 19 zeros:

>>> Float(0.3, 20)
0.29999999999999998890

If you want a 20-digit value of the decimal 0.3 (not the floating point approximation of 0.3) you should send the 0.3 as a string. The underlying representation is still binary but a higher precision than Python’s float is used:

>>> Float('0.3', 20)
0.30000000000000000000

Although you can increase the precision of an existing Float using Float it will not increase the accuracy – the underlying value is not changed:

>>> def show(f):  # binary rep of Float
...     from diofant import Mul, Pow
...     s, m, e, b = f._mpf_
...     v = Mul(int(m), Pow(2, int(e), evaluate=False), evaluate=False)
...     print(f'{v} at prec={f._prec}')
...
>>> t = Float('0.3', 3)
>>> show(t)
4915/2**14 at prec=13
>>> show(Float(t, 20))  # higher prec, not higher accuracy
4915/2**14 at prec=70
>>> show(Float(t, 2))  # lower prec
307/2**10 at prec=10

ceiling()[source]: Compute ceiling of self.

epsilon_eq(other, epsilon='1e-15')[source]: Test approximate equality.

floor()[source]: Compute floor of self.

property num: Return mpmath representation.

Rational

class diofant.core.numbers.Rational(p, q=1)[source]

Represents integers and rational numbers (p/q) of any size.

Examples

>>> Rational(3)
3
>>> Rational(1, 2)
1/2

Rational is unprejudiced in accepting input. If a float is passed, the underlying value of the binary representation will be returned:

>>> Rational(.5)
1/2
>>> Rational(.2)
3602879701896397/18014398509481984

If the simpler representation of the float is desired then consider limiting the denominator to the desired value or convert the float to a string (which is roughly equivalent to limiting the denominator to 10**12):

>>> Rational(str(.2))
1/5
>>> Rational(.2).limit_denominator(10**12)
1/5

An arbitrarily precise Rational is obtained when a string literal is passed:

>>> Rational('1.23')
123/100
>>> Rational('1e-2')
1/100
>>> Rational('.1')
1/10

The conversion of floats to expressions or simple fractions can be handled with nsimplify:

>>> nsimplify(.3)  # numbers that have a simple form
3/10

But if the input does not reduce to a literal Rational, an error will be raised:

>>> Rational(pi)
Traceback (most recent call last):
...
TypeError: invalid input: pi

Low-level access numerator and denominator:

>>> r = Rational(3, 4)
>>> r
3/4
>>> r.numerator
3
>>> r.denominator
4

Note that these properties return integers (not Diofant Integers) so some care is needed when using them in expressions:

>>> r.numerator/r.denominator
0.75

as_content_primitive(radical=False)[source]

Return the tuple (R, self/R) where R is the positive Rational extracted from self.

Examples

>>> Rational(-3, 2).as_content_primitive()
(3/2, -1)

factors(limit=None, use_trial=True, use_rho=False, use_pm1=False, verbose=False, visual=False)[source]: A wrapper to factorint which return factors of self that are smaller than limit (or cheap to compute). Special methods of factoring are disabled by default so that only trial division is used.

gcd(other)[source]: Compute GCD of \(self\) and \(other\).

lcm(other)[source]: Compute LCM of \(self\) and \(other\).

limit_denominator(max_denominator=1000000)[source]

Closest Rational to self with denominator at most max_denominator.

>>> Rational('3.141592653589793').limit_denominator(10)
22/7
>>> Rational('3.141592653589793').limit_denominator(100)
311/99

Integer

class diofant.core.numbers.Integer(i)[source]

Represents integer numbers.

property is_composite

Test if self is a positive integer that has at least one positive divisor other than 1 or the number itself.

References

https://en.wikipedia.org/wiki/Composite_number

property is_even

Test if self can have only values from the set of even integers.

See also

is_odd

References

https://en.wikipedia.org/wiki/Parity_%28mathematics%29

property is_imaginary

Test if self is an imaginary number.

I.e. that it can be written as a real number multiplied by the imaginary unit I.

References

https://en.wikipedia.org/wiki/Imaginary_number

property is_nonzero: Test if self is nonzero.

See also

is_zero

property is_odd

Test if self can have only values from the set of odd integers.

See also

is_even

References

https://en.wikipedia.org/wiki/Parity_%28mathematics%29

property is_prime

Test if self is a natural number greater than 1 that has no positive divisors other than 1 and itself.

References

https://en.wikipedia.org/wiki/Prime_number

property is_zero: Test if self is zero.

See also

is_nonzero

NumberSymbol

class diofant.core.numbers.NumberSymbol[source]

Base class for symbolic numbers.

approximation_interval(number_cls)[source]: Return an interval with number_cls endpoints that contains the value of NumberSymbol. If not implemented, then return None.

mod_inverse

diofant.core.numbers.mod_inverse(a, m)[source]

Return the number c such that, ( a * c ) % m == 1 where c has the same sign as a. If no such value exists, a ValueError is raised.

Examples

Suppose we wish to find multiplicative inverse x of 3 modulo 11. This is the same as finding x such that 3 * x = 1 (mod 11). One value of x that satisfies this congruence is 4. Because 3 * 4 = 12 and 12 = 1 mod(11). This is the value return by mod_inverse:

>>> mod_inverse(3, 11)
4
>>> mod_inverse(-3, 11)
-4

When there is a common factor between the numerators of a and m the inverse does not exist:

>>> mod_inverse(2, 4)
Traceback (most recent call last):
...
ValueError: inverse of 2 mod 4 does not exist

>>> mod_inverse(Integer(2)/7, Integer(5)/2)
7/2

References

integer_digits

diofant.core.numbers.integer_digits(n, b)[source]: Gives a list of the base \(b\) digits in the integer \(n\).

Zero

class diofant.core.numbers.Zero(*args, **kwargs)[source]

The number zero.

Zero is a singleton, and can be accessed by S.Zero

Examples

>>> Integer(0) is S.Zero
True
>>> 1/S.Zero
zoo

References

https://en.wikipedia.org/wiki/Zero

One

class diofant.core.numbers.One(*args, **kwargs)[source]

The number one.

One is a singleton, and can be accessed by S.One.

Examples

>>> Integer(1) is S.One
True

References

https://en.wikipedia.org/wiki/1_%28number%29

NegativeOne

class diofant.core.numbers.NegativeOne(*args, **kwargs)[source]

The number negative one.

NegativeOne is a singleton, and can be accessed by S.NegativeOne.

Examples

>>> Integer(-1) is S.NegativeOne
True

See also

One

References

https://en.wikipedia.org/wiki/%E2%88%921_%28number%29

Half

class diofant.core.numbers.Half(*args, **kwargs)[source]

The rational number 1/2.

Half is a singleton, and can be accessed by S.Half.

Examples

>>> Rational(1, 2) is S.Half
True

References

https://en.wikipedia.org/wiki/One_half

NaN

class diofant.core.numbers.NaN(*args, **kwargs)[source]

Not a Number.

This serves as a place holder for numeric values that are indeterminate. Most operations on NaN, produce another NaN. Most indeterminate forms, such as 0/0 or oo - oo` produce NaN. Two exceptions are ``0**0 and oo**0, which all produce 1 (this is consistent with Python’s float).

NaN is loosely related to floating point nan, which is defined in the IEEE 754 floating point standard, and corresponds to the Python float('nan'). Differences are noted below.

NaN is mathematically not equal to anything else, even NaN itself. This explains the initially counter-intuitive results with Eq and == in the examples below.

NaN is not comparable so inequalities raise a TypeError. This is in constrast with floating point nan where all inequalities are false.

NaN is a singleton, and can be accessed by nan.

Examples

>>> nan is nan
True
>>> oo - oo
nan
>>> nan + 1
nan
>>> Eq(nan, nan)   # mathematical equality
false
>>> nan == nan     # structural equality
True

References

https://en.wikipedia.org/wiki/NaN

Infinity

class diofant.core.numbers.Infinity(*args, **kwargs)[source]

Positive infinite quantity.

In real analysis the symbol \(\infty\) denotes an unbounded limit: \(x\to\infty\) means that \(x\) grows without bound.

Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled \(+\infty\) and \(-\infty\) can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers.

Infinity is a singleton, and can be accessed by oo, or can be imported as oo.

Examples

>>> 1 + oo
oo
>>> 42/oo
0
>>> limit(exp(x), x, oo)
oo

See also

NegativeInfinity, NaN

References

https://en.wikipedia.org/wiki/Infinity

NegativeInfinity

class diofant.core.numbers.NegativeInfinity(*args, **kwargs)[source]

Negative infinite quantity.

NegativeInfinity is a singleton, and can be accessed by -oo.

See also

Infinity

ComplexInfinity

class diofant.core.numbers.ComplexInfinity(*args, **kwargs)[source]

Complex infinity.

In complex analysis the symbol \(\tilde\infty\), called “complex infinity”, represents a quantity with infinite magnitude, but undetermined complex phase.

ComplexInfinity is a singleton, and can be accessed by as zoo.

Examples

>>> zoo + 42
zoo
>>> 42/zoo
0
>>> zoo + zoo
nan
>>> zoo*zoo
zoo

See also

Infinity

Exp1

class diofant.core.numbers.Exp1(*args, **kwargs)[source]

The \(e\) constant.

The transcendental number \(e = 2.718281828\ldots\) is the base of the natural logarithm and of the exponential function, \(e = \exp(1)\). Sometimes called Euler’s number or Napier’s constant.

Exp1 is a singleton, and can be imported as E.

Examples

>>> E is exp(1)
True
>>> log(E)
1

References

https://en.wikipedia.org/wiki/E_%28mathematical_constant%29

ImaginaryUnit

class diofant.core.numbers.ImaginaryUnit(*args, **kwargs)[source]

The imaginary unit, \(i = \sqrt{-1}\).

I is a singleton, and can be imported as I.

Examples

>>> sqrt(-1)
I
>>> I*I
-1
>>> 1/I
-I

References

https://en.wikipedia.org/wiki/Imaginary_unit

Pi

class diofant.core.numbers.Pi(*args, **kwargs)[source]

The \(\pi\) constant.

The transcendental number \(\pi = 3.141592654\ldots\) represents the ratio of a circle’s circumference to its diameter, the area of the unit circle, the half-period of trigonometric functions, and many other things in mathematics.

Pi is a singleton, and can be imported as pi.

Examples

>>> pi > 3
true
>>> pi.is_irrational
True
>>> sin(x + 2*pi)
sin(x)
>>> integrate(exp(-x**2), (x, -oo, oo))
sqrt(pi)

References

https://en.wikipedia.org/wiki/Pi

EulerGamma

class diofant.core.numbers.EulerGamma(*args, **kwargs)[source]

The Euler-Mascheroni constant.

