Conversion of symbolic expressions to other types

This module provides routines for converting new symbolic expressions to other types. Primarily, it provides a class Converter which will walk the expression tree and make calls to methods overridden by subclasses.

class sage.symbolic.expression_conversions.AlgebraicConverter(field)

Bases: sage.symbolic.expression_conversions.Converter

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: a = AlgebraicConverter(QQbar)
sage: a.field
Algebraic Field
sage: a.reciprocal_trig_functions['cot']
tan
arithmetic(ex, operator)

Convert a symbolic expression to an algebraic number.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: f = 2^(1/2)
sage: a = AlgebraicConverter(QQbar)
sage: a.arithmetic(f, f.operator())
1.414213562373095?
composition(ex, operator)

Coerce to an algebraic number.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: a = AlgebraicConverter(QQbar)
sage: a.composition(exp(I*pi/3, hold=True), exp)
0.500000000000000? + 0.866025403784439?*I
sage: a.composition(sin(pi/7), sin)
0.4338837391175581? + 0.?e-18*I
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import AlgebraicConverter
sage: a = AlgebraicConverter(QQbar)
sage: f = SR(2)
sage: a.pyobject(f, f.pyobject())
2
sage: _.parent()
Algebraic Field
class sage.symbolic.expression_conversions.Converter(use_fake_div=False)

Bases: object

If use_fake_div is set to True, then the converter will try to replace expressions whose operator is operator.mul with the corresponding expression whose operator is operator.truediv.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import Converter
sage: c = Converter(use_fake_div=True)
sage: c.use_fake_div
True
arithmetic(ex, operator)

The input to this method is a symbolic expression and the infix operator corresponding to that expression. Typically, one will convert all of the arguments and then perform the operation afterward.

composition(ex, operator)

The input to this method is a symbolic expression and its operator. This method will get called when you have a symbolic function application.

derivative(ex, operator)

The input to this method is a symbolic expression which corresponds to a relation.

get_fake_div(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import Converter
sage: c = Converter(use_fake_div=True)
sage: c.get_fake_div(sin(x)/x)
FakeExpression([sin(x), x], <built-in function truediv>)
sage: c.get_fake_div(-1*sin(x))
FakeExpression([sin(x)], <built-in function neg>)
sage: c.get_fake_div(-x)
FakeExpression([x], <built-in function neg>)
sage: c.get_fake_div((2*x^3+2*x-1)/((x-2)*(x+1)))
FakeExpression([2*x^3 + 2*x - 1, FakeExpression([x + 1, x - 2], <built-in function mul>)], <built-in function truediv>)

Check if trac ticket #8056 is fixed, i.e., if numerator is 1.:

sage: c.get_fake_div(1/pi/x)
FakeExpression([1, FakeExpression([pi, x], <built-in function mul>)], <built-in function truediv>)
pyobject(ex, obj)

The input to this method is the result of calling pyobject() on a symbolic expression.

Note

Note that if a constant such as pi is encountered in the expression tree, its corresponding pyobject which is an instance of sage.symbolic.constants.Pi will be passed into this method. One cannot do arithmetic using such an object.

relation(ex, operator)

The input to this method is a symbolic expression which corresponds to a relation.

symbol(ex)

The input to this method is a symbolic expression which corresponds to a single variable. For example, this method could be used to return a generator for a polynomial ring.

class sage.symbolic.expression_conversions.ExpressionTreeWalker(ex)

