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Computing Characteristic Polynomials of Matrices of Structured Polynomials

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Computer Algebra in Scientific Computing (CASC 2016)

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Abstract

We present a parallel modular algorithm for finding characteristic polynomials of matrices with integer coefficient bivariate polynomials. For each prime, evaluation and interpolation gives us the bridge between polynomial matrices and matrices over a finite field so that the Hessenberg algorithm can be used.

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada

    Marshall Law & Michael Monagan

Authors
  1. Marshall Law
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  2. Michael Monagan
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Corresponding authors

Correspondence to Marshall Law or Michael Monagan .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Appendices

A Appendix

A1: 16 x 16 matrix

$$ \left[ \begin{array}{cccccccccccccccc} {x}^{8}&{}{x}^{5}y&{}{x}^{5}y&{}{x}^{4}{y}^{2}&{}{x}^{5}y&{}{x}^{ 2}{y}^{2}&{}{x}^{4}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{5}y&{}{x}^{4}{y}^{2}&{}{x}^{2}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x }^{4}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{3}{y}^{3}&{}{x}^{4}{y}^{4}\\ {x}^{7}&{}{x}^{6}y&{}{ x}^{4}y&{}{x}^{5}{y}^{2}&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{3}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{4}y&{}{x}^{5}{y} ^{2}&{}x{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{3}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{2}{y}^{3}&{}{x}^{5}{y}^{4} \\ {x}^{7}&{}{x}^{4}y&{}{x}^{6}y&{}{x}^{5}{y}^{2}&{}{x}^{4}y&{}x{y}^{2}&{}{x}^{5}{y}^{2}&{}{ x}^{4}{y}^{3}&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{3}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{3}{y}^{2}&{}{x}^{2}{y}^{3 }&{}{x}^{4}{y}^{3}&{}{x}^{5}{y}^{4}\\ {x}^{6}&{}{x}^{5}y&{}{x}^{5}y&{}{x}^{6}{y}^{2}&{}{x} ^{3}y&{}{x}^{2}{y}^{2}&{}{x}^{4}{y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{3}y&{}{x}^{4}{y}^{2}&{}{x}^{2}{y}^{2}&{}{x}^{5 }{y}^{3}&{}{x}^{2}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{3}{y}^{3}&{}{x}^{6}{y}^{4}\\ {x}^{7} &{}{x}^{4}y&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{6}y&{}{x}^{3}{y}^{2}&{}{x}^{5}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{4}y &{}{x}^{3}{y}^{2}&{}x{y}^{2}&{}{x}^{2}{y}^{3}&{}{x}^{5}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{4}{y}^{3}&{}{x}^{5}{y}^ {4}\\ {x}^{6}&{}{x}^{5}y&{}{x}^{3}y&{}{x}^{4}{y}^{2}&{}{x}^{5}y&{}{x}^{4}{y}^{2}&{}{x}^{4} {y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{3}y&{}{x}^{4}{y}^{2}&{}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{4}{y}^{2}&{}{x}^{5}{y}^ {3}&{}{x}^{3}{y}^{3}&{}{x}^{6}{y}^{4}\\ {x}^{6}&{}{x}^{3}y&{}{x}^{5}y&{}{x}^{4}{y}^{2}&{}{ x}^{5}y&{}{x}^{2}{y}^{2}&{}{x}^{6}{y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{3}y&{}{x}^{2}{y}^{2}&{}{x}^{2}{y}^{2}&{}{x}^ {3}{y}^{3}&{}{x}^{4}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{5}{y}^{3}&{}{x}^{6}{y}^{4}\\ {x}^{ 5}&{}{x}^{4}y&{}{x}^{4}y&{}{x}^{5}{y}^{2}&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{5}{y}^{2}&{}{x}^{6}{y}^{3}&{}{x}^{2 }y&{}{x}^{3}{y}^{2}&{}x{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{3}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{4}{y}^{3}&{}{x}^{7}{y }^{4}\\ {x}^{7}&{}{x}^{4}y&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{4}y&{}x{y}^{2}&{}{x}^{3}{y}^ {2}&{}{x}^{2}{y}^{3}&{}{x}^{6}y&{}{x}^{5}{y}^{2}&{}{x}^{3}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{5}{y}^{2}&{}{x}^{4}{ y}^{3}&{}{x}^{4}{y}^{3}&{}{x}^{5}{y}^{4}\\ {x}^{6}&{}{x}^{5}y&{}{x}^{3}y&{}{x}^{4}{y}^{2 }&{}{x}^{3}y&{}{x}^{2}{y}^{2}&{}{x}^{2}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{5}y&{}{x}^{6}{y}^{2}&{}{x}^{2}{y}^{2}&{}{ x}^{5}{y}^{3}&{}{x}^{4}{y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{3}{y}^{3}&{}{x}^{6}{y}^{4}\\ {x }^{6}&{}{x}^{3}y&{}{x}^{5}y&{}{x}^{4}{y}^{2}&{}{x}^{3}y&{}{y}^{2}&{}{x}^{4}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{5}y&{}{ x}^{4}{y}^{2}&{}{x}^{4}{y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{4}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{5}{y}^{3}&{}{x}^{6} {y}^{4}\\ {x}^{5}&{}{x}^{4}y&{}{x}^{4}y&{}{x}^{5}{y}^{2}&{}{x}^{2}y&{}x{y}^{2}&{}{x}^{3}{y }^{2}&{}{x}^{4}{y}^{3}&{}{x}^{4}y&{}{x}^{5}{y}^{2}&{}{x}^{3}{y}^{2}&{}{x}^{6}{y}^{3}&{}{x}^{3}{y}^{2}&{}{x}^{4 }{y}^{3}&{}{x}^{4}{y}^{3}&{}{x}^{7}{y}^{4}\\ {x}^{6}&{}{x}^{3}y&{}{x}^{3}y&{}{x}^{2}{y}^ {2}&{}{x}^{5}y&{}{x}^{2}{y}^{2}&{}{x}^{4}{y}^{2}&{}{x}^{3}{y}^{3}&{}{x}^{5}y&{}{x}^{4}{y}^{2}&{}{x}^{2}{y}^{2} &{}{x}^{3}{y}^{3}&{}{x}^{6}{y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{5}{y}^{3}&{}{x}^{6}{y}^{4}\\ {x}^{5}&{}{x}^{4}y&{}{x}^{2}y&{}{x}^{3}{y}^{2}&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{3}{y}^{2}&{}{x}^{4}{y}^{3}&{}{ x}^{4}y&{}{x}^{5}{y}^{2}&{}x{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{5}{y}^{2}&{}{x}^{6}{y}^{3}&{}{x}^{4}{y}^{3}&{}{x}^ {7}{y}^{4}\\ {x}^{5}&{}{x}^{2}y&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{4}y&{}x{y}^{2}&{}{x}^{5 }{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{4}y&{}{x}^{3}{y}^{2}&{}{x}^{3}{y}^{2}&{}{x}^{4}{y}^{3}&{}{x}^{5}{y}^{2}&{}{x} ^{4}{y}^{3}&{}{x}^{6}{y}^{3}&{}{x}^{7}{y}^{4}\\ {x}^{4}&{}{x}^{3}y&{}{x}^{3}y&{}{x}^{4}{ y}^{2}&{}{x}^{3}y&{}{x}^{2}{y}^{2}&{}{x}^{4}{y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{3}y&{}{x}^{4}{y}^{2}&{}{x}^{2}{y}^ {2}&{}{x}^{5}{y}^{3}&{}{x}^{4}{y}^{2}&{}{x}^{5}{y}^{3}&{}{x}^{5}{y}^{3}&{}{x}^{8}{y}^{4}\end{array} \right] $$