\(\gamma = 0.5772157\ldots\) (also called Euler’s constant) is a mathematical constant recurring in analysis and number theory. It is defined as the limiting difference between the harmonic series and the natural logarithm:

\[\gamma = \lim\limits_{n\to\infty} \left(\sum\limits_{k=1}^n\frac{1}{k} - \ln n\right)\]

Examples

>>> EulerGamma.is_irrational
>>> EulerGamma > 0
true
>>> EulerGamma > 1
false

References

https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant

Catalan

class diofant.core.numbers.Catalan(*args, **kwargs)[source]

Catalan’s constant.

\(K = 0.91596559\ldots\) is given by the infinite series

\[K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\]

Examples

>>> Catalan.is_irrational
>>> Catalan > 0
true
>>> Catalan > 1
false

References

https://en.wikipedia.org/wiki/Catalan%27s_constant

GoldenRatio

class diofant.core.numbers.GoldenRatio(*args, **kwargs)[source]

The golden ratio, \(\phi\).

\(\phi = \frac{1 + \sqrt{5}}{2}\) is algebraic number. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities, i.e. their maximum.

Examples

>>> GoldenRatio > 1
true
>>> GoldenRatio.expand(func=True)
1/2 + sqrt(5)/2
>>> GoldenRatio.is_irrational
True

References

https://en.wikipedia.org/wiki/Golden_ratio

power

Pow

class diofant.core.power.Pow(b, e, evaluate=None)[source]

Defines the expression x**y as “x raised to a power y”.

For complex numbers \(x\) and \(y\), Pow gives the principal value of \(exp(y*log(x))\).

Singleton definitions involving (0, 1, -1, oo, -oo, I, -I):

expr	value	reason
z**0	1	Although arguments over 0**0 exist, see [2].
z**1	z
(-oo)**(-1)	0
(-1)**-1	-1
0**-1	zoo	This is not strictly true, as 0**-1 may be undefined, but is convenient in some contexts where the base is assumed to be positive.
1**-1	1
oo**-1	0
0**oo	0	Because for all complex numbers z near 0, z**oo -> 0.
0**-oo	zoo	This is not strictly true, as 0**oo may be oscillating between positive and negative values or rotating in the complex plane. It is convenient, however, when the base is positive.
1oo 1-oo	nan	Because there are various cases where lim(x(t),t)=1, lim(y(t),t)=oo (or -oo), but lim(x(t)**y(t), t) != 1. See [3].
z**zoo	nan	No limit for z**t for t -> zoo.
(-1)oo (-1)(-oo)	nan	Because of oscillations in the limit.
oo**oo	oo
oo**-oo	0
(-oo)oo (-oo)-oo	nan
ooI (-oo)I	nan	ooe could probably be best thought of as the limit of xe for real x as x tends to oo. If e is I, then the limit does not exist and nan is used to indicate that.
oo(1+I) (-oo)(1+I)	zoo	If the real part of e is positive, then the limit of abs(x**e) is oo. So the limit value is zoo.
oo(-1+I) -oo(-1+I)	0	If the real part of e is negative, then the limit is 0.

Because symbolic computations are more flexible that floating point calculations and we prefer to never return an incorrect answer, we choose not to conform to all IEEE 754 conventions. This helps us avoid extra test-case code in the calculation of limits.

References

as_base_exp()[source]

Return base and exp of self.

If base is 1/Integer, then return Integer, -exp. If this extra processing is not needed, the base and exp properties will give the raw arguments

Examples

>>> p = Pow(Rational(1, 2), 2, evaluate=False)
>>> p.as_base_exp()
(2, -2)
>>> p.args
(1/2, 2)

as_content_primitive(radical=False)[source]

Return the tuple (R, self/R) where R is the positive Rational extracted from self.

Examples

>>> sqrt(4 + 4*sqrt(2)).as_content_primitive()
(2, sqrt(1 + sqrt(2)))
>>> sqrt(3 + 3*sqrt(2)).as_content_primitive()
(1, sqrt(3)*sqrt(1 + sqrt(2)))

>>> ((2*x + 2)**2).as_content_primitive()
(4, (x + 1)**2)
>>> (4**((1 + y)/2)).as_content_primitive()
(2, 4**(y/2))
>>> (3**((1 + y)/2)).as_content_primitive()
(1, 3**((y + 1)/2))
>>> (3**((5 + y)/2)).as_content_primitive()
(9, 3**((y + 1)/2))
>>> eq = 3**(2 + 2*x)
>>> powsimp(eq) == eq
True
>>> eq.as_content_primitive()
(9, 3**(2*x))
>>> powsimp(Mul(*_))
3**(2*x + 2)

>>> eq = (2 + 2*x)**y
>>> s = expand_power_base(eq)
>>> s.is_Mul, s
(False, (2*x + 2)**y)
>>> eq.as_content_primitive()
(1, (2*(x + 1))**y)
>>> s = expand_power_base(_[1])
>>> s.is_Mul, s
(True, 2**y*(x + 1)**y)

as_real_imag(deep=True, **hints)[source]: Returns real and imaginary parts of self

See also

diofant.core.expr.Expr.as_real_imag

property base: Returns base of the power expression.

classmethod class_key()[source]: Nice order of classes.

property exp: Returns exponent of the power expression.

integer_nthroot

diofant.core.power.integer_nthroot(y, n)[source]

Return a tuple containing x = floor(y**(1/n)) and a boolean indicating whether the result is exact (that is, whether x**n == y).

>>> integer_nthroot(16, 2)
(4, True)
>>> integer_nthroot(26, 2)
(5, False)

mul

Mul

class diofant.core.mul.Mul(*args, **options)[source]

Symbolic multiplication class.

as_base_exp()[source]: Return base and exp of self.

See also

diofant.core.expr.Expr.as_base_exp

as_coeff_Mul(rational=False)[source]: Efficiently extract the coefficient of a product.

as_coeff_mul(*deps, **kwargs)[source]: Return the tuple (c, args) where self is written as a Mul.

See also

diofant.core.expr.Expr.as_coeff_mul

as_content_primitive(radical=False)[source]

Return the tuple (R, self/R) where R is the positive Rational extracted from self.

Examples

>>> (-3*sqrt(2)*(2 - 2*sqrt(2))).as_content_primitive()
(6, -sqrt(2)*(-sqrt(2) + 1))

as_ordered_factors(order=None)[source]

Transform an expression into an ordered list of factors.

Examples

>>> (2*x*y*sin(x)*cos(x)).as_ordered_factors()
[2, x, y, sin(x), cos(x)]

as_powers_dict()[source]: Return self as a dictionary of factors with each factor being treated as a power.

See also

diofant.core.expr.Expr.as_powers_dict

as_real_imag(deep=True, **hints)[source]: Returns real and imaginary parts of self

See also

diofant.core.expr.Expr.as_real_imag

as_two_terms()[source]

Return head and tail of self.

This is the most efficient way to get the head and tail of an expression.

if you want only the head, use self.args[0];
if you want to process the arguments of the tail then use self.as_coef_mul() which gives the head and a tuple containing the arguments of the tail when treated as a Mul.
if you want the coefficient when self is treated as an Add then use self.as_coeff_add()[0]

>>> (3*x*y).as_two_terms()
(3, x*y)

classmethod class_key()[source]: Nice order of classes.

classmethod flatten(seq)[source]

Return commutative, noncommutative and order arguments by combining related terms.

Notes

In an expression like a*b*c, python process this through diofant as Mul(Mul(a, b), c). This can have undesirable consequences.
- Sometimes terms are not combined as one would like: {c.f. sympy/sympy#4596}
>>> 2*(x + 1) # this is the 2-arg Mul behavior 2*x + 2 >>> y*(x + 1)*2 2*y*(x + 1) >>> 2*(x + 1)*y # 2-arg result will be obtained first y*(2*x + 2) >>> Mul(2, x + 1, y) # all 3 args simultaneously processed 2*y*(x + 1) >>> 2*((x + 1)*y) # parentheses can control this behavior 2*y*(x + 1)
Powers with compound bases may not find a single base to combine with unless all arguments are processed at once. Post-processing may be necessary in such cases. {c.f. sympy/sympy#5728}
>>> a = sqrt(x*sqrt(y)) >>> a**3 (x*sqrt(y))**(3/2) >>> Mul(a, a, a) (x*sqrt(y))**(3/2) >>> a*a*a x*sqrt(y)*sqrt(x*sqrt(y)) >>> _.subs({a.base: z}).subs({z: a.base}) (x*sqrt(y))**(3/2)
- If more than two terms are being multiplied then all the previous terms will be re-processed for each new argument. So if each of a, b and c were Mul expression, then a*b*c (or building up the product with *=) will process all the arguments of a and b twice: once when a*b is computed and again when c is multiplied.
  
  Using Mul(a, b, c) will process all arguments once.
The results of Mul are cached according to arguments, so flatten will only be called once for Mul(a, b, c). If you can structure a calculation so the arguments are most likely to be repeats then this can save time in computing the answer. For example, say you had a Mul, M, that you wished to divide by d[i] and multiply by n[i] and you suspect there are many repeats in n. It would be better to compute M*n[i]/d[i] rather than M/d[i]*n[i] since every time n[i] is a repeat, the product, M*n[i] will be returned without flattening – the cached value will be returned. If you divide by the d[i] first (and those are more unique than the n[i]) then that will create a new Mul, M/d[i] the args of which will be traversed again when it is multiplied by n[i].

{c.f. sympy/sympy#5706}

This consideration is moot if the cache is turned off.

The validity of the above notes depends on the implementation details of Mul and flatten which may change at any time. Therefore, you should only consider them when your code is highly performance sensitive.

add

Add

class diofant.core.add.Add(*args, **options)[source]

Symbolic addition class.

as_coeff_Add(rational=False)[source]: Efficiently extract the coefficient of a summation.

as_coeff_add(*deps)[source]

Returns a tuple (coeff, args) where self is treated as an Add and coeff is the Number term and args is a tuple of all other terms.

Examples

>>> (7 + 3*x).as_coeff_add()
(7, (3*x,))
>>> (7*x).as_coeff_add()
(0, (7*x,))

as_coefficients_dict()[source]

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

>>> (3*x + x*y + 4).as_coefficients_dict()
{1: 4, x: 3, x*y: 1}
>>> _[y]
0
>>> (3*y*x).as_coefficients_dict()
{x*y: 3}

as_content_primitive(radical=False)[source]

Return the tuple (R, self/R) where R is the positive Rational extracted from self. If radical is True (default is False) then common radicals will be removed and included as a factor of the primitive expression.

Examples

>>> (3 + 3*sqrt(2)).as_content_primitive()
(3, 1 + sqrt(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)))

as_real_imag(deep=True, **hints)[source]

Returns a tuple representing a complex number.