Bases: sage.symbolic.expression_conversions.Converter

A class that walks the tree. Mainly for subclassing.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: from sage.symbolic.random_tests import random_expr
sage: ex = sin(atan(0,hold=True)+hypergeometric((1,),(1,),x))
sage: s = ExpressionTreeWalker(ex)
sage: bool(s() == ex)
True
sage: foo = random_expr(20, nvars=2)
sage: s = ExpressionTreeWalker(foo)
sage: bool(s() == foo)
True
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: foo = function('foo')
sage: f = x*foo(x) + pi/foo(x)
sage: s = ExpressionTreeWalker(f)
sage: bool(s.arithmetic(f, f.operator()) == f)
True
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: foo = function('foo')
sage: f = foo(atan2(0, 0, hold=True))
sage: s = ExpressionTreeWalker(f)
sage: bool(s.composition(f, f.operator()) == f)
True
derivative(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: foo = function('foo')
sage: f = foo(x).diff(x)
sage: s = ExpressionTreeWalker(f)
sage: bool(s.derivative(f, f.operator()) == f)
True
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: f = SR(2)
sage: s = ExpressionTreeWalker(f)
sage: bool(s.pyobject(f, f.pyobject()) == f.pyobject())
True
relation(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: foo = function('foo')
sage: eq = foo(x) == x
sage: s = ExpressionTreeWalker(eq)
sage: s.relation(eq, eq.operator()) == eq
True
symbol(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: s = ExpressionTreeWalker(x)
sage: bool(s.symbol(x) == x)
True
tuple(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import ExpressionTreeWalker
sage: foo = function('foo')
sage: f = hypergeometric((1,2,3,),(x,),x)
sage: s = ExpressionTreeWalker(f)
sage: bool(s() == f)
True
class sage.symbolic.expression_conversions.FakeExpression(operands, operator)

Bases: object

Pynac represents \(x/y\) as \(xy^{-1}\). Often, tree-walkers would prefer to see divisions instead of multiplications and negative exponents. To allow for this (since Pynac internally doesn’t have division at all), there is a possibility to pass use_fake_div=True; this will rewrite an Expression into a mixture of Expression and FakeExpression nodes, where the FakeExpression nodes are used to represent divisions. These nodes are intended to act sufficiently like Expression nodes that tree-walkers won’t care about the difference.

operands()

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FakeExpression
sage: import operator; x,y = var('x,y')
sage: f = FakeExpression([x, y], operator.truediv)
sage: f.operands()
[x, y]
operator()

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FakeExpression
sage: import operator; x,y = var('x,y')
sage: f = FakeExpression([x, y], operator.truediv)
sage: f.operator()
<built-in function truediv>
pyobject()

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FakeExpression
sage: import operator; x,y = var('x,y')
sage: f = FakeExpression([x, y], operator.truediv)
sage: f.pyobject()
Traceback (most recent call last):
...
TypeError: self must be a numeric expression
class sage.symbolic.expression_conversions.FastCallableConverter(ex, etb)

Bases: sage.symbolic.expression_conversions.Converter

EXAMPLES:

sage: from sage.symbolic.expression_conversions import FastCallableConverter
sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x'])
sage: f = FastCallableConverter(x+2, etb)
sage: f.ex
x + 2
sage: f.etb
<sage.ext.fast_callable.ExpressionTreeBuilder object at 0x...>
sage: f.use_fake_div
True
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: var('x,y')
(x, y)
sage: (x+y)._fast_callable_(etb)
add(v_0, v_1)
sage: (-x)._fast_callable_(etb)
neg(v_0)
sage: (x+y+x^2)._fast_callable_(etb)
add(add(ipow(v_0, 2), v_0), v_1)
composition(ex, function)

Given an ExpressionTreeBuilder, return an Expression representing this value.

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: x,y = var('x,y')
sage: sin(sqrt(x+y))._fast_callable_(etb)
sin(sqrt(add(v_0, v_1)))
sage: arctan2(x,y)._fast_callable_(etb)
{arctan2}(v_0, v_1)
pyobject(ex, obj)

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x'])
sage: pi._fast_callable_(etb)
pi
sage: etb = ExpressionTreeBuilder(vars=['x'], domain=RDF)
sage: pi._fast_callable_(etb)
3.141592653589793
relation(ex, operator)

EXAMPLES:

sage: ff = fast_callable(x == 2, vars=['x'])
sage: ff(2)
0
sage: ff(4)
2
sage: ff = fast_callable(x < 2, vars=['x'])
Traceback (most recent call last):
...
NotImplementedError
symbol(ex)

Given an ExpressionTreeBuilder, return an Expression representing this value.

