Polynomial
Polynomial()A Symbolica polynomial with rational coefficients.
Methods
| Name | Description |
|---|---|
| adjoin | Adjoin the coefficient ring of this polynomial R[a] with b, whose minimal polynomial |
| approximate_roots | Approximate all complex roots of a univariate polynomial, given a maximal number of iterations |
| coefficient_list | Get the coefficient list, optionally in the variables vars. |
| contains | Check if the polynomial contains the given variable. |
| content | Get the content, i.e., the GCD of the coefficients. |
| degree | Get the degree of the polynomial in var. |
| derivative | Take a derivative in x. |
| evaluate | Evaluate the polynomial at point input. |
| evaluate_complex | Evaluate the polynomial at point input with complex input. |
| extended_gcd | Compute the extended GCD of two polynomials, yielding the GCD and the Bezout coefficients s and t |
| factor | Factorize the polynomial. |
| factor_square_free | Compute the square-free factorization of the polynomial. |
| format | Convert the polynomial into a human-readable string, with tunable settings. |
| gcd | Compute the greatest common divisor (GCD) of two or more polynomials. |
| get_variables | Get the list of variables in the internal ordering of the polynomial. |
| groebner_basis | Compute the Groebner basis of a polynomial system. |
| integrate | Integrate the polynomial in x. |
| interpolate | Perform Newton interpolation in the variable x given the sample points |
| isolate_roots | Isolate the real roots of the polynomial. The result is a list of intervals with rational bounds that contain exactly one root, |
| lcoeff | Get the leading coefficient. |
| nterms | Get the number of terms. |
| parse | Parse a polynomial with rational coefficients from a string. |
| primitive | Get the primitive part of the polynomial, i.e., the polynomial divided |
| quot_rem | Divide self by rhs, returning the quotient and remainder. |
| reduce | Completely reduce the polynomial w.r.t the polynomials gs. |
| reorder | Set a new variable ordering for the polynomial. |
| replace | Replace the variable x with a polynomial v. |
| resultant | Compute the resultant of two polynomials with respect to the variable var. |
| simplify_algebraic_number | Find the minimal polynomial for the algebraic number represented by this polynomial |
| to_expression | Convert the polynomial to an expression. |
| to_finite_field | Convert the coefficients of the polynomial to a finite field with prime prime. |
| to_latex | Convert the polynomial into a LaTeX string. |
| to_number_field | Convert the polynomial to a number field defined by the minimal polynomial minimal_poly. |
adjoin
Polynomial.adjoin(b, new_symbol=None)Adjoin the coefficient ring of this polynomial R[a] with b, whose minimal polynomial is R[a][b] and form R[b]. Also return the new representation of a and b.
b must be irreducible over R and R[a]; this is not checked.
If new_symbol is provided, the variable of the new extension will be renamed to it. Otherwise, the variable of the new extension will be the same as that of b.
Examples
>>> from symbolica import *
>>> sqrt2 = P('a^2-2')
>>> sqrt23 = P('b^2-a-3')
>>> (min_poly, rep2, rep23) = sqrt2.adjoin(sqrt23)
>>>
>>> # convert to number field
>>> a = P('a^2+b').replace(S('a'), rep2).replace(S('b'), rep23).to_number_field(min_poly)approximate_roots
Polynomial.approximate_roots(max_iterations, tolerance)Approximate all complex roots of a univariate polynomial, given a maximal number of iterations and a given tolerance. Returns the roots and their multiplicity.
Examples
>>> p = E('x^10+9x^7+4x^3+2x+1').to_polynomial()
>>> for (r, m) in p.approximate_roots(1000, 1e-10):
>>> print(r, m)coefficient_list
Polynomial.coefficient_list(xs)Get the coefficient list, optionally in the variables xs.
Examples
>>> from symbolica import *
>>> x = S('x')
>>> p = E('x*y+2*x+x^2').to_polynomial()
>>> for n, pp in p.coefficient_list(x):
>>> print(n, pp)contains
Polynomial.contains(var)Check if the polynomial contains the given variable.
content
Polynomial.content()Get the content, i.e., the GCD of the coefficients.