Examples

>>> (7 + 9*I).as_real_imag()
(7, 9)
>>> ((1 + I)/(1 - I)).as_real_imag()
(0, 1)
>>> ((1 + 2*I)*(1 + 3*I)).as_real_imag()
(-5, 5)

as_two_terms()[source]

Return head and tail of self.

This is the most efficient way to get the head and tail of an expression.

if you want only the head, use self.args[0];
if you want to process the arguments of the tail then use self.as_coef_add() which gives the head and a tuple containing the arguments of the tail when treated as an Add.
if you want the coefficient when self is treated as a Mul then use self.as_coeff_mul()[0]

>>> (3*x*y).as_two_terms()
(3, x*y)

classmethod class_key()[source]: Nice order of classes.

classmethod flatten(seq)[source]

Takes the sequence “seq” of nested Adds and returns a flatten list.

Returns: (commutative_part, noncommutative_part, order_symbols)

Applies associativity, all terms are commutable with respect to addition.

See also

diofant.core.mul.Mul.flatten

getO()[source]: Returns the additive O(..) symbol.

See also

diofant.core.expr.Expr.getO

primitive()[source]

Return (R, self/R) where R is the Rational GCD of self.

R is collected only from the leading coefficient of each term.

Examples

>>> (2*x + 4*y).primitive()
(2, x + 2*y)

>>> (2*x/3 + 4*y/9).primitive()
(2/9, 3*x + 2*y)

>>> (2*x/3 + 4.2*y).primitive()
(1/3, 2*x + 12.6*y)

No subprocessing of term factors is performed:

>>> ((2 + 2*x)*x + 2).primitive()
(1, x*(2*x + 2) + 2)

Recursive subprocessing can be done with the as_content_primitive() method:

>>> ((2 + 2*x)*x + 2).as_content_primitive()
(2, x*(x + 1) + 1)

See also

diofant.polys.polytools.primitive

removeO()[source]: Removes the additive O(..) symbol.

See also

diofant.core.expr.Expr.removeO

mod

Mod

class diofant.core.mod.Mod(p, q)[source]

Represents a modulo operation on symbolic expressions.

Receives two arguments, dividend p and divisor q.

The convention used is the same as Python’s: the remainder always has the same sign as the divisor.

Examples

>>> x**2 % y
x**2%y
>>> _.subs({x: 5, y: 6})
1

relational

Rel

diofant.core.relational.Rel[source]: alias of Relational

Eq

diofant.core.relational.Eq[source]: alias of Equality

Ne

diofant.core.relational.Ne[source]: alias of Unequality

Lt

diofant.core.relational.Lt[source]: alias of StrictLessThan

Le

diofant.core.relational.Le[source]: alias of LessThan

Gt

diofant.core.relational.Gt[source]: alias of StrictGreaterThan

Ge

diofant.core.relational.Ge[source]: alias of GreaterThan

Relational

class diofant.core.relational.Relational(lhs, rhs=0, rop=None, **assumptions)[source]

Base class for all relation types.

Subclasses of Relational should generally be instantiated directly, but Relational can be instantiated with a valid \(rop\) value to dispatch to the appropriate subclass.

Parameters:: rop (str or None) – Indicates what subclass to instantiate. Valid values can be found in the keys of Relational.ValidRelationalOperator.

Examples

>>> Rel(y, x+x**2, '==')
Eq(y, x**2 + x)

as_set()[source]

Rewrites univariate inequality in terms of real sets

Examples

>>> (x > 0).as_set()
(0, oo]
>>> Eq(x, 0).as_set()
{0}

property canonical

Return a canonical form of the relational.

The rules for the canonical form, in order of decreasing priority are:

Number on right if left is not a Number;
Symbol on the left;
Gt/Ge changed to Lt/Le;
Lt/Le are unchanged;
Eq and Ne get ordered args.

equals(other, failing_expression=False)[source]: Return True if the sides of the relationship are mathematically identical and the type of relationship is the same. If failing_expression is True, return the expression whose truth value was unknown.

property lhs: The left-hand side of the relation.

property reversed

Return the relationship with sides (and sign) reversed.

Examples

>>> Eq(x, 1)
Eq(x, 1)
>>> _.reversed
Eq(1, x)
>>> x < 1
x < 1
>>> _.reversed
1 > x

property rhs: The right-hand side of the relation.

Equality

class diofant.core.relational.Equality(lhs, rhs=0, **options)[source]

An equal relation between two objects.

Represents that two objects are equal. If they can be easily shown to be definitively equal (or unequal), this will reduce to True (or False). Otherwise, the relation is maintained as an unevaluated Equality object. Use the simplify function on this object for more nontrivial evaluation of the equality relation.

As usual, the keyword argument evaluate=False can be used to prevent any evaluation.

Examples

>>> Eq(y, x + x**2)
Eq(y, x**2 + x)
>>> Eq(2, 5)
false
>>> Eq(2, 5, evaluate=False)
Eq(2, 5)
>>> _.doit()
false
>>> Eq(exp(x), exp(x).rewrite(cos))
Eq(E**x, sinh(x) + cosh(x))
>>> simplify(_)
true

See also

diofant.logic.boolalg.Equivalent: for representing equality between two boolean expressions

Notes

This class is not the same as the == operator. The == operator tests for exact structural equality between two expressions; this class compares expressions mathematically.

If either object defines an \(_eval_Eq\) method, it can be used in place of the default algorithm. If \(lhs._eval_Eq(rhs)\) or \(rhs._eval_Eq(lhs)\) returns anything other than None, that return value will be substituted for the Equality. If None is returned by \(_eval_Eq\), an Equality object will be created as usual.

GreaterThan

class diofant.core.relational.GreaterThan(lhs, rhs=0, **options)[source]

Class representations of inequalities.

The *Than classes represent unequal relationships, where the left-hand side is generally bigger or smaller than the right-hand side. For example, the GreaterThan class represents an unequal relationship where the left-hand side is at least as big as the right side, if not bigger. In mathematical notation:

lhs >= rhs

In total, there are four *Than classes, to represent the four inequalities:

Class Name	Symbol
GreaterThan	(>=)
LessThan	(<=)
StrictGreaterThan	(>)
StrictLessThan	(<)

All classes take two arguments, lhs and rhs.

Signature Example	Math equivalent
GreaterThan(lhs, rhs)	lhs >= rhs
LessThan(lhs, rhs)	lhs <= rhs
StrictGreaterThan(lhs, rhs)	lhs > rhs
StrictLessThan(lhs, rhs)	lhs < rhs

In addition to the normal .lhs and .rhs of Relations, *Than inequality objects also have the .lts and .gts properties, which represent the “less than side” and “greater than side” of the operator. Use of .lts and .gts in an algorithm rather than .lhs and .rhs as an assumption of inequality direction will make more explicit the intent of a certain section of code, and will make it similarly more robust to client code changes:

>>> e = GreaterThan(x, 1)
>>> e
x >= 1
>>> f'{e.gts} >= {e.lts} is the same as {e.lts} <= {e.gts}'
'x >= 1 is the same as 1 <= x'

Examples

One generally does not instantiate these classes directly, but uses various convenience methods:

>>> e1 = Ge(x, 2)  # Ge is a convenience wrapper
>>> print(e1)
x >= 2

>>> print(f'{Ge(x, 2)}\n{Gt(x, 2)}\n{Le(x, 2)}\n{Lt(x, 2)}')
x >= 2
x > 2
x <= 2
x < 2

Another option is to use the Python inequality operators (>=, >, <=, <) directly. Their main advantage over the Ge, Gt, Le, and Lt counterparts, is that one can write a more “mathematical looking” statement rather than littering the math with oddball function calls. However there are certain (minor) caveats of which to be aware (search for ‘gotcha’, below).

>>> e2 = x >= 2
>>> print(e2)
x >= 2
>>> print(f'e1: {e1},    e2: {e2}')
e1: x >= 2,    e2: x >= 2
>>> e1 == e2
True

However, it is also perfectly valid to instantiate a *Than class less succinctly and less conveniently:

>>> print(f"{Rel(x, 1, '>=')}\n{Relational(x, 1, '>=')}\n{GreaterThan(x, 1)}")
x >= 1
x >= 1
x >= 1

>>> print(f"{Rel(x, 1, '>')}\n{Relational(x, 1, '>')}\n{StrictGreaterThan(x, 1)}")
x > 1
x > 1
x > 1

>>> print(f"{Rel(x, 1, '<=')}\n{Relational(x, 1, '<=')}\n{LessThan(x, 1)}")
x <= 1
x <= 1
x <= 1

>>> print(f"{Rel(x, 1, '<')}\n{Relational(x, 1, '<')}\n{StrictLessThan(x, 1)}")
x < 1
x < 1
x < 1

Notes

There are a couple of “gotchas” when using Python’s operators.

The first enters the mix when comparing against a literal number as the lhs argument. Due to the order that Python decides to parse a statement, it may not immediately find two objects comparable. For example, to evaluate the statement (1 < x), Python will first recognize the number 1 as a native number, and then that x is not a native number. At this point, because a native Python number does not know how to compare itself with a Diofant object Python will try the reflective operation, (x > 1). Unfortunately, there is no way available to Diofant to recognize this has happened, so the statement (1 < x) will turn silently into (x > 1).

>>> e1 = x > 1
>>> e2 = x >= 1
>>> e3 = x < 1
>>> e4 = x <= 1
>>> e5 = 1 > x
>>> e6 = 1 >= x
>>> e7 = 1 < x
>>> e8 = 1 <= x
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
x > 1     x >= 1
x < 1     x <= 1
x < 1     x <= 1
x > 1     x >= 1

If the order of the statement is important (for visual output to the console, perhaps), one can work around this annoyance in a couple ways: (1) “sympify” the literal before comparison, (2) use one of the wrappers, or (3) use the less succinct methods described above:

>>> e1 = Integer(1) > x
>>> e2 = Integer(1) >= x
>>> e3 = Integer(1) < x
>>> e4 = Integer(1) <= x
>>> e5 = Gt(1, x)
>>> e6 = Ge(1, x)
>>> e7 = Lt(1, x)
>>> e8 = Le(1, x)
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
1 > x     1 >= x
1 < x     1 <= x
1 > x     1 >= x
1 < x     1 <= x

The other gotcha is with chained inequalities. Occasionally, one may be tempted to write statements like:

>>> x < y < z
Traceback (most recent call last):
...
TypeError: symbolic boolean expression has no truth value.

Due to an implementation detail or decision of Python, to create a chained inequality, the only method currently available is to make use of And:

>>> And(x < y, y < z)
(x < y) & (y < z)

LessThan

class diofant.core.relational.LessThan(lhs, rhs=0, **options)[source]

Class representations of inequalities.