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: x, y, z = var('x,y,z')
sage: x._fast_callable_(etb)
v_0
sage: y._fast_callable_(etb)
v_1
sage: z._fast_callable_(etb)
Traceback (most recent call last):
...
ValueError: Variable 'z' not found
tuple(ex)

Given a symbolic tuple, return its elements as a Python list.

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x'])
sage: SR._force_pyobject((2, 3, x^2))._fast_callable_(etb)
[2, 3, x^2]
class sage.symbolic.expression_conversions.FastFloatConverter(ex, *vars)

Bases: sage.symbolic.expression_conversions.Converter

Returns an object which provides fast floating point evaluation of the symbolic expression ex. This is an class used internally and is not meant to be used directly.

See sage.ext.fast_eval for more information.

EXAMPLES:

sage: x,y,z = var('x,y,z')
sage: f = 1 + sin(x)/x + sqrt(z^2+y^2)/cosh(x)
sage: ff = f._fast_float_('x', 'y', 'z')
sage: f(x=1.0,y=2.0,z=3.0).n()
4.1780638977...
sage: ff(1.0,2.0,3.0)
4.1780638977...

Using _fast_float_ without specifying the variable names is deprecated:

sage: f = x._fast_float_()
doctest:...: DeprecationWarning: Substitution using
function-call syntax and unnamed arguments is deprecated
and will be removed from a future release of Sage; you
can use named arguments instead, like EXPR(x=..., y=...)
See http://trac.sagemath.org/5930 for details.
sage: f(1.2)
1.2

Using _fast_float_ on a function which is the identity is Using _fast_float_ on a function which is the identity is now supported (see trac ticket #10246):

sage: f = symbolic_expression(x).function(x)
sage: f._fast_float_(x)
<sage.ext.fast_eval.FastDoubleFunc object at ...>
sage: f(22)
22
arithmetic(ex, operator)

EXAMPLES:

sage: x,y = var('x,y')
sage: f = x*x-y
sage: ff = f._fast_float_('x','y')
sage: ff(2,3)
1.0

sage: a = x + 2*y
sage: f = a._fast_float_('x', 'y')
sage: f(1,0)
1.0
sage: f(0,1)
2.0

sage: f = sqrt(x)._fast_float_('x'); f.op_list()
['load 0', 'call sqrt(1)']

sage: f = (1/2*x)._fast_float_('x'); f.op_list()
['load 0', 'push 0.5', 'mul']
composition(ex, operator)

EXAMPLES:

sage: f = sqrt(x)._fast_float_('x')
sage: f(2)
1.41421356237309...
sage: y = var('y')
sage: f = sqrt(x+y)._fast_float_('x', 'y')
sage: f(1,1)
1.41421356237309...
sage: f = sqrt(x+2*y)._fast_float_('x', 'y')
sage: f(2,0)
1.41421356237309...
sage: f(0,1)
1.41421356237309...
pyobject(ex, obj)

EXAMPLES:

sage: f = SR(2)._fast_float_()
sage: f(3)
2.0
relation(ex, operator)

EXAMPLES:

sage: ff = fast_float(x == 2, 'x')
sage: ff(2)
0.0
sage: ff(4)
2.0
sage: ff = fast_float(x < 2, 'x')
Traceback (most recent call last):
...
NotImplementedError
symbol(ex)

EXAMPLES:

sage: f = x._fast_float_('x', 'y')
sage: f(1,2)
1.0
sage: f = x._fast_float_('y', 'x')
sage: f(1,2)
2.0
class sage.symbolic.expression_conversions.FriCASConverter

Bases: sage.symbolic.expression_conversions.InterfaceInit

Converts any expression to FriCAS.

