Dense univariate polynomials over \(\RR\), implemented using MPFR#

class sage.rings.polynomial.polynomial_real_mpfr_dense.PolynomialRealDense[source]#

Bases: Polynomial

change_ring(R)[source]#

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RR['x'], [-2, 0, 1.5])
sage: f.change_ring(QQ)
3/2*x^2 - 2
sage: f.change_ring(RealField(10))
1.5*x^2 - 2.0
sage: f.change_ring(RealField(100))
1.5000000000000000000000000000*x^2 - 2.0000000000000000000000000000

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RR['x'], [-Integer(2), Integer(0), RealNumber('1.5')])
>>> f.change_ring(QQ)
3/2*x^2 - 2
>>> f.change_ring(RealField(Integer(10)))
1.5*x^2 - 2.0
>>> f.change_ring(RealField(Integer(100)))
1.5000000000000000000000000000*x^2 - 2.0000000000000000000000000000
degree()[source]#

Return the degree of the polynomial.

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RR['x'], [1, 2, 3]); f
3.00000000000000*x^2 + 2.00000000000000*x + 1.00000000000000
sage: f.degree()
2

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RR['x'], [Integer(1), Integer(2), Integer(3)]); f
3.00000000000000*x^2 + 2.00000000000000*x + 1.00000000000000
>>> f.degree()
2
integral()[source]#

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RR['x'], [3, pi, 1])                          # needs sage.symbolic
sage: f.integral()                                                          # needs sage.symbolic
0.333333333333333*x^3 + 1.57079632679490*x^2 + 3.00000000000000*x

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RR['x'], [Integer(3), pi, Integer(1)])                          # needs sage.symbolic
>>> f.integral()                                                          # needs sage.symbolic
0.333333333333333*x^3 + 1.57079632679490*x^2 + 3.00000000000000*x
list(copy=True)[source]#

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RR['x'], [1, 0, -2]); f
-2.00000000000000*x^2 + 1.00000000000000
sage: f.list()
[1.00000000000000, 0.000000000000000, -2.00000000000000]

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RR['x'], [Integer(1), Integer(0), -Integer(2)]); f
-2.00000000000000*x^2 + 1.00000000000000
>>> f.list()
[1.00000000000000, 0.000000000000000, -2.00000000000000]
quo_rem(other)[source]#

Return the quotient with remainder of self by other.

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RR['x'], [-2, 0, 1])
sage: g = PolynomialRealDense(RR['x'], [5, 1])
sage: q, r = f.quo_rem(g)
sage: q
x - 5.00000000000000
sage: r
23.0000000000000
sage: q*g + r == f
True
sage: fg = f*g
sage: fg.quo_rem(f)
(x + 5.00000000000000, 0)
sage: fg.quo_rem(g)
(x^2 - 2.00000000000000, 0)

sage: # needs sage.symbolic
sage: f = PolynomialRealDense(RR['x'], range(5))
sage: g = PolynomialRealDense(RR['x'], [pi,3000,4])
sage: q, r = f.quo_rem(g)
sage: g*q + r == f
True

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RR['x'], [-Integer(2), Integer(0), Integer(1)])
>>> g = PolynomialRealDense(RR['x'], [Integer(5), Integer(1)])
>>> q, r = f.quo_rem(g)
>>> q
x - 5.00000000000000
>>> r
23.0000000000000
>>> q*g + r == f
True
>>> fg = f*g
>>> fg.quo_rem(f)
(x + 5.00000000000000, 0)
>>> fg.quo_rem(g)
(x^2 - 2.00000000000000, 0)

>>> # needs sage.symbolic
>>> f = PolynomialRealDense(RR['x'], range(Integer(5)))
>>> g = PolynomialRealDense(RR['x'], [pi,Integer(3000),Integer(4)])
>>> q, r = f.quo_rem(g)
>>> g*q + r == f
True
reverse(degree=None)[source]#

Return reverse of the input polynomial thought as a polynomial of degree degree.

If \(f\) is a degree-\(d\) polynomial, its reverse is \(x^d f(1/x)\).

INPUT:

  • degree (None or an integer) – if specified, truncate or zero pad the list of coefficients to this degree before reversing it.