The *Than classes represent unequal relationships, where the left-hand side is generally bigger or smaller than the right-hand side. For example, the GreaterThan class represents an unequal relationship where the left-hand side is at least as big as the right side, if not bigger. In mathematical notation:

lhs >= rhs

In total, there are four *Than classes, to represent the four inequalities:

Class Name	Symbol
GreaterThan	(>=)
LessThan	(<=)
StrictGreaterThan	(>)
StrictLessThan	(<)

All classes take two arguments, lhs and rhs.

Signature Example	Math equivalent
GreaterThan(lhs, rhs)	lhs >= rhs
LessThan(lhs, rhs)	lhs <= rhs
StrictGreaterThan(lhs, rhs)	lhs > rhs
StrictLessThan(lhs, rhs)	lhs < rhs

In addition to the normal .lhs and .rhs of Relations, *Than inequality objects also have the .lts and .gts properties, which represent the “less than side” and “greater than side” of the operator. Use of .lts and .gts in an algorithm rather than .lhs and .rhs as an assumption of inequality direction will make more explicit the intent of a certain section of code, and will make it similarly more robust to client code changes:

>>> e = GreaterThan(x, 1)
>>> e
x >= 1
>>> f'{e.gts} >= {e.lts} is the same as {e.lts} <= {e.gts}'
'x >= 1 is the same as 1 <= x'

Examples

One generally does not instantiate these classes directly, but uses various convenience methods:

>>> e1 = Ge(x, 2)  # Ge is a convenience wrapper
>>> print(e1)
x >= 2

>>> print(f'{Ge(x, 2)}\n{Gt(x, 2)}\n{Le(x, 2)}\n{Lt(x, 2)}')
x >= 2
x > 2
x <= 2
x < 2

Another option is to use the Python inequality operators (>=, >, <=, <) directly. Their main advantage over the Ge, Gt, Le, and Lt counterparts, is that one can write a more “mathematical looking” statement rather than littering the math with oddball function calls. However there are certain (minor) caveats of which to be aware (search for ‘gotcha’, below).

>>> e2 = x >= 2
>>> print(e2)
x >= 2
>>> print(f'e1: {e1},    e2: {e2}')
e1: x >= 2,    e2: x >= 2
>>> e1 == e2
True

However, it is also perfectly valid to instantiate a *Than class less succinctly and less conveniently:

>>> print(f"{Rel(x, 1, '>=')}\n{Relational(x, 1, '>=')}\n{GreaterThan(x, 1)}")
x >= 1
x >= 1
x >= 1

>>> print(f"{Rel(x, 1, '>')}\n{Relational(x, 1, '>')}\n{StrictGreaterThan(x, 1)}")
x > 1
x > 1
x > 1

>>> print(f"{Rel(x, 1, '<=')}\n{Relational(x, 1, '<=')}\n{LessThan(x, 1)}")
x <= 1
x <= 1
x <= 1

>>> print(f"{Rel(x, 1, '<')}\n{Relational(x, 1, '<')}\n{StrictLessThan(x, 1)}")
x < 1
x < 1
x < 1

Notes

There are a couple of “gotchas” when using Python’s operators.

The first enters the mix when comparing against a literal number as the lhs argument. Due to the order that Python decides to parse a statement, it may not immediately find two objects comparable. For example, to evaluate the statement (1 < x), Python will first recognize the number 1 as a native number, and then that x is not a native number. At this point, because a native Python number does not know how to compare itself with a Diofant object Python will try the reflective operation, (x > 1). Unfortunately, there is no way available to Diofant to recognize this has happened, so the statement (1 < x) will turn silently into (x > 1).

>>> e1 = x > 1
>>> e2 = x >= 1
>>> e3 = x < 1
>>> e4 = x <= 1
>>> e5 = 1 > x
>>> e6 = 1 >= x
>>> e7 = 1 < x
>>> e8 = 1 <= x
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
x > 1     x >= 1
x < 1     x <= 1
x < 1     x <= 1
x > 1     x >= 1

If the order of the statement is important (for visual output to the console, perhaps), one can work around this annoyance in a couple ways: (1) “sympify” the literal before comparison, (2) use one of the wrappers, or (3) use the less succinct methods described above:

>>> e1 = Integer(1) > x
>>> e2 = Integer(1) >= x
>>> e3 = Integer(1) < x
>>> e4 = Integer(1) <= x
>>> e5 = Gt(1, x)
>>> e6 = Ge(1, x)
>>> e7 = Lt(1, x)
>>> e8 = Le(1, x)
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
1 > x     1 >= x
1 < x     1 <= x
1 > x     1 >= x
1 < x     1 <= x

The other gotcha is with chained inequalities. Occasionally, one may be tempted to write statements like:

>>> x < y < z
Traceback (most recent call last):
...
TypeError: symbolic boolean expression has no truth value.

Due to an implementation detail or decision of Python, to create a chained inequality, the only method currently available is to make use of And:

>>> And(x < y, y < z)
(x < y) & (y < z)

Unequality

class diofant.core.relational.Unequality(lhs, rhs=0, **options)[source]

An unequal relation between two objects.

Represents that two objects are not equal. If they can be shown to be definitively equal, this will reduce to False; if definitively unequal, this will reduce to True. Otherwise, the relation is maintained as an Unequality object.

Examples

>>> Ne(y, x+x**2)
Ne(y, x**2 + x)

See also

Equality

Notes

This class is not the same as the != operator. The != operator tests for exact structural equality between two expressions; this class compares expressions mathematically.

This class is effectively the inverse of Equality. As such, it uses the same algorithms, including any available \(_eval_Eq\) methods.

StrictGreaterThan

class diofant.core.relational.StrictGreaterThan(lhs, rhs=0, **options)[source]

Class representations of inequalities.

The *Than classes represent unequal relationships, where the left-hand side is generally bigger or smaller than the right-hand side. For example, the GreaterThan class represents an unequal relationship where the left-hand side is at least as big as the right side, if not bigger. In mathematical notation:

lhs >= rhs

In total, there are four *Than classes, to represent the four inequalities:

Class Name	Symbol
GreaterThan	(>=)
LessThan	(<=)
StrictGreaterThan	(>)
StrictLessThan	(<)

All classes take two arguments, lhs and rhs.

Signature Example	Math equivalent
GreaterThan(lhs, rhs)	lhs >= rhs
LessThan(lhs, rhs)	lhs <= rhs
StrictGreaterThan(lhs, rhs)	lhs > rhs
StrictLessThan(lhs, rhs)	lhs < rhs

In addition to the normal .lhs and .rhs of Relations, *Than inequality objects also have the .lts and .gts properties, which represent the “less than side” and “greater than side” of the operator. Use of .lts and .gts in an algorithm rather than .lhs and .rhs as an assumption of inequality direction will make more explicit the intent of a certain section of code, and will make it similarly more robust to client code changes:

>>> e = GreaterThan(x, 1)
>>> e
x >= 1
>>> f'{e.gts} >= {e.lts} is the same as {e.lts} <= {e.gts}'
'x >= 1 is the same as 1 <= x'

Examples

One generally does not instantiate these classes directly, but uses various convenience methods:

>>> e1 = Ge(x, 2)  # Ge is a convenience wrapper
>>> print(e1)
x >= 2

>>> print(f'{Ge(x, 2)}\n{Gt(x, 2)}\n{Le(x, 2)}\n{Lt(x, 2)}')
x >= 2
x > 2
x <= 2
x < 2

Another option is to use the Python inequality operators (>=, >, <=, <) directly. Their main advantage over the Ge, Gt, Le, and Lt counterparts, is that one can write a more “mathematical looking” statement rather than littering the math with oddball function calls. However there are certain (minor) caveats of which to be aware (search for ‘gotcha’, below).

>>> e2 = x >= 2
>>> print(e2)
x >= 2
>>> print(f'e1: {e1},    e2: {e2}')
e1: x >= 2,    e2: x >= 2
>>> e1 == e2
True

However, it is also perfectly valid to instantiate a *Than class less succinctly and less conveniently:

>>> print(f"{Rel(x, 1, '>=')}\n{Relational(x, 1, '>=')}\n{GreaterThan(x, 1)}")
x >= 1
x >= 1
x >= 1

>>> print(f"{Rel(x, 1, '>')}\n{Relational(x, 1, '>')}\n{StrictGreaterThan(x, 1)}")
x > 1
x > 1
x > 1

>>> print(f"{Rel(x, 1, '<=')}\n{Relational(x, 1, '<=')}\n{LessThan(x, 1)}")
x <= 1
x <= 1
x <= 1

>>> print(f"{Rel(x, 1, '<')}\n{Relational(x, 1, '<')}\n{StrictLessThan(x, 1)}")
x < 1
x < 1
x < 1

Notes

There are a couple of “gotchas” when using Python’s operators.

The first enters the mix when comparing against a literal number as the lhs argument. Due to the order that Python decides to parse a statement, it may not immediately find two objects comparable. For example, to evaluate the statement (1 < x), Python will first recognize the number 1 as a native number, and then that x is not a native number. At this point, because a native Python number does not know how to compare itself with a Diofant object Python will try the reflective operation, (x > 1). Unfortunately, there is no way available to Diofant to recognize this has happened, so the statement (1 < x) will turn silently into (x > 1).

>>> e1 = x > 1
>>> e2 = x >= 1
>>> e3 = x < 1
>>> e4 = x <= 1
>>> e5 = 1 > x
>>> e6 = 1 >= x
>>> e7 = 1 < x
>>> e8 = 1 <= x
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
x > 1     x >= 1
x < 1     x <= 1
x < 1     x <= 1
x > 1     x >= 1

If the order of the statement is important (for visual output to the console, perhaps), one can work around this annoyance in a couple ways: (1) “sympify” the literal before comparison, (2) use one of the wrappers, or (3) use the less succinct methods described above:

>>> e1 = Integer(1) > x
>>> e2 = Integer(1) >= x
>>> e3 = Integer(1) < x
>>> e4 = Integer(1) <= x
>>> e5 = Gt(1, x)
>>> e6 = Ge(1, x)
>>> e7 = Lt(1, x)
>>> e8 = Le(1, x)
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
1 > x     1 >= x
1 < x     1 <= x
1 > x     1 >= x
1 < x     1 <= x

The other gotcha is with chained inequalities. Occasionally, one may be tempted to write statements like:

>>> x < y < z
Traceback (most recent call last):
...
TypeError: symbolic boolean expression has no truth value.

Due to an implementation detail or decision of Python, to create a chained inequality, the only method currently available is to make use of And:

>>> And(x < y, y < z)
(x < y) & (y < z)

StrictLessThan

class diofant.core.relational.StrictLessThan(lhs, rhs=0, **options)[source]

Class representations of inequalities.