EXAMPLES:

sage: var('x,y')
(x, y)
sage: f = exp(x^2) - arcsin(pi+x)/y
sage: f._fricas_()                                                      # optional - fricas
     2
    x
y %e   - asin(x + %pi)
----------------------
           y
derivative(ex, operator)

Convert the derivative of self in FriCAS.

INPUT:

  • ex – a symbolic expression
  • operator – operator

EXAMPLES:

sage: var('x,y,z')
(x, y, z)
sage: f = function("F")
sage: f(x)._fricas_()                                               # optional - fricas
F(x)
sage: diff(f(x,y,z), x, z, x)._fricas_()                            # optional - fricas
F      (x,y,z)
 ,1,1,3

Check that trac ticket #25838 is fixed:

sage: var('x')
x
sage: F = function('F')
sage: integrate(F(x), x, algorithm="fricas")                        # optional - fricas
integral(F(x), x)

sage: integrate(diff(F(x), x)*sin(F(x)), x, algorithm="fricas")     # optional - fricas
-cos(F(x))
class sage.symbolic.expression_conversions.HoldRemover(ex, exclude=None)

Bases: sage.symbolic.expression_conversions.ExpressionTreeWalker

A class that walks the tree and evaluates every operator that is not in a given list of exceptions.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import HoldRemover
sage: ex = sin(pi*cos(0, hold=True), hold=True); ex
sin(pi*cos(0))
sage: h = HoldRemover(ex)
sage: h()
0
sage: h = HoldRemover(ex, [sin])
sage: h()
sin(pi)
sage: h = HoldRemover(ex, [cos])
sage: h()
sin(pi*cos(0))
sage: ex = atan2(0, 0, hold=True) + hypergeometric([1,2], [3,4], 0, hold=True)
sage: h = HoldRemover(ex, [atan2])
sage: h()
arctan2(0, 0) + 1
sage: h = HoldRemover(ex, [hypergeometric])
sage: h()
NaN + hypergeometric((1, 2), (3, 4), 0)
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import HoldRemover
sage: ex = sin(pi*cos(0, hold=True), hold=True); ex
sin(pi*cos(0))
sage: h = HoldRemover(ex)
sage: h()
0
class sage.symbolic.expression_conversions.InterfaceInit(interface)

Bases: sage.symbolic.expression_conversions.Converter

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: a = pi + 2
sage: m(a)
'(%pi)+(2)'
sage: m(sin(a))
'sin((%pi)+(2))'
sage: m(exp(x^2) + pi + 2)
'(%pi)+(exp((_SAGE_VAR_x)^(2)))+(2)'
arithmetic(ex, operator)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.arithmetic(x+2, sage.symbolic.operators.add_vararg)
'(_SAGE_VAR_x)+(2)'
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.composition(sin(x), sin)
'sin(_SAGE_VAR_x)'
sage: m.composition(ceil(x), ceil)
'ceiling(_SAGE_VAR_x)'

sage: m = InterfaceInit(mathematica)
sage: m.composition(sin(x), sin)
'Sin[x]'
derivative(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: f = function('f')
sage: a = f(x).diff(x); a
diff(f(x), x)
sage: print(m.derivative(a, a.operator()))
diff('f(_SAGE_VAR_x), _SAGE_VAR_x, 1)
sage: b = f(x).diff(x, x)
sage: print(m.derivative(b, b.operator()))
diff('f(_SAGE_VAR_x), _SAGE_VAR_x, 2)

