Dense univariate polynomials over \(\RR\), implemented using MPFR¶
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class
sage.rings.polynomial.polynomial_real_mpfr_dense.PolynomialRealDense¶ Bases:
sage.rings.polynomial.polynomial_element.Polynomial-
change_ring(R)¶ 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
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degree()¶ 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
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integral()¶ EXAMPLES:
sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense sage: f = PolynomialRealDense(RR['x'], [3, pi, 1]) sage: f.integral() 0.333333333333333*x^3 + 1.57079632679490*x^2 + 3.00000000000000*x
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list(copy=True)¶ 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]
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quo_rem(other)¶ Return the quotient with remainder of
selfbyother.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: 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
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reverse()¶ Returns \(x^d f(1/x)\) where \(d\) is the degree of \(f\).
EXAMPLES:
sage: from sage.rings.polynomial.polynomial_real_mpfr_dense import PolynomialRealDense sage: f = PolynomialRealDense(RR['x'], [-3, pi, 0, 1]) sage: f.reverse() -3.00000000000000*x^3 + 3.14159265358979*x^2 + 1.00000000000000
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shift(n)¶ 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
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truncate(n)¶ 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
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truncate_abs(bound)¶ 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
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sage.rings.polynomial.polynomial_real_mpfr_dense.make_PolynomialRealDense(parent, data)¶ 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