The *Than classes represent unequal relationships, where the left-hand side is generally bigger or smaller than the right-hand side. For example, the GreaterThan class represents an unequal relationship where the left-hand side is at least as big as the right side, if not bigger. In mathematical notation:

lhs >= rhs

In total, there are four *Than classes, to represent the four inequalities:

Class Name	Symbol
GreaterThan	(>=)
LessThan	(<=)
StrictGreaterThan	(>)
StrictLessThan	(<)

All classes take two arguments, lhs and rhs.

Signature Example	Math equivalent
GreaterThan(lhs, rhs)	lhs >= rhs
LessThan(lhs, rhs)	lhs <= rhs
StrictGreaterThan(lhs, rhs)	lhs > rhs
StrictLessThan(lhs, rhs)	lhs < rhs

In addition to the normal .lhs and .rhs of Relations, *Than inequality objects also have the .lts and .gts properties, which represent the “less than side” and “greater than side” of the operator. Use of .lts and .gts in an algorithm rather than .lhs and .rhs as an assumption of inequality direction will make more explicit the intent of a certain section of code, and will make it similarly more robust to client code changes:

>>> e = GreaterThan(x, 1)
>>> e
x >= 1
>>> f'{e.gts} >= {e.lts} is the same as {e.lts} <= {e.gts}'
'x >= 1 is the same as 1 <= x'

Examples

One generally does not instantiate these classes directly, but uses various convenience methods:

>>> e1 = Ge(x, 2)  # Ge is a convenience wrapper
>>> print(e1)
x >= 2

>>> print(f'{Ge(x, 2)}\n{Gt(x, 2)}\n{Le(x, 2)}\n{Lt(x, 2)}')
x >= 2
x > 2
x <= 2
x < 2

Another option is to use the Python inequality operators (>=, >, <=, <) directly. Their main advantage over the Ge, Gt, Le, and Lt counterparts, is that one can write a more “mathematical looking” statement rather than littering the math with oddball function calls. However there are certain (minor) caveats of which to be aware (search for ‘gotcha’, below).

>>> e2 = x >= 2
>>> print(e2)
x >= 2
>>> print(f'e1: {e1},    e2: {e2}')
e1: x >= 2,    e2: x >= 2
>>> e1 == e2
True

However, it is also perfectly valid to instantiate a *Than class less succinctly and less conveniently:

>>> print(f"{Rel(x, 1, '>=')}\n{Relational(x, 1, '>=')}\n{GreaterThan(x, 1)}")
x >= 1
x >= 1
x >= 1

>>> print(f"{Rel(x, 1, '>')}\n{Relational(x, 1, '>')}\n{StrictGreaterThan(x, 1)}")
x > 1
x > 1
x > 1

>>> print(f"{Rel(x, 1, '<=')}\n{Relational(x, 1, '<=')}\n{LessThan(x, 1)}")
x <= 1
x <= 1
x <= 1

>>> print(f"{Rel(x, 1, '<')}\n{Relational(x, 1, '<')}\n{StrictLessThan(x, 1)}")
x < 1
x < 1
x < 1

Notes

There are a couple of “gotchas” when using Python’s operators.

The first enters the mix when comparing against a literal number as the lhs argument. Due to the order that Python decides to parse a statement, it may not immediately find two objects comparable. For example, to evaluate the statement (1 < x), Python will first recognize the number 1 as a native number, and then that x is not a native number. At this point, because a native Python number does not know how to compare itself with a Diofant object Python will try the reflective operation, (x > 1). Unfortunately, there is no way available to Diofant to recognize this has happened, so the statement (1 < x) will turn silently into (x > 1).

>>> e1 = x > 1
>>> e2 = x >= 1
>>> e3 = x < 1
>>> e4 = x <= 1
>>> e5 = 1 > x
>>> e6 = 1 >= x
>>> e7 = 1 < x
>>> e8 = 1 <= x
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
x > 1     x >= 1
x < 1     x <= 1
x < 1     x <= 1
x > 1     x >= 1

If the order of the statement is important (for visual output to the console, perhaps), one can work around this annoyance in a couple ways: (1) “sympify” the literal before comparison, (2) use one of the wrappers, or (3) use the less succinct methods described above:

>>> e1 = Integer(1) > x
>>> e2 = Integer(1) >= x
>>> e3 = Integer(1) < x
>>> e4 = Integer(1) <= x
>>> e5 = Gt(1, x)
>>> e6 = Ge(1, x)
>>> e7 = Lt(1, x)
>>> e8 = Le(1, x)
>>> print('%s     %s\n'*4 % (e1, e2, e3, e4, e5, e6, e7, e8))
1 > x     1 >= x
1 < x     1 <= x
1 > x     1 >= x
1 < x     1 <= x

The other gotcha is with chained inequalities. Occasionally, one may be tempted to write statements like:

>>> x < y < z
Traceback (most recent call last):
...
TypeError: symbolic boolean expression has no truth value.

Due to an implementation detail or decision of Python, to create a chained inequality, the only method currently available is to make use of And:

>>> And(x < y, y < z)
(x < y) & (y < z)

multidimensional

vectorize

class diofant.core.multidimensional.vectorize(*mdargs)[source]

Generalizes a function taking scalars to accept multidimensional arguments.

Examples

>>> @vectorize(0)
... def vsin(x):
...     return sin(x)

>>> vsin([1, x, y])
[sin(1), sin(x), sin(y)]

>>> @vectorize(0, 1)
... def vdiff(f, y):
...     return diff(f, y)

>>> vdiff([f(x, y), g(x, y)], [x, y])
[[Derivative(f(x, y), x), Derivative(f(x, y), y)],
 [Derivative(g(x, y), x), Derivative(g(x, y), y)]]

function

Lambda

class diofant.core.function.Lambda(variables, expr)[source]

Lambda(x, expr) represents a lambda function similar to Python’s ‘lambda x: expr’. A function of several variables is written as Lambda((x, y, …), expr).

A simple example:

>>> f = Lambda(x, x**2)
>>> f(4)
16

For multivariate functions, use:

>>> f2 = Lambda((x, y, z, t), x + y**z + t**z)
>>> f2(1, 2, 3, 4)
73

A handy shortcut for lots of arguments:

>>> p = x, y, z
>>> f = Lambda(p, x + y*z)
>>> f(*p)
x + y*z

property expr: The return value of the function.

property free_symbols: Return from the atoms of self those which are free symbols.

See also

diofant.core.basic.Basic.free_symbols

property variables: The variables used in the internal representation of the function.

WildFunction

class diofant.core.function.WildFunction(*args)[source]

A WildFunction function matches any function (with its arguments).

Examples

>>> F = WildFunction('F')
>>> F.nargs
Naturals0()
>>> x.match(F)
>>> F.match(F)
{F_: F_}
>>> f(x).match(F)
{F_: f(x)}
>>> cos(x).match(F)
{F_: cos(x)}
>>> f(x, y).match(F)
{F_: f(x, y)}

To match functions with a given number of arguments, set nargs to the desired value at instantiation:

>>> F = WildFunction('F', nargs=2)
>>> F.nargs
{2}
>>> f(x).match(F)
>>> f(x, y).match(F)
{F_: f(x, y)}

To match functions with a range of arguments, set nargs to a tuple containing the desired number of arguments, e.g. if nargs = (1, 2) then functions with 1 or 2 arguments will be matched.

>>> F = WildFunction('F', nargs=(1, 2))
>>> F.nargs
{1, 2}
>>> f(x).match(F)
{F_: f(x)}
>>> f(x, y).match(F)
{F_: f(x, y)}
>>> f(x, y, 1).match(F)

Derivative

class diofant.core.function.Derivative(expr, *args, **assumptions)[source]

Carries out differentiation of the given expression with respect to symbols.

expr must define ._eval_derivative(symbol) method that returns the differentiation result. This function only needs to consider the non-trivial case where expr contains symbol and it should call the diff() method internally (not _eval_derivative); Derivative should be the only one to call _eval_derivative.

Simplification of high-order derivatives:

Because there can be a significant amount of simplification that can be done when multiple differentiations are performed, results will be automatically simplified in a fairly conservative fashion unless the keyword simplify is set to False.

>>> e = sqrt((x + 1)**2 + x)
>>> diff(e, (x, 5), simplify=False).count_ops()
136
>>> diff(e, (x, 5)).count_ops()
30

Ordering of variables:

If evaluate is set to True and the expression can not be evaluated, the list of differentiation symbols will be sorted, that is, the expression is assumed to have continuous derivatives up to the order asked. This sorting assumes that derivatives wrt Symbols commute, derivatives wrt non-Symbols commute, but Symbol and non-Symbol derivatives don’t commute with each other.

Derivative wrt non-Symbols:

This class also allows derivatives wrt non-Symbols that have _diff_wrt set to True, such as Function and Derivative. When a derivative wrt a non- Symbol is attempted, the non-Symbol is temporarily converted to a Symbol while the differentiation is performed.

Note that this may seem strange, that Derivative allows things like f(g(x)).diff(g(x)), or even f(cos(x)).diff(cos(x)). The motivation for allowing this syntax is to make it easier to work with variational calculus (i.e., the Euler-Lagrange method). The best way to understand this is that the action of derivative with respect to a non-Symbol is defined by the above description: the object is substituted for a Symbol and the derivative is taken with respect to that. This action is only allowed for objects for which this can be done unambiguously, for example Function and Derivative objects. Note that this leads to what may appear to be mathematically inconsistent results. For example:

>>> (2*cos(x)).diff(cos(x))
2
>>> (2*sqrt(1 - sin(x)**2)).diff(cos(x))
0

This appears wrong because in fact 2*cos(x) and 2*sqrt(1 - sin(x)**2) are identically equal. However this is the wrong way to think of this. Think of it instead as if we have something like this:

>>> from diofant.abc import s
>>> def f(u):
...     return 2*u
...
>>> def g(u):
...     return 2*sqrt(1 - u**2)
...
>>> f(cos(x))
2*cos(x)
>>> g(sin(x))
2*sqrt(-sin(x)**2 + 1)
>>> f(c).diff(c)
2
>>> f(c).diff(c)
2
>>> g(s).diff(c)
0
>>> g(sin(x)).diff(cos(x))
0

Here, the Symbols c and s act just like the functions cos(x) and sin(x), respectively. Think of 2*cos(x) as f(c).subs({c: cos(x)}) (or f(c) at c = cos(x)) and 2*sqrt(1 - sin(x)**2) as g(s).subs({s: sin(x)}) (or g(s) at s = sin(x)), where f(u) == 2*u and g(u) == 2*sqrt(1 - u**2). Here, we define the function first and evaluate it at the function, but we can actually unambiguously do this in reverse in Diofant, because expr.subs({Function: Symbol}) is well-defined: just structurally replace the function everywhere it appears in the expression.