We can also convert expressions where the argument is not just a variable, but the result is an “at” expression using temporary variables:

sage: y = var('y')
sage: t = (f(x*y).diff(x))/y
sage: t
D[0](f)(x*y)
sage: m.derivative(t, t.operator())
"at(diff('f(_SAGE_VAR_t0), _SAGE_VAR_t0, 1), [_SAGE_VAR_t0 = (_SAGE_VAR_x)*(_SAGE_VAR_y)])"
sage: f = function('f')(x)
sage: df = f.diff(x); df
diff(f(x), x)
sage: maxima(df)
'diff('f(_SAGE_VAR_x),_SAGE_VAR_x,1)
sage: a = df.subs(x=exp(x)); a
D[0](f)(e^x)
sage: b = maxima(a); b
%at('diff('f(_SAGE_VAR_t0),_SAGE_VAR_t0,1),_SAGE_VAR_t0=%e^_SAGE_VAR_x)
sage: bool(b.sage() == a)
True
sage: a = df.subs(x=4); a
D[0](f)(4)
sage: b = maxima(a); b
%at('diff('f(_SAGE_VAR_t0),_SAGE_VAR_t0,1),_SAGE_VAR_t0=4)
sage: bool(b.sage() == a)
True

It also works with more than one variable. Note the preferred syntax function('f')(x, y) to create a general symbolic function of more than one variable:

sage: x, y = var('x y')
sage: f = function('f')(x, y)
sage: f_x = f.diff(x); f_x
diff(f(x, y), x)
sage: maxima(f_x)
'diff('f(_SAGE_VAR_x,_SAGE_VAR_y),_SAGE_VAR_x,1)
sage: a = f_x.subs(x=4); a
D[0](f)(4, y)
sage: b = maxima(a); b
%at('diff('f(_SAGE_VAR_t0,_SAGE_VAR_y),_SAGE_VAR_t0,1),_SAGE_VAR_t0=4)
sage: bool(b.sage() == a)
True
sage: a = f_x.subs(x=4).subs(y=8); a
D[0](f)(4, 8)
sage: b = maxima(a); b
%at('diff('f(_SAGE_VAR_t0,8),_SAGE_VAR_t0,1),_SAGE_VAR_t0=4)
sage: bool(b.sage() == a)
True

Test a special case (trac ticket #16697):

sage: x,y = var('x,y')
sage: (gamma_inc(x,y).diff(x))
diff(gamma(x, y), x)
sage: (gamma_inc(x,x+1).diff(x)).simplify()
-(x + 1)^(x - 1)*e^(-x - 1) + D[0](gamma)(x, x + 1)
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: ii = InterfaceInit(gp)
sage: f = 2+I
sage: ii.pyobject(f, f.pyobject())
'I + 2'

sage: ii.pyobject(SR(2), 2)
'2'

sage: ii.pyobject(pi, pi.pyobject())
'Pi'
relation(ex, operator)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.relation(x==3, operator.eq)
'_SAGE_VAR_x = 3'
sage: m.relation(x==3, operator.lt)
'_SAGE_VAR_x < 3'
symbol(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: m.symbol(x)
'_SAGE_VAR_x'
sage: f(x) = x
sage: m.symbol(f)
'_SAGE_VAR_x'
sage: ii = InterfaceInit(gp)
sage: ii.symbol(x)
'x'
tuple(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import InterfaceInit
sage: m = InterfaceInit(maxima)
sage: t = SR._force_pyobject((3, 4, e^x))
sage: m.tuple(t)
'[3,4,exp(_SAGE_VAR_x)]'
class sage.symbolic.expression_conversions.LaurentPolynomialConverter(ex, base_ring=None, ring=None)

Bases: sage.symbolic.expression_conversions.PolynomialConverter

A converter from symbolic expressions to Laurent polynomials.

See laurent_polynomial() for details.

class sage.symbolic.expression_conversions.PolynomialConverter(ex, base_ring=None, ring=None)

Bases: sage.symbolic.expression_conversions.Converter

A converter from symbolic expressions to polynomials.