Examples
>>> from symbolica import *
>>> p = E('3x^2+6x+9').to_polynomial()
>>> print(p.content())degree
Polynomial.degree(var)Get the degree of the polynomial in var.
derivative
Polynomial.derivative(x)Take a derivative in x.
Examples
>>> from symbolica import *
>>> x = S('x')
>>> p = E('x^2+2').to_polynomial()
>>> print(p.derivative(x))evaluate
Polynomial.evaluate(inputs)Evaluate the polynomial at point input.
Examples
>>> from symbolica import *
>>> P('x*y+2*x+x^2').evaluate([2., 3.])Yields 14.0.
evaluate_complex
Polynomial.evaluate_complex(inputs)Evaluate the polynomial at point input with complex input.
Examples
>>> from symbolica import *
>>> P('x*y+2*x+x^2').evaluate([2+1j, 3+2j])Yields 11+13j.
extended_gcd
Polynomial.extended_gcd(rhs)Compute the extended GCD of two polynomials, yielding the GCD and the Bezout coefficients s and t such that self * s + rhs * t = gcd(self, rhs).
Examples
>>> from symbolica import *
>>> E('(1+x)(20+x)').to_polynomial().extended_gcd(E('x^2+2').to_polynomial())yields (1, 1/67-7/402*x, 47/134+7/402*x).
factor
Polynomial.factor()Factorize the polynomial.
Examples
>>> from symbolica import *
>>> p = E('(x+1)(x+2)(x+3)(x+4)(x+5)(x^2+6)(x^3+7)(x+8)(x^4+9)(x^5+x+10)').expand().to_polynomial()
>>> print('Factorization of {}:'.format(p))
>>> for f, exp in p.factor():
>>> print(' ({})^{}'.format(f, exp))factor_square_free
Polynomial.factor_square_free()Compute the square-free factorization of the polynomial.
Examples
>>> from symbolica import *
>>> p = E('3*(2*x^2+y)(x^3+y)^2(1+4*y)^2(1+x)').expand().to_polynomial()
>>> print('Square-free factorization of {}:'.format(p))
>>> for f, exp in p.factor_square_free():
>>> print(' ({})^{}'.format(f, exp))format
Polynomial.format(
mode=PrintMode.Symbolica,
terms_on_new_line=False,
color_top_level_sum=True,
color_builtin_symbols=True,
print_finite_field=True,
symmetric_representation_for_finite_field=False,
explicit_rational_polynomial=False,
number_thousands_separator=None,
multiplication_operator='*',
double_star_for_exponentiation=False,
square_brackets_for_function=False,
num_exp_as_superscript=True,
precision=None,
show_namespaces=False,
max_terms=None,
custom_print_mode=None,
)Convert the polynomial into a human-readable string, with tunable settings.
Examples
>>> p = FiniteFieldPolynomial.parse("3*x^2+2*x+7*x^3", ['x'], 11)
>>> print(p.format(symmetric_representation_for_finite_field=True))Yields z³⁴+x^(x+2)+y⁴+f(x,x²)+128_378_127_123 z^(2/3) w² x⁻¹ y⁻¹+3/5.
gcd
Polynomial.gcd(rhs)Compute the greatest common divisor (GCD) of two polynomials.
get_variables
Polynomial.get_variables()Get the list of variables in the internal ordering of the polynomial.
groebner_basis
Polynomial.groebner_basis(_cls, system, grevlex=True, print_stats=False)Compute the Groebner basis of a polynomial system.
If grevlex=True, reverse graded lexicographical ordering is used, otherwise the ordering is lexicographical.
If print_stats=True intermediate statistics will be printed.
Examples
>>> basis = Polynomial.groebner_basis(
>>> [E("a b c d - 1").to_polynomial(),
>>> E("a b c + a b d + a c d + b c d").to_polynomial(),
>>> E("a b + b c + a d + c d").to_polynomial(),
>>> E("a + b + c + d").to_polynomial()],
>>> grevlex=True,
>>> print_stats=True
>>> )
>>> for p in basis:
>>> print(p)integrate
Polynomial.integrate(x)Integrate the polynomial in x.
Examples
>>> from symbolica import *
>>> x = S('x')
>>> p = E('x^2+2').to_polynomial()
>>> print(p.integrate(x))interpolate
Polynomial.interpolate(_cls, x, sample_points, values)Perform Newton interpolation in the variable x given the sample points sample_points and the values values.