This is the same notational convenience used in the Euler-Lagrange method when one says F(t, f(t), f’(t)).diff(f(t)). What is actually meant is that the expression in question is represented by some F(t, u, v) at u = f(t) and v = f’(t), and F(t, f(t), f’(t)).diff(f(t)) simply means F(t, u, v).diff(u) at u = f(t).

We do not allow derivatives to be taken with respect to expressions where this is not so well defined. For example, we do not allow expr.diff(x*y) because there are multiple ways of structurally defining where x*y appears in an expression, some of which may surprise the reader (for example, a very strict definition would have that (x*y*z).diff(x*y) == 0).

>>> (x*y*z).diff(x*y)
Traceback (most recent call last):
...
ValueError: Can't differentiate wrt the variable: x*y, 1

Note that this definition also fits in nicely with the definition of the chain rule. Note how the chain rule in Diofant is defined using unevaluated Subs objects:

>>> f, g = symbols('f g', cls=Function)
>>> f(2*g(x)).diff(x)
2*Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1, 2*g(x)))
>>> f(g(x)).diff(x)
Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1, g(x)))

Finally, note that, to be consistent with variational calculus, and to ensure that the definition of substituting a Function for a Symbol in an expression is well-defined, derivatives of functions are assumed to not be related to the function. In other words, we have:

>>> diff(f(x), x).diff(f(x))
0

The same is true for derivatives of different orders:

>>> diff(f(x), (x, 2)).diff(diff(f(x), (x, 1)))
0
>>> diff(f(x), (x, 1)).diff(diff(f(x), (x, 2)))
0

Note, any class can allow derivatives to be taken with respect to itself.

Examples

Some basic examples:

>>> Derivative(x**2, x, evaluate=True)
2*x
>>> Derivative(Derivative(f(x, y), x), y)
Derivative(f(x, y), x, y)
>>> Derivative(f(x), (x, 3))
Derivative(f(x), x, x, x)
>>> Derivative(f(x, y), y, x, evaluate=True)
Derivative(f(x, y), x, y)

Now some derivatives wrt functions:

>>> Derivative(f(x)**2, f(x), evaluate=True)
2*f(x)
>>> Derivative(f(g(x)), x, evaluate=True)
Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1, g(x)))

doit(**hints)[source]: Evaluate objects that are not evaluated by default.

See also

diofant.core.basic.Basic.doit

doit_numerically(z0)[source]

Evaluate the derivative at z numerically.

When we can represent derivatives at a point, this should be folded into the normal evalf. For now, we need a special method.

property expr: Return expression.

property free_symbols: Return from the atoms of self those which are free symbols.

See also

diofant.core.basic.Basic.free_symbols

property variables: Return tuple of symbols, wrt derivative is taken.

diff

diofant.core.function.diff(f, *args, **kwargs)[source]

Differentiate f with respect to symbols.

This is just a wrapper to unify .diff() and the Derivative class; its interface is similar to that of integrate(). You can use the same shortcuts for multiple variables as with Derivative. For example, diff(f(x), x, x, x) and diff(f(x), (x, 3)) both return the third derivative of f(x).

You can pass evaluate=False to get an unevaluated Derivative class. Note that if there are 0 symbols (such as diff(f(x), (x, 0)), then the result will be the function (the zeroth derivative), even if evaluate=False.

Examples

>>> diff(sin(x), x)
cos(x)
>>> diff(f(x), x, x, x)
Derivative(f(x), x, x, x)
>>> diff(f(x), (x, 3))
Derivative(f(x), x, x, x)
>>> diff(sin(x)*cos(y), (x, 2), (y, 2))
sin(x)*cos(y)

>>> type(diff(sin(x), x))
cos
>>> type(diff(sin(x), x, evaluate=False))
<class 'diofant.core.function.Derivative'>
>>> type(diff(sin(x), (x, 0)))
sin
>>> type(diff(sin(x), (x, 0), evaluate=False))
sin

>>> diff(sin(x))
cos(x)
>>> diff(sin(x*y))
Traceback (most recent call last):
...
ValueError: specify differentiation variables to differentiate sin(x*y)

Note that diff(sin(x)) syntax is meant only for convenience in interactive sessions and should be avoided in library code.

References

https://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html

FunctionClass

class diofant.core.function.FunctionClass(*args, **kwargs)[source]

Base class for function classes. FunctionClass is a subclass of type.

Use Function(‘<function name>’ [ , signature ]) to create undefined function classes.

property nargs

Return a set of the allowed number of arguments for the function.

Examples

If the function can take any number of arguments, the set of whole numbers is returned:

>>> Function('f').nargs
Naturals0()

If the function was initialized to accept one or more arguments, a corresponding set will be returned:

>>> Function('f', nargs=1).nargs
{1}
>>> Function('f', nargs=(2, 1)).nargs
{1, 2}

The undefined function, after application, also has the nargs attribute; the actual number of arguments is always available by checking the args attribute:

>>> f(1).nargs
Naturals0()
>>> len(f(1).args)
1

Function

class diofant.core.function.Function(*args)[source]

Base class for applied mathematical functions.

It also serves as a constructor for undefined function classes.

Examples

First example shows how to use Function as a constructor for undefined function classes:

>>> g = g(x)
>>> f
f
>>> f(x)
f(x)
>>> g
g(x)
>>> f(x).diff(x)
Derivative(f(x), x)
>>> g.diff(x)
Derivative(g(x), x)

In the following example Function is used as a base class for MyFunc that represents a mathematical function MyFunc. Suppose that it is well known, that MyFunc(0) is 1 and MyFunc at infinity goes to 0, so we want those two simplifications to occur automatically. Suppose also that MyFunc(x) is real exactly when x is real. Here is an implementation that honours those requirements:

>>> class MyFunc(Function):
...
...     @classmethod
...     def eval(cls, x):
...         if x.is_Number:
...             if x == 0:
...                 return Integer(1)
...             elif x is oo:
...                 return Integer(0)
...
...     def _eval_is_real(self):
...         return self.args[0].is_real
...
>>> MyFunc(0) + sin(0)
1
>>> MyFunc(oo)
0
>>> MyFunc(3.54).evalf()  # Not yet implemented for MyFunc.
MyFunc(3.54)
>>> MyFunc(I).is_real
False

In order for MyFunc to become useful, several other methods would need to be implemented. See source code of some of the already implemented functions for more complete examples.

Also, if the function can take more than one argument, then nargs must be defined, e.g. if MyFunc can take one or two arguments then,

>>> class MyFunc(Function):
...     nargs = (1, 2)
...
>>>

classmethod class_key()[source]: Nice order of classes.

fdiff(argindex=1)[source]: Returns the first derivative of the function.

Note

Not all functions are the same

Diofant defines many functions (like cos and factorial). It also allows the user to create generic functions which act as argument holders. Such functions are created just like symbols:

>>> f = Function('f')
>>> f(2) + f(x)
f(2) + f(x)

If you want to see which functions appear in an expression you can use the atoms method:

>>> e = (f(x) + cos(x) + 2)
>>> e.atoms(Function)
{f(x), cos(x)}

If you just want the function you defined, not Diofant functions, the thing to search for is AppliedUndef:

>>> from diofant.core.function import AppliedUndef
>>> e.atoms(AppliedUndef)
{f(x)}

Subs

class diofant.core.function.Subs(expr, *args, **assumptions)[source]

Represents unevaluated substitutions of an expression.

Subs receives at least 2 arguments: an expression, a pair of old and new expression to substitute or several such pairs.

Subs objects are generally useful to represent unevaluated derivatives calculated at a point.

The variables may be expressions, but they are subjected to the limitations of subs(), so it is usually a good practice to use only symbols for variables, since in that case there can be no ambiguity.

There’s no automatic expansion - use the method .doit() to effect all possible substitutions of the object and also of objects inside the expression.

When evaluating derivatives at a point that is not a symbol, a Subs object is returned. One is also able to calculate derivatives of Subs objects - in this case the expression is always expanded (for the unevaluated form, use Derivative()).

Examples

>>> e = Subs(f(x).diff(x), (x, y))
>>> e.subs({y: 0})
Subs(Derivative(f(x), x), (x, 0))
>>> e.subs({f: sin}).doit()
cos(y)

>>> Subs(f(x)*sin(y) + z, (x, 0), (y, 1))
Subs(z + f(x)*sin(y), (x, 0), (y, 1))
>>> _.doit()
z + f(0)*sin(1)

doit(**hints)[source]: Evaluate objects that are not evaluated by default.

See also

diofant.core.basic.Basic.doit

evalf(dps=15, **options)[source]: Evaluate the given formula to an accuracy of dps decimal digits.

See also

diofant.core.evalf.EvalfMixin.evalf

property expr: The expression on which the substitution operates.

property free_symbols: Return from the atoms of self those which are free symbols.

See also

diofant.core.basic.Basic.free_symbols

property point: The values for which the variables are to be substituted.

property variables: The variables to be evaluated.

expand

diofant.core.function.expand(e, deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints)[source]

Expand an expression using methods given as hints.

Hints evaluated unless explicitly set to False are: basic, log, multinomial, mul, power_base, and power_exp. The following hints are supported but not applied unless set to True: complex, func, and trig. In addition, the following meta-hints are supported by some or all of the other hints: frac, numer, denom, modulus, and force. deep is supported by all hints. Additionally, subclasses of Expr may define their own hints or meta-hints.

Parameters:

basic (boolean, optional) – This hint is used for any special rewriting of an object that should be done automatically (along with the other hints like mul) when expand is called. This is a catch-all hint to handle any sort of expansion that may not be described by the existing hint names.
deep (boolean, optional) – If deep is set to True (the default), things like arguments of functions are recursively expanded. Use deep=False to only expand on the top level.
mul (boolean, optional) – Distributes multiplication over addition (``):
```
>>> (y*(x + z)).expand(mul=True)
x*y + y*z
```

multinomial (boolean, optional) – Expand (x + y + …)**n where n is a positive integer.

>>> ((x + y + z)**2).expand(multinomial=True)
x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2

power_exp (boolean, optional) – Expand addition in exponents into multiplied bases.

>>> exp(x + y).expand(power_exp=True)
E**x*E**y
>>> (2**(x + y)).expand(power_exp=True)
2**x*2**y

power_base (boolean, optional) – Split powers of multiplied bases.

This only happens by default if assumptions allow, or if the force meta-hint is used:
```
>>> ((x*y)**z).expand(power_base=True)
(x*y)**z
>>> ((x*y)**z).expand(power_base=True, force=True)
x**z*y**z
>>> ((2*y)**z).expand(power_base=True)
2**z*y**z
```
Note that in some cases where this expansion always holds, Diofant performs it automatically:
```
>>> (x*y)**2
x**2*y**2
```
log (boolean, optional) – Pull out power of an argument as a coefficient and split logs products into sums of logs.