See polynomial() for details.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: x, y = var('x,y')
sage: p = PolynomialConverter(x+y, base_ring=QQ)
sage: p.base_ring
Rational Field
sage: p.ring
Multivariate Polynomial Ring in x, y over Rational Field

sage: p = PolynomialConverter(x, base_ring=QQ)
sage: p.base_ring
Rational Field
sage: p.ring
Univariate Polynomial Ring in x over Rational Field

sage: p = PolynomialConverter(x, ring=QQ['x,y'])
sage: p.base_ring
Rational Field
sage: p.ring
Multivariate Polynomial Ring in x, y over Rational Field

sage: p = PolynomialConverter(x+y, ring=QQ['x'])
Traceback (most recent call last):
...
TypeError: y is not a variable of Univariate Polynomial Ring in x over Rational Field
arithmetic(ex, operator)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import PolynomialConverter

sage: x, y = var('x, y')
sage: p = PolynomialConverter(x, base_ring=RR)
sage: p.arithmetic(pi+e, operator.add)
5.85987448204884
sage: p.arithmetic(x^2, operator.pow)
x^2

sage: p = PolynomialConverter(x+y, base_ring=RR)
sage: p.arithmetic(x*y+y^2, operator.add)
x*y + y^2

sage: p = PolynomialConverter(y^(3/2), ring=SR['x'])
sage: p.arithmetic(y^(3/2), operator.pow)
y^(3/2)
sage: _.parent()
Symbolic Ring
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: a = sin(2)
sage: p = PolynomialConverter(a*x, base_ring=RR)
sage: p.composition(a, a.operator())
0.909297426825682
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: p = PolynomialConverter(x, base_ring=QQ)
sage: f = SR(2)
sage: p.pyobject(f, f.pyobject())
2
sage: _.parent()
Rational Field
relation(ex, op)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import PolynomialConverter

sage: x, y = var('x, y')
sage: p = PolynomialConverter(x, base_ring=RR)

sage: p.relation(x==3, operator.eq)
x - 3.00000000000000
sage: p.relation(x==3, operator.lt)
Traceback (most recent call last):
...
ValueError: Unable to represent as a polynomial

sage: p = PolynomialConverter(x - y, base_ring=QQ)
sage: p.relation(x^2 - y^3 + 1 == x^3, operator.eq)
-x^3 - y^3 + x^2 + 1
symbol(ex)

Returns a variable in the polynomial ring.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import PolynomialConverter
sage: p = PolynomialConverter(x, base_ring=QQ)
sage: p.symbol(x)
x
sage: _.parent()
Univariate Polynomial Ring in x over Rational Field
sage: y = var('y')
sage: p = PolynomialConverter(x*y, ring=SR['x'])
sage: p.symbol(y)
y
class sage.symbolic.expression_conversions.RingConverter(R, subs_dict=None)

Bases: sage.symbolic.expression_conversions.Converter

A class to convert expressions to other rings.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF, subs_dict={x:2})
sage: R.ring
Real Interval Field with 53 bits of precision
sage: R.subs_dict
{x: 2}
sage: R(pi+e)
5.85987448204884?
sage: loads(dumps(R))
<sage.symbolic.expression_conversions.RingConverter object at 0x...>
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: P.<z> = ZZ[]
sage: R = RingConverter(P, subs_dict={x:z})
sage: a = 2*x^2 + x + 3
sage: R(a)
2*z^2 + z + 3
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF)
sage: R(cos(2))
-0.4161468365471424?
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF)
sage: R(SR(5/2))
2.5000000000000000?
symbol(ex)

All symbols appearing in the expression must either appear in subs_dict or be convertable by the ring’s element constructor in order for the conversion to be successful.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import RingConverter
sage: R = RingConverter(RIF, subs_dict={x:2})
sage: R(x+pi)
5.141592653589794?

sage: R = RingConverter(RIF)
sage: R(x+pi)
Traceback (most recent call last):
...
TypeError: unable to simplify to a real interval approximation

sage: R = RingConverter(QQ['x'])
sage: R(x^2+x)
x^2 + x
sage: R(x^2+x).parent()
Univariate Polynomial Ring in x over Rational Field
class sage.symbolic.expression_conversions.SubstituteFunction(ex, original, new)

