A sympy/sage Module for Computing Polynomial Remainder Sequences: [preprint] (original) (raw)
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Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x]
2016
In this paper we present two new methods for computing the subresultant polynomial remainder sequence (prs) of two polynomials f, g ∈ Z[x]. We are now able to also correctly compute the Euclidean and modified Euclidean prs of f, g by using either of the functions employed by our methods to compute the remainder polynomials. Another innovation is that we are able to obtain subresultant prs’s in Z[x] by employing the function rem(f, g, x) to compute the remainder polynomials in [x]. This is achieved by our method subresultants_amv_q (f, g, x), which is somewhat slow due to the inherent higher cost of com- putations in the field of rationals. To improve in speed, our second method, subresultants_amv(f, g, x), computes the remainder polynomials in the ring Z[x] by employing the function rem_z(f, g, x); the time complexity and performance of this method are very competitive. Our methods are two different implementations of Theorem 1 (Section 3), which establishes a one-to-one corresponde...
Theory of multiple polynomial remainder sequence
Publications of the Research Institute for Mathematical Sciences, 1984
Given a set of polynomials {P£, D (#), • • • > Po m) U)}j with coefficients in an integral domain I, we can generate a sequence of sets of remainders {P^(x) 9 ..., Pf w) U)}, x=l,2 v ..
Matrix computation of subresultant polynomial remainder sequences in integral domains
Reliable Computing, 1995
We present an improved variant of the matrix-triangularization subresultant prs method [2] for the computation of a greatest common divisor of two polynomials A and B (of degrees m and n, respectively) along with their polynomial remainder sequence. It is improved in the sense that we obtain complete theoretical results, independent of Van Vleck's theorem [13] (which is not always true [1], [6]), and, instead of transforming a matrix of order 2•max(m, n) [2], we are now transforming a matrix of order m + n. An example is also included to clarify the concepts.
Recursive polynomial remainder sequence and its subresultants
Journal of Algebra, 2008
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a "nested" expression, i.e. a Sylvester matrix whose elements are themselves determinants.
On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials
Serdica Journal of Computing, 2015
In 1917 Pell (1) and Gordon used sylvester2, Sylvester’s little known and hardly ever used matrix of 1853, to compute(2) the coefficients of a Sturmian remainder — obtained in applying in Q[x], Sturm’s algorithm on two polynomials f, g ∈ Z[x] of degree n — in terms of the determinants (3) of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. (4) In this paper we extend the work by Pell and Gordon and show how to compute (2) the coefficients of an Euclidean remainder — obtained in finding in Q[x], the greatest common divisor of f, g ∈ Z[x] of degree n — in terms of the determinants (5) of the corresponding submatrices of sylvester1, Sylvester’s widely known and used matrix of 1840. (1) See the link http://en.wikipedia.org/wiki/Anna\_Johnson\_Pell\_Wheeler for her biography (2) Both for complete and incomplete sequences, as defined in the sequel. (3) Also known as modified subresultants. (4...
Subresultants in Recursive Polynomial Remainder Sequence
2008
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate "recursively" with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresultants to be determinants representing the coefficients in recursive PRS by coefficients of initial polynomials. Finally, we discuss usage of recursive subresultants in approximate algebraic computation, which motivates the present work.
2014
In 1971 using pseudo-divisions that is, by working in [x] Brown and Traub computed Euclid's polynomial remainder sequences (prs's) and (proper) subresultant prs's using sylvester1, the most widely known form of Sylvester's matrix, whose determinant defines the resultant of two polynomials. In this paper we use, for the first time in the literature, the Pell-Gordon Theorem of 1917, and sylvester2, a little known form of Sylvester's matrix of 1853 to initially compute Sturm sequences in [x] without pseudo-divisions that is, by working in É[x]. We then extend our work in É[x] and, despite the fact that the absolute value of the determinant of sylvester2 equals the absolute value of the resultant, we construct modified subresultant prs's, which may differ from the proper ones only in sign.
2015
In 1917 Pell 1 and Gordon used sylvester2, Sylvester's little known and hardly ever used matrix of 1853, to compute 2 the coefficients of a Sturmian remainder-obtained in applying in É[x], Sturm's algorithm on two polynomials f, g ∈ [x] of degree n-in terms of the determinants 3 of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. 4 In this paper we extend the work by Pell and Gordon and show how to compute 2 the coefficients of an Euclidean remainder-obtained in finding in É[x], the greatest common divisor of f, g ∈ [x] of degree n-in terms of the determinants 5 of the corresponding submatrices of sylvester1, Sylvester's widely known and used matrix of 1840. See the link http://en.wikipedia.org/wiki/Anna\_Johnson\_Pell\_Wheeler for her biography. 2 Both for complete and incomplete sequences, as defined in the sequel. 3 Also known as modified subresultants. 4 Using determinan...
Recursive Polynomial Remainder Sequence and the Nested Subresultants
Lecture Notes in Computer Science, 2005
We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for investigation of the recursive PRS. Finally, we discuss usage of the reduced nested subresultant in approximate algebraic computation, which motivates the present work.