Note that these only work if the arguments of the log function have the proper assumptions–the arguments must be positive and the exponents must be real–or else the force hint must be True:
```
>>> log(x**2*y).expand(log=True)
log(x**2*y)
>>> log(x**2*y).expand(log=True, force=True)
2*log(x) + log(y)
>>> x, y = symbols('x y', positive=True)
>>> log(x**2*y).expand(log=True)
2*log(x) + log(y)
```
complex (boolean, optional) – Split an expression into real and imaginary parts.
```
>>> x, y = symbols('x y')
>>> (x + y).expand(complex=True)
re(x) + re(y) + I*im(x) + I*im(y)
>>> cos(x).expand(complex=True)
-I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))
```
Note that this is just a wrapper around as_real_imag(). Most objects that wish to redefine _eval_expand_complex() should consider redefining as_real_imag() instead.
func (boolean : optional) – Expand other functions.
```
>>> gamma(x + 1).expand(func=True)
x*gamma(x)
```
trig (boolean, optional) – Do trigonometric expansions.
```
>>> cos(x + y).expand(trig=True)
-sin(x)*sin(y) + cos(x)*cos(y)
>>> sin(2*x).expand(trig=True)
2*sin(x)*cos(x)
```
Note that the forms of sin(n*x) and cos(n*x) in terms of sin(x) and cos(x) are not unique, due to the identity \(\sin^2(x) + \cos^2(x) = 1\). The current implementation uses the form obtained from Chebyshev polynomials, but this may change.
force (boolean, optional) – If the force hint is used, assumptions about variables will be ignored in making the expansion.

Notes

You can shut off unwanted methods:

>>> (exp(x + y)*(x + y)).expand()
E**x*E**y*x + E**x*E**y*y
>>> (exp(x + y)*(x + y)).expand(power_exp=False)
E**(x + y)*x + E**(x + y)*y
>>> (exp(x + y)*(x + y)).expand(mul=False)
E**x*E**y*(x + y)

Use deep=False to only expand on the top level:

>>> exp(x + exp(x + y)).expand()
E**x*E**(E**x*E**y)
>>> exp(x + exp(x + y)).expand(deep=False)
E**(E**(x + y))*E**x

Hints are applied in an arbitrary, but consistent order (in the current implementation, they are applied in alphabetical order, except multinomial comes before mul, but this may change). Because of this, some hints may prevent expansion by other hints if they are applied first. For example, mul may distribute multiplications and prevent log and power_base from expanding them. Also, if mul is applied before multinomial, the expression might not be fully distributed. The solution is to use the various expand_hint helper functions or to use hint=False to this function to finely control which hints are applied. Here are some examples:
```
>>> x, y, z = symbols('x y z', positive=True)

>>> expand(log(x*(y + z)))
log(x) + log(y + z)
```
Here, we see that log was applied before mul. To get the mul expanded form, either of the following will work:
```
>>> expand_mul(log(x*(y + z)))
log(x*y + x*z)
>>> expand(log(x*(y + z)), log=False)
log(x*y + x*z)
```
A similar thing can happen with the power_base hint:
```
>>> expand((x*(y + z))**x)
(x*y + x*z)**x
```
To get the power_base expanded form, either of the following will work:
```
>>> expand((x*(y + z))**x, mul=False)
x**x*(y + z)**x
>>> expand_power_base((x*(y + z))**x)
x**x*(y + z)**x

>>> expand((x + y)*y/x)
y + y**2/x
```
The parts of a rational expression can be targeted:
```
>>> expand((x + y)*y/x/(x + 1), frac=True)
(x*y + y**2)/(x**2 + x)
>>> expand((x + y)*y/x/(x + 1), numer=True)
(x*y + y**2)/(x*(x + 1))
>>> expand((x + y)*y/x/(x + 1), denom=True)
y*(x + y)/(x**2 + x)
```

The modulus meta-hint can be used to reduce the coefficients of an expression post-expansion:

>>> expand((3*x + 1)**2)
9*x**2 + 6*x + 1
>>> expand((3*x + 1)**2, modulus=5)
4*x**2 + x + 1

Either expand() the function or .expand() the method can be used. Both are equivalent:

>>> expand((x + 1)**2)
x**2 + 2*x + 1
>>> ((x + 1)**2).expand()
x**2 + 2*x + 1

Objects can define their own expand hints by defining _eval_expand_hint(). The function should take the form:
```
def _eval_expand_hint(self, **hints):
    # Only apply the method to the top-level expression
    ...
```
See also the example below. Objects should define _eval_expand_hint() methods only if hint applies to that specific object. The generic _eval_expand_hint() method defined in Expr will handle the no-op case.

Each hint should be responsible for expanding that hint only. Furthermore, the expansion should be applied to the top-level expression only. expand() takes care of the recursion that happens when deep=True.

You should only call _eval_expand_hint() methods directly if you are 100% sure that the object has the method, as otherwise you are liable to get unexpected AttributeError’s. Note, again, that you do not need to recursively apply the hint to args of your object: this is handled automatically by expand(). _eval_expand_hint() should generally not be used at all outside of an _eval_expand_hint() method. If you want to apply a specific expansion from within another method, use the public expand() function, method, or expand_hint() functions.

In order for expand to work, objects must be rebuildable by their args, i.e., obj.func(*obj.args) == obj must hold.

Expand methods are passed **hints so that expand hints may use ‘metahints’–hints that control how different expand methods are applied. For example, the force=True hint described above that causes expand(log=True) to ignore assumptions is such a metahint. The deep meta-hint is handled exclusively by expand() and is not passed to _eval_expand_hint() methods.

Note that expansion hints should generally be methods that perform some kind of ‘expansion’. For hints that simply rewrite an expression, use the .rewrite() API.

Examples

>>> class MyClass(Expr):
...     def __new__(cls, *args):
...         args = sympify(args)
...         return Expr.__new__(cls, *args)
...
...     def _eval_expand_double(self, **hints):
...         # Doubles the args of MyClass.
...         # If there more than four args, doubling is not performed,
...         # unless force=True is also used (False by default).
...         force = hints.pop('force', False)
...         if not force and len(self.args) > 4:
...             return self
...         return self.func(*(self.args + self.args))
...
>>> a = MyClass(1, 2, MyClass(3, 4))
>>> a
MyClass(1, 2, MyClass(3, 4))
>>> a.expand(double=True)
MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4))
>>> a.expand(double=True, deep=False)
MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4))

>>> b = MyClass(1, 2, 3, 4, 5)
>>> b.expand(double=True)
MyClass(1, 2, 3, 4, 5)
>>> b.expand(double=True, force=True)
MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5)

References

https://mathworld.wolfram.com/Multiple-AngleFormulas.html

PoleError

class diofant.core.function.PoleError[source]: Raised when an expansion pole is encountered.

count_ops

diofant.core.function.count_ops(expr, visual=False)[source]

Return a representation (integer or expression) of the operations in expr.

If visual is False (default) then the sum of the coefficients of the visual expression will be returned.

If visual is True then the number of each type of operation is shown with the core class types (or their virtual equivalent) multiplied by the number of times they occur.

If expr is an iterable, the sum of the op counts of the items will be returned.

Examples

Although there isn’t a SUB object, minus signs are interpreted as either negations or subtractions:

>>> (x - y).count_ops(visual=True)
SUB
>>> (-x).count_ops(visual=True)
NEG

Here, there are two Adds and a Pow:

>>> (1 + a + b**2).count_ops(visual=True)
2*ADD + POW

In the following, an Add, Mul, Pow and two functions:

>>> (sin(x)*x + sin(x)**2).count_ops(visual=True)
ADD + MUL + POW + 2*SIN

for a total of 5:

>>> (sin(x)*x + sin(x)**2).count_ops(visual=False)
5

Note that “what you type” is not always what you get. The expression 1/x/y is translated by diofant into 1/(x*y) so it gives a DIV and MUL rather than two DIVs:

>>> (1/x/y).count_ops(visual=True)
DIV + MUL

The visual option can be used to demonstrate the difference in operations for expressions in different forms. Here, the Horner representation is compared with the expanded form of a polynomial:

>>> eq = x*(1 + x*(2 + x*(3 + x)))
>>> count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True)
-MUL + 3*POW

The count_ops function also handles iterables:

>>> count_ops([x, sin(x), None, True, x + 2], visual=False)
2
>>> count_ops([x, sin(x), None, True, x + 2], visual=True)
ADD + SIN
>>> count_ops({x: sin(x), x + 2: y + 1}, visual=True)
2*ADD + SIN

expand_mul

diofant.core.function.expand_mul(expr, deep=True)[source]

Wrapper around expand that only uses the mul hint. See the expand docstring for more information.

Examples

>>> x, y = symbols('x y', positive=True)
>>> expand_mul(exp(x+y)*(x+y)*log(x*y**2))
E**(x + y)*x*log(x*y**2) + E**(x + y)*y*log(x*y**2)

expand_log

diofant.core.function.expand_log(expr, deep=True, force=False)[source]

Wrapper around expand that only uses the log hint. See the expand docstring for more information.

Examples

>>> x, y = symbols('x y', positive=True)
>>> expand_log(exp(x+y)*(x+y)*log(x*y**2))
E**(x + y)*(x + y)*(log(x) + 2*log(y))

expand_func

diofant.core.function.expand_func(expr, deep=True)[source]

Wrapper around expand that only uses the func hint. See the expand docstring for more information.

Examples

>>> expand_func(gamma(x + 2))
x*(x + 1)*gamma(x)

expand_trig

diofant.core.function.expand_trig(expr, deep=True)[source]

Wrapper around expand that only uses the trig hint. See the expand docstring for more information.

Examples

>>> expand_trig(sin(x+y)*(x+y))
(x + y)*(sin(x)*cos(y) + sin(y)*cos(x))

expand_complex

diofant.core.function.expand_complex(expr, deep=True)[source]

Wrapper around expand that only uses the complex hint. See the expand docstring for more information.

Examples

>>> expand_complex(exp(z))
E**re(z)*I*sin(im(z)) + E**re(z)*cos(im(z))
>>> expand_complex(sqrt(I))
sqrt(2)/2 + sqrt(2)*I/2

See also

diofant.core.expr.Expr.as_real_imag

expand_multinomial

diofant.core.function.expand_multinomial(expr, deep=True)[source]

Wrapper around expand that only uses the multinomial hint. See the expand docstring for more information.