Bases: sage.symbolic.expression_conversions.ExpressionTreeWalker

A class that walks the tree and replaces occurrences of a function with another.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: s(1/foo(foo(x)) + foo(2))
1/bar(bar(x)) + bar(2)
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: f = foo(x)
sage: s.composition(f, f.operator())
bar(x)
sage: f = foo(foo(x))
sage: s.composition(f, f.operator())
bar(bar(x))
sage: f = sin(foo(x))
sage: s.composition(f, f.operator())
sin(bar(x))
sage: f = foo(sin(x))
sage: s.composition(f, f.operator())
bar(sin(x))
derivative(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SubstituteFunction
sage: foo = function('foo'); bar = function('bar')
sage: s = SubstituteFunction(foo(x), foo, bar)
sage: f = foo(x).diff(x)
sage: s.derivative(f, f.operator())
diff(bar(x), x)
class sage.symbolic.expression_conversions.SympyConverter

Bases: sage.symbolic.expression_conversions.Converter

Converts any expression to SymPy.

EXAMPLES:

sage: import sympy
sage: var('x,y')
(x, y)
sage: f = exp(x^2) - arcsin(pi+x)/y
sage: f._sympy_()
exp(x**2) - asin(x + pi)/y
sage: _._sage_()
-arcsin(pi + x)/y + e^(x^2)

sage: sympy.sympify(x) # indirect doctest
x
arithmetic(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: f = x + 2
sage: s.arithmetic(f, f.operator())
x + 2
composition(ex, operator)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: f = sin(2)
sage: s.composition(f, f.operator())
sin(2)
sage: type(_)
sin
sage: f = arcsin(2)
sage: s.composition(f, f.operator())
asin(2)
derivative(ex, operator)

Convert the derivative of self in sympy.

INPUT:

  • ex – a symbolic expression
  • operator – operator
pyobject(ex, obj)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: f = SR(2)
sage: s.pyobject(f, f.pyobject())
2
sage: type(_)
<class 'sympy.core.numbers.Integer'>
relation(ex, op)

EXAMPLES:

sage: import operator
sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: s.relation(x == 3, operator.eq)
Eq(x, 3)
sage: s.relation(pi < 3, operator.lt)
pi < 3
sage: s.relation(x != pi, operator.ne)
Ne(x, pi)
sage: s.relation(x > 0, operator.gt)
x > 0
symbol(ex)

EXAMPLES:

sage: from sage.symbolic.expression_conversions import SympyConverter
sage: s = SympyConverter()
sage: s.symbol(x)
x
sage: type(_)
<class 'sympy.core.symbol.Symbol'>
sage.symbolic.expression_conversions.algebraic(ex, field)

Returns the symbolic expression ex as a element of the algebraic field field.

EXAMPLES:

sage: a = SR(5/6)
sage: AA(a)
5/6
sage: type(AA(a))
<class 'sage.rings.qqbar.AlgebraicReal'>
sage: QQbar(a)
5/6
sage: type(QQbar(a))
<class 'sage.rings.qqbar.AlgebraicNumber'>
sage: QQbar(i)
I
sage: AA(golden_ratio)
1.618033988749895?
sage: QQbar(golden_ratio)
1.618033988749895?
sage: QQbar(sin(pi/3))
0.866025403784439?

sage: QQbar(sqrt(2) + sqrt(8))
4.242640687119285?
sage: AA(sqrt(2) ^ 4) == 4
True
sage: AA(-golden_ratio)
-1.618033988749895?
sage: QQbar((2*I)^(1/2))
1 + 1*I
sage: QQbar(e^(pi*I/3))
0.50000000000000000? + 0.866025403784439?*I

sage: AA(x*sin(0))
0
sage: QQbar(x*sin(0))
0
sage.symbolic.expression_conversions.fast_callable(ex, etb)

Given an ExpressionTreeBuilder etb, return an Expression representing the symbolic expression ex.