A2: \( f_6(x,y) \) of 16 x 16 matrix

$$ ( 2x^{16}+4x^{14} ) y^{12}+ ( 4x^{18}+32x^{16}+28x^{14}+8x^{12} ) y^{11}+ $$
$$ ( x^{20}+34x^{18}+149x^{16}+188x^{14}+43x^{12}-22x^ {10}+3x^{8} ) y^{10}+ $$
$$ ( 16x^{20}+128x^{18}+452x^{16}+568x^{14}+268x^{12}-32x^{10}-72x^{8}-8x^{6} ) y^{9}+ $$
$$ ( 52x^{20}+348x^{18}+910{ x}^{16}+1172x^{14}+704x^{12}+68x^{10}-136x^{8}-120x^{6}-28x^{4} ) y^{8}+ $$
$$ ( 112x^{20}+596x^{18}+1344x^{16}+1788x^{14}+1224x^{12}+216x^{10}-220 x^{8}-184x^{6}-92x^{4}-32x^{2} ) y^{7}+ $$
$$ ( 133x^{20}+734x^{18}+1551x^{16}+1948x^{14}+1476x^{12}+428x^{10}-320x^{8}-276x^{6}-81x^{4}-34x ^{2}-15 ) y^{6}+ $$
$$ ( 112x^{20}+596x^{18}+1344x^{16}+1788x^{14}+1224x^{12}+216x^{10}-220x^{8}-184x^{6}-92x^{4}-32x^{2} ) y^{5}+ $$
$$ ( 52x^{ 20}+348x^{18}+910x^{16}+1172x^{14}+704x^{12}+68x^{10}-136x^{8}-120x^{6}-28x^{4} ) y^{4}+ $$
$$ ( 16x^{20}+128x^{18}+452x^{16}+568x^{14}+268x^{12 }-32x^{10}-72x^{8}-8x^{6} ) y^{3}+ $$
$$ ( x^{20}+34x^{18}+149x^{16}+188x^{14}+43x^{12}-22x^{10}+3x^{8} ) y^{2}+ $$
$$ ( 4x^{18}+32x^{16}+28{ x}^{14}+8x^{12} ) y+ ( 2x^{16}+4x^{14} ) y^0 $$

A3: First two coefficients of \( C(\lambda ,x,y) \) for 16 by 16 matrix

$$ c_0(x,y) = {x}^{32}{y}^{32} \left( x^2-1 \right) ^{32} $$
$$ c_1(x,y) = -{x}^{32}{y}^{28} \left( x^2-1 \right) ^{28} \left( 2\,{x}^{4}{y}^{2}+4\,{x}^{2}{y}^{3}+4\,{x}^{2}{y}^{2}+{y}^{4}+4\,{x}^{2}y+1 \right) $$

B1: Time 16 by 16 on Maple

figure a

B2: Time 16 by 16 on Magma

figure b

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Law, M., Monagan, M. (2016). Computing Characteristic Polynomials of Matrices of Structured Polynomials. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_22

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_22

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-45641-6

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