Examples

>>> x, y = symbols('x y', positive=True)
>>> expand_multinomial((x + exp(x + 1))**2)
2*E**(x + 1)*x + E**(2*x + 2) + x**2

expand_power_exp

diofant.core.function.expand_power_exp(expr, deep=True)[source]

Wrapper around expand that only uses the power_exp hint.

Examples

>>> expand_power_exp(x**(y + 2))
x**2*x**y

See also

expand

expand_power_base

diofant.core.function.expand_power_base(expr, deep=True, force=False)[source]

Wrapper around expand that only uses the power_base hint.

A wrapper to expand(power_base=True) which separates a power with a base that is a Mul into a product of powers, without performing any other expansions, provided that assumptions about the power’s base and exponent allow.

deep=False (default is True) will only apply to the top-level expression.

force=True (default is False) will cause the expansion to ignore assumptions about the base and exponent. When False, the expansion will only happen if the base is non-negative or the exponent is an integer.

>>> (x*y)**2
x**2*y**2

>>> (2*x)**y
(2*x)**y
>>> expand_power_base(_)
2**y*x**y

>>> expand_power_base((x*y)**z)
(x*y)**z
>>> expand_power_base((x*y)**z, force=True)
x**z*y**z
>>> expand_power_base(sin((x*y)**z), deep=False)
sin((x*y)**z)
>>> expand_power_base(sin((x*y)**z), force=True)
sin(x**z*y**z)

>>> expand_power_base((2*sin(x))**y + (2*cos(x))**y)
2**y*sin(x)**y + 2**y*cos(x)**y

>>> expand_power_base((2*exp(y))**x)
2**x*(E**y)**x

>>> expand_power_base((2*cos(x))**y)
2**y*cos(x)**y

Notice that sums are left untouched. If this is not the desired behavior, apply full expand() to the expression:

>>> expand_power_base(((x+y)*z)**2)
z**2*(x + y)**2
>>> (((x+y)*z)**2).expand()
x**2*z**2 + 2*x*y*z**2 + y**2*z**2

>>> expand_power_base((2*y)**(1+z))
2**(z + 1)*y**(z + 1)
>>> ((2*y)**(1+z)).expand()
2*2**z*y*y**z

See also

expand

nfloat

diofant.core.function.nfloat(expr, n=15, exponent=False)[source]

Make all Rationals in expr Floats except those in exponents (unless the exponents flag is set to True).

Examples

>>> nfloat(x**4 + x/2 + cos(pi/3) + 1 + sqrt(y))
x**4 + 0.5*x + sqrt(y) + 1.5
>>> nfloat(x**4 + sqrt(y), exponent=True)
x**4.0 + y**0.5

evalf

class diofant.core.evalf.EvalfMixin[source]

Mixin class adding evalf capability.

evalf(dps=15, subs=None, maxn=110, chop=False, strict=True, quad=None)[source]

Evaluate the given formula to an accuracy of dps decimal 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=110)

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=True)

quad=<str>
Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad=’osc’.

PrecisionExhausted

class diofant.core.evalf.PrecisionExhausted[source]: Raised when precision is exhausted.

N

diofant.core.evalf.N(x, dps=15, **options)[source]

Calls x.evalf(dps, **options).

Examples

>>> Sum(1/k**k, (k, 1, oo))
Sum(k**(-k), (k, 1, oo))
>>> N(_, 4)
1.291

See also

diofant.core.evalf.EvalfMixin.evalf

containers

Tuple

class diofant.core.containers.Tuple(*args)[source]

Wrapper around the builtin tuple object

The Tuple is a subclass of Basic, so that it works well in the Diofant framework. The wrapped tuple is available as self.args, but you can also access elements or slices with [:] syntax.

>>> Tuple(a, b, c)[1:]
(b, c)
>>> Tuple(a, b, c).subs({a: d})
(d, b, c)

index(value, start=None, stop=None)[source]

Return first index of value.

Raises ValueError if the value is not present.

tuple_count(value)[source]: T.count(value) -> int – return number of occurrences of value.

Dict

class diofant.core.containers.Dict(*args)[source]

Wrapper around the builtin dict object

The Dict is a subclass of Basic, so that it works well in the Diofant framework. Because it is immutable, it may be included in sets, but its values must all be given at instantiation and cannot be changed afterwards. Otherwise it behaves identically to the Python dict.

>>> D = Dict({1: 'one', 2: 'two'})
>>> for key in D:
...     if key == 1:
...         print(f'{key} {D[key]}')
1 one

The args are sympified so the 1 and 2 are Integers and the values are Symbols. Queries automatically sympify args so the following work:

>>> 1 in D
True
>>> D.has('one')  # searches keys and values
True
>>> 'one' in D  # not in the keys
False
>>> D[1]
one

property args: Returns a tuple of arguments of ‘self’.

See also

diofant.core.basic.Basic.args

get(key, default=None)[source]: Return the value for key if key is in the dictionary, else default.

items()[source]: Returns a set-like object providing a view on Dict’s items.

keys()[source]: Returns a set-like object providing a view on Dict’s keys.

values()[source]: Returns a set-like object providing a view on Dict’s values.

compatibility

Reimplementations of constructs introduced in later versions of Python than we support. Also some functions that are needed Diofant-wide and are located here for easy import.

class diofant.core.compatibility.NotIterable[source]: Use this as mixin when creating a class which is not supposed to return true when iterable() is called on its instances. I.e. avoid infinite loop when calling e.g. list() on the instance

diofant.core.compatibility.as_int(n)[source]

Convert the argument to a builtin integer.

The return value is guaranteed to be equal to the input. ValueError is raised if the input has a non-integral value.

Examples

>>> 3.0
3.0
>>> as_int(3.0)  # convert to int and test for equality
3
>>> int(sqrt(10))
3
>>> as_int(sqrt(10))
Traceback (most recent call last):
...
ValueError: ... is not an integer

diofant.core.compatibility.is_sequence(i, include=None)[source]

Return a boolean indicating whether i is a sequence in the Diofant sense. If anything that fails the test below should be included as being a sequence for your application, set ‘include’ to that object’s type; multiple types should be passed as a tuple of types.

Note: although generators can generate a sequence, they often need special handling to make sure their elements are captured before the generator is exhausted, so these are not included by default in the definition of a sequence.

See also

iterable

Examples

>>> from types import GeneratorType
>>> is_sequence([])
True
>>> is_sequence(set())
False
>>> is_sequence('abc')
False
>>> is_sequence('abc', include=str)
True
>>> generator = (c for c in 'abc')
>>> is_sequence(generator)
False
>>> is_sequence(generator, include=(str, GeneratorType))
True

diofant.core.compatibility.iterable(i, exclude=(<class 'str'>, <class 'dict'>, <class 'diofant.core.compatibility.NotIterable'>))[source]

Return a boolean indicating whether i is Diofant iterable. True also indicates that the iterator is finite, i.e. you e.g. call list(…) on the instance.

When Diofant is working with iterables, it is almost always assuming that the iterable is not a string or a mapping, so those are excluded by default. If you want a pure Python definition, make exclude=None. To exclude multiple items, pass them as a tuple.

See also

is_sequence

Examples

>>> things = [[1], (1,), {1}, Tuple(1), (j for j in [1, 2]), {1: 2}, '1', 1]
>>> for i in things:
...     print(f'{iterable(i)} {type(i)}')
True <... 'list'>
True <... 'tuple'>
True <... 'set'>
True <class 'diofant.core.containers.Tuple'>
True <... 'generator'>
False <... 'dict'>
False <... 'str'>
False <... 'int'>

>>> iterable({}, exclude=None)
True
>>> iterable({}, exclude=str)
True
>>> iterable('no', exclude=str)
False

exprtools

gcd_terms

diofant.core.exprtools.gcd_terms(terms, isprimitive=False, clear=True, fraction=True)[source]

Compute the GCD of terms and put them together.

terms can be an expression or a non-Basic sequence of expressions which will be handled as though they are terms from a sum.

If isprimitive is True the _gcd_terms will not run the primitive method on the terms.

clear controls the removal of integers from the denominator of an Add expression. When True (default), all numerical denominator will be cleared; when False the denominators will be cleared only if all terms had numerical denominators other than 1.

fraction, when True (default), will put the expression over a common denominator.

Examples

>>> gcd_terms((x + 1)**2*y + (x + 1)*y**2)
y*(x + 1)*(x + y + 1)
>>> gcd_terms(x/2 + 1)
(x + 2)/2
>>> gcd_terms(x/2 + 1, clear=False)
x/2 + 1
>>> gcd_terms(x/2 + y/2, clear=False)
(x + y)/2
>>> gcd_terms(x/2 + 1/x)
(x**2 + 2)/(2*x)
>>> gcd_terms(x/2 + 1/x, fraction=False)
(x + 2/x)/2
>>> gcd_terms(x/2 + 1/x, fraction=False, clear=False)
x/2 + 1/x

>>> gcd_terms(x/2/y + 1/x/y)
(x**2 + 2)/(2*x*y)
>>> gcd_terms(x/2/y + 1/x/y, fraction=False, clear=False)
(x + 2/x)/(2*y)

The clear flag was ignored in this case because the returned expression was a rational expression, not a simple sum.

factor_terms

diofant.core.exprtools.factor_terms(expr, radical=False, clear=False, fraction=False, sign=True)[source]

Remove common factors from terms in all arguments without changing the underlying structure of the expr. No expansion or simplification (and no processing of non-commutatives) is performed.

If radical=True then a radical common to all terms will be factored out of any Add sub-expressions of the expr.

If clear=False (default) then coefficients will not be separated from a single Add if they can be distributed to leave one or more terms with integer coefficients.

If fraction=True (default is False) then a common denominator will be constructed for the expression.

If sign=True (default) then even if the only factor in common is a -1, it will be factored out of the expression.

Examples

>>> factor_terms(x + x*(2 + 4*y)**3)
x*(8*(2*y + 1)**3 + 1)
>>> A = Symbol('A', commutative=False)
>>> factor_terms(x*A + x*A + x*y*A)
x*(y*A + 2*A)

When clear is False, a rational will only be factored out of an Add expression if all terms of the Add have coefficients that are fractions:

>>> factor_terms(x/2 + 1, clear=False)
x/2 + 1
>>> factor_terms(x/2 + 1, clear=True)
(x + 2)/2

This only applies when there is a single Add that the coefficient multiplies:

>>> factor_terms(x*y/2 + y, clear=True)
y*(x + 2)/2
>>> factor_terms(x*y/2 + y, clear=False) == _
True

If a -1 is all that can be factored out, to not factor it out, the flag sign must be False:

>>> factor_terms(-x - y)
-(x + y)
>>> factor_terms(-x - y, sign=False)
-x - y
>>> factor_terms(-2*x - 2*y, sign=False)
-2*(x + y)