EXAMPLES:

sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: x,y = var('x,y')
sage: f = y+2*x^2
sage: f._fast_callable_(etb)
add(mul(ipow(v_0, 2), 2), v_1)

sage: f = (2*x^3+2*x-1)/((x-2)*(x+1))
sage: f._fast_callable_(etb)
div(add(add(mul(ipow(v_0, 3), 2), mul(v_0, 2)), -1), mul(add(v_0, 1), add(v_0, -2)))
sage.symbolic.expression_conversions.fast_float(ex, *vars)

Returns an object which provides fast floating point evaluation of the symbolic expression ex.

See sage.ext.fast_eval for more information.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import fast_float
sage: f = sqrt(x+1)
sage: ff = fast_float(f, 'x')
sage: ff(1.0)
1.4142135623730951
sage.symbolic.expression_conversions.laurent_polynomial(ex, base_ring=None, ring=None)

Return a Laurent polynomial from the symbolic expression ex.

INPUT:

  • ex – a symbolic expression
  • base_ring, ring – Either a base_ring or a Laurent polynomial ring can be specified for the parent of result. If just a base_ring is given, then the variables of the base_ring will be the variables of the expression ex.

OUTPUT:

A Laurent polynomial.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import laurent_polynomial
sage: f = x^2 + 2/x
sage: laurent_polynomial(f, base_ring=QQ)
2*x^-1 + x^2
sage: _.parent()
Univariate Laurent Polynomial Ring in x over Rational Field

sage: laurent_polynomial(f, ring=LaurentPolynomialRing(QQ, 'x, y'))
x^2 + 2*x^-1
sage: _.parent()
Multivariate Laurent Polynomial Ring in x, y over Rational Field

sage: x, y = var('x, y')
sage: laurent_polynomial(x + 1/y^2, ring=LaurentPolynomialRing(QQ, 'x, y'))
x + y^-2
sage: _.parent()
Multivariate Laurent Polynomial Ring in x, y over Rational Field
sage.symbolic.expression_conversions.polynomial(ex, base_ring=None, ring=None)

Return a polynomial from the symbolic expression ex.

INPUT:

  • ex – a symbolic expression
  • base_ring, ring – Either a base_ring or a polynomial ring can be specified for the parent of result. If just a base_ring is given, then the variables of the base_ring will be the variables of the expression ex.

OUTPUT:

A polynomial.

EXAMPLES:

sage: from sage.symbolic.expression_conversions import polynomial
sage: f = x^2 + 2
sage: polynomial(f, base_ring=QQ)
x^2 + 2
sage: _.parent()
Univariate Polynomial Ring in x over Rational Field

sage: polynomial(f, ring=QQ['x,y'])
x^2 + 2
sage: _.parent()
Multivariate Polynomial Ring in x, y over Rational Field

sage: x, y = var('x, y')
sage: polynomial(x + y^2, ring=QQ['x,y'])
y^2 + x
sage: _.parent()
Multivariate Polynomial Ring in x, y over Rational Field

sage: s,t=var('s,t')
sage: expr=t^2-2*s*t+1
sage: expr.polynomial(None,ring=SR['t'])
t^2 - 2*s*t + 1
sage: _.parent()
Univariate Polynomial Ring in t over Symbolic Ring

sage: polynomial(x*y, ring=SR['x'])
y*x

sage: polynomial(y - sqrt(x), ring=SR['y'])
y - sqrt(x)
sage: _.list()
[-sqrt(x), 1]

The polynomials can have arbitrary (constant) coefficients so long as they coerce into the base ring:

sage: polynomial(2^sin(2)*x^2 + exp(3), base_ring=RR)
1.87813065119873*x^2 + 20.0855369231877