Examples
>>> x, y = S('x', 'y')
>>> a = Polynomial.interpolate(
>>> x, [4, 5], [(y**2+5).to_polynomial(), (y**3).to_polynomial()])
>>> print(a)yields 25-5*x+5*y^2-y^2*x-4*y^3+y^3*x.
isolate_roots
Polynomial.isolate_roots(refine=None)Isolate the real roots of the polynomial. The result is a list of intervals with rational bounds that contain exactly one root, and the multiplicity of that root. Optionally, the intervals can be refined to a given precision.
Examples
>>> from symbolica import *
>>> p = E('2016+5808*x+5452*x^2+1178*x^3+-753*x^4+-232*x^5+41*x^6').to_polynomial()
>>> for a, b, n in p.isolate_roots():
>>> print('({},{}): {}'.format(a, b, n))yields
(-56/45,-77/62): 1
(-98/79,-119/96): 1
(-119/96,-21/17): 1
(-7/6,0): 1
(0,6): 1
(6,12): 1lcoeff
Polynomial.lcoeff()Get the leading coefficient.
Examples
>>> from symbolica import Expression
>>> p = E('3x^2+6x+9').to_polynomial().lcoeff()
>>> print(p)Yields 3.
nterms
Polynomial.nterms()Get the number of terms in the polynomial.
parse
Polynomial.parse(_cls, input, vars, default_namespace='python')Parse a polynomial with integer coefficients from a string. The input must be written in an expanded format and a list of all the variables must be provided.
If these requirements are too strict, use Expression.to_polynomial() or RationalPolynomial.parse() instead.
Examples
>>> e = Polynomial.parse('3*x^2+y+y*4', ['x', 'y'])Raises
| Name | Type | Description |
|---|---|---|
| ValueError | If the input is not a valid Symbolica polynomial. |
primitive
Polynomial.primitive()Get the primitive part of the polynomial, i.e., the polynomial divided by its content.
Examples
>>> from symbolica import Expression
>>> p = E('3x^2+6x+9').to_polynomial().primitive()
>>> print(p)Yields x^2+2*x+3.
quot_rem
Polynomial.quot_rem(rhs)Divide self by rhs, returning the quotient and remainder.
reduce
Polynomial.reduce(gs, grevlex=True)Completely reduce the polynomial w.r.t the polynomials gs.
If grevlex=True, reverse graded lexicographical ordering is used, otherwise the ordering is lexicographical.
Examples
>>> E('y^2+x').to_polynomial().reduce([E('x').to_polynomial()])yields y^2
reorder
Polynomial.reorder(vars)Reorder the polynomial in-place to use the given variable order.
replace
Polynomial.replace(x, v)Replace the variable x with a polynomial v.
Examples
>>> from symbolica import *
>>> x = S('x')
>>> p = E('x*y+2*x+x^2').to_polynomial()
>>> r = E('y+1').to_polynomial())
>>> p.replace(x, r)resultant
Polynomial.resultant(rhs, var)Compute the resultant of two polynomials with respect to the variable var.
simplify_algebraic_number
Polynomial.simplify_algebraic_number(minimal_poly)Find the minimal polynomial for the algebraic number represented by this polynomial expressed in the number field defined by minimal_poly.
Examples
>>> from symbolica import *
>>> (min_poly, rep2, rep23) = P('a^2-2').adjoin(P('b^2-3'))
>>> rep2.simplify_algebraic_number(min_poly)Yields b^2-2.
to_expression
Polynomial.to_expression()Convert the polynomial to an expression.
Examples
>>> from symbolica import *
>>> e = E('x*y+2*x+x^2')
>>> p = e.to_polynomial()
>>> print((e - p.to_expression()).expand())to_finite_field
Polynomial.to_finite_field(prime)Convert the coefficients of the polynomial to a finite field with prime prime.
to_latex
Polynomial.to_latex()Convert the polynomial into a LaTeX string.
to_number_field
Polynomial.to_number_field(minimal_poly)Convert the polynomial to a number field defined by the minimal polynomial minimal_poly.
Examples
>>> from symbolica import *
>>> a = P('a').to_number_field(P('a^2-2'))
>>> print(a * a)Yields 2.