Polynomial
Polynomial()
A Symbolica polynomial with rational coefficients.
Methods
| Name | Description |
|---|---|
| coefficient_list | Get the coefficient list, optionally in the variables xs. |
| content | Get the content, i.e., the GCD of the coefficients. |
| derivative | Take a derivative in x. |
| factor | Factorize the polynomial. |
| factor_square_free | Compute the square-free factorization of the polynomial. |
| gcd | Compute the greatest common divisor (GCD) of two polynomials. |
| get_var_list | 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, |
| nterms | Get the number of terms in the polynomial. |
| parse | Parse a polynomial with integer coefficients from a string. |
| pretty_str | Convert the polynomial into a human-readable string, with tunable settings. |
| quot_rem | Divide self by rhs, returning the quotient and remainder. |
| replace | Replace the variable x with a polynomial v. |
| resultant | Compute the resultant of two polynomials with respect to the variable var. |
| 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_integer_polynomial | Convert the polynomial to a polynomial with integer coefficients, if possible. |
| to_latex | Convert the polynomial into a LaTeX string. |
coefficient_list
Polynomial.coefficient_list(xs)
Get the coefficient list, optionally in the variables xs.
Examples
>>> from symbolica import Expression
>>> x = Expression.symbol('x')
>>> p = Expression.parse('x*y+2*x+x^2').to_polynomial()
>>> for n, pp in p.coefficient_list(x):
>>> print(n, pp)content
Polynomial.content()
Get the content, i.e., the GCD of the coefficients.
Examples
>>> from symbolica import Expression
>>> p = Expression.parse('3x^2+6x+9').to_polynomial()
>>> print(p.content())derivative
Polynomial.derivative(x)
Take a derivative in x.
Examples
>>> from symbolica import Expression
>>> x = Expression.symbol('x')
>>> p = Expression.parse('x^2+2').to_polynomial()
>>> print(p.derivative(x))factor
Polynomial.factor()
Factorize the polynomial.
Examples
>>> from symbolica import Expression
>>> p = Expression.parse('(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 Expression
>>> p = Expression.parse('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))gcd
Polynomial.gcd(rhs)
Compute the greatest common divisor (GCD) of two polynomials.
get_var_list
Polynomial.get_var_list()
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(
>>> [Expression.parse("a b c d - 1").to_polynomial(),
>>> Expression.parse("a b c + a b d + a c d + b c d").to_polynomial(),
>>> Expression.parse("a b + b c + a d + c d").to_polynomial(),
>>> Expression.parse("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 Expression
>>> x = Expression.symbol('x')
>>> p = Expression.parse('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 Expression
>>> p = Expression.parse('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): 1nterms
Polynomial.nterms()
Get the number of terms in the polynomial.
parse
Polynomial.parse(_cls, input, vars)
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
| Type | Description |
|---|---|
| ValueError | If the input is not a valid Symbolica polynomial. |
pretty_str
Polynomial.pretty_str(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, latex=False)
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.pretty_str(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.
quot_rem
Polynomial.quot_rem(rhs)
Divide self by rhs, returning the quotient and remainder.
replace
Polynomial.replace(x, v)
Replace the variable x with a polynomial v.
Examples
>>> from symbolica import Expression
>>> x = Expression.symbol('x')
>>> p = Expression.parse('x*y+2*x+x^2').to_polynomial()
>>> r = Expression.parse('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.
to_expression
Polynomial.to_expression()
Convert the polynomial to an expression.
Examples
>>> from symbolica import Expression
>>> e = Expression.parse('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_integer_polynomial
Polynomial.to_integer_polynomial()
Convert the polynomial to a polynomial with integer coefficients, if possible.
to_latex
Polynomial.to_latex()
Convert the polynomial into a LaTeX string.