EXAMPLES:

sage: # needs sage.symbolic
sage: f = RR['x']([-3, pi, 0, 1])
sage: f.reverse()
-3.00000000000000*x^3 + 3.14159265358979*x^2 + 1.00000000000000
sage: f.reverse(2)
-3.00000000000000*x^2 + 3.14159265358979*x
sage: f.reverse(5)
-3.00000000000000*x^5 + 3.14159265358979*x^4 + x^2

>>> from sage.all import *
>>> # needs sage.symbolic
>>> f = RR['x']([-Integer(3), pi, Integer(0), Integer(1)])
>>> f.reverse()
-3.00000000000000*x^3 + 3.14159265358979*x^2 + 1.00000000000000
>>> f.reverse(Integer(2))
-3.00000000000000*x^2 + 3.14159265358979*x
>>> f.reverse(Integer(5))
-3.00000000000000*x^5 + 3.14159265358979*x^4 + x^2
shift(n)[source]#

Returns this polynomial multiplied by the power \(x^n\). If \(n\) is negative, terms below \(x^n\) will be discarded. Does not change this polynomial.

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RR['x'], [1, 2, 3]); f
3.00000000000000*x^2 + 2.00000000000000*x + 1.00000000000000
sage: f.shift(10)
3.00000000000000*x^12 + 2.00000000000000*x^11 + x^10
sage: f.shift(-1)
3.00000000000000*x + 2.00000000000000
sage: f.shift(-10)
0

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RR['x'], [Integer(1), Integer(2), Integer(3)]); f
3.00000000000000*x^2 + 2.00000000000000*x + 1.00000000000000
>>> f.shift(Integer(10))
3.00000000000000*x^12 + 2.00000000000000*x^11 + x^10
>>> f.shift(-Integer(1))
3.00000000000000*x + 2.00000000000000
>>> f.shift(-Integer(10))
0
truncate(n)[source]#

Returns the polynomial of degree \(< n\) which is equivalent to self modulo \(x^n\).

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RealField(10)['x'], [1, 2, 4, 8])
sage: f.truncate(3)
4.0*x^2 + 2.0*x + 1.0
sage: f.truncate(100)
8.0*x^3 + 4.0*x^2 + 2.0*x + 1.0
sage: f.truncate(1)
1.0
sage: f.truncate(0)
0

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RealField(Integer(10))['x'], [Integer(1), Integer(2), Integer(4), Integer(8)])
>>> f.truncate(Integer(3))
4.0*x^2 + 2.0*x + 1.0
>>> f.truncate(Integer(100))
8.0*x^3 + 4.0*x^2 + 2.0*x + 1.0
>>> f.truncate(Integer(1))
1.0
>>> f.truncate(Integer(0))
0
truncate_abs(bound)[source]#

Truncate all high order coefficients below bound.

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
sage: f = PolynomialRealDense(RealField(10)['x'], [10^-k for k in range(10)])
sage: f
1.0e-9*x^9 + 1.0e-8*x^8 + 1.0e-7*x^7 + 1.0e-6*x^6 + 0.000010*x^5
 + 0.00010*x^4 + 0.0010*x^3 + 0.010*x^2 + 0.10*x + 1.0
sage: f.truncate_abs(0.5e-6)
1.0e-6*x^6 + 0.000010*x^5 + 0.00010*x^4 + 0.0010*x^3 + 0.010*x^2 + 0.10*x + 1.0
sage: f.truncate_abs(10.0)
0
sage: f.truncate_abs(1e-100) == f
True

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense
>>> f = PolynomialRealDense(RealField(Integer(10))['x'], [Integer(10)**-k for k in range(Integer(10))])
>>> f
1.0e-9*x^9 + 1.0e-8*x^8 + 1.0e-7*x^7 + 1.0e-6*x^6 + 0.000010*x^5
 + 0.00010*x^4 + 0.0010*x^3 + 0.010*x^2 + 0.10*x + 1.0
>>> f.truncate_abs(RealNumber('0.5e-6'))
1.0e-6*x^6 + 0.000010*x^5 + 0.00010*x^4 + 0.0010*x^3 + 0.010*x^2 + 0.10*x + 1.0
>>> f.truncate_abs(RealNumber('10.0'))
0
>>> f.truncate_abs(RealNumber('1e-100')) == f
True
sage.rings.polynomial.polynomial_real_mpfr_dense.make_PolynomialRealDense(parent, data)[source]#

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import make_PolynomialRealDense
sage: make_PolynomialRealDense(RR['x'], [1,2,3])
3.00000000000000*x^2 + 2.00000000000000*x + 1.00000000000000

>>> from sage.all import *
>>> from sage.rings.polynomial.polynomial_real_mpfr_dense import make_PolynomialRealDense
>>> make_PolynomialRealDense(RR['x'], [Integer(1),Integer(2),Integer(3)])
3.00000000000000*x^2 + 2.00000000000000*x + 1.00000000000000