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Fast and backward stable computation of roots of polynomials

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Thomas Mach

The document discusses the development of a fast and stable algorithm for computing polynomial roots, particularly through an enhanced implementation of Francis's algorithm. The authors propose an O(n) representation that leverages specific matrix structures to improve computational efficiency while maintaining backward stability. The work includes numerical experiments and comparisons with existing methods, emphasizing the contributions made by the authors in this area.

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1 . . . 1 a0 a1 ... an1 Fast and stable computation of the roots of polynomials Jared L. Aurentz1, Thomas Mach2, Raf Vandebril2, and David S. Watkins1 1Dept. of Mathematics, Washington State University 2Dept. Computer Science, KU Leuven Structured Numerical Linear and Multilinear Algebra: Analysis, Algorithms and Applications 11 September 2014 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 1/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Collaborators This is joint work with David S. Watkins (WSU) Jared L. Aurentz (WSU) Raf Vandebril (KU Leuven) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 2/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Problem p(x) = xn + an1xn1 + an2xn2 + + a0 = 0 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 3/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Problem p(x) = xn + an1xn1 + an2xn2 + + a0 = 0 Companion matrix A = 2 66666664 a0 1 a1 1 a2 . . . ... 1 an2 1 an1 3 77777775 Get the zeros of p by computing the eigenvalues of A. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 3/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Cost If structure not exploited: If structure exploited: J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 4/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Cost If structure not exploited: O(n2) storage O(n3) ops Francis's implicitly-shifted QR algorithm Matlab's roots command. If structure exploited: O(n) storage O(n2) ops data-sparse representation + Francis's algorithm several methods proposed including some by my coauthors (fast but potentially unstable) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 4/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Some of the Competitors Chandrasekaran, Gu, Xia, Zhu (2007) Bini, Boito, Eidelman, Gemignani, Gohberg (2010) Boito, Eidelman, Gemignani, Gohberg (2012) Fortran codes available Quasiseparable generator representation we use essentially 2 2 unitary matrices results on backward stability J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 5/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Contribution We present yet another O(n) representation Francis's algorithm in O(n) ops/iteration Fortran codes (we're faster) normwise almost backward stable (We can prove it.) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 6/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure companion matrix is unitary-plus-rank-one A = 2 0 0 ei 1 0 6664 . . . ... 1 0 3 7775+ 2 6664 0 0 ei a0 0 0 a1 ... ... ... 0 0 an1 3 7775 preserved by unitary similarities J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure companion matrix is unitary-plus-rank-one A = 2 0 0 ei 1 0 6664 . . . ... 1 0 3 7775+ 2 6664 0 0 ei a0 0 0 a1 ... ... ... 0 0 an1 3 7775 preserved by unitary similarities companion matrix is also upper Hessenberg. preserved by Francis's algorithm J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure companion matrix is unitary-plus-rank-one A = 2 0 0 ei 1 0 6664 . . . ... 1 0 3 7775+ 2 6664 0 0 ei a0 0 0 a1 ... ... ... 0 0 an1 3 7775 preserved by unitary similarities companion matrix is also upper Hessenberg. preserved by Francis's algorithm We exploit this structure. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure Chandrasekaran, Gu, Xia, Zhu (2007) Compute the companion's QR factorization A = QR Q is upper Hessenberg and unitary. R is upper triangular and unitary-plus-rank-one. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 8/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure Chandrasekaran, Gu, Xia, Zhu (2007) Compute the companion's QR factorization A = QR Q is upper Hessenberg and unitary. R is upper triangular and unitary-plus-rank-one. We do this too. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 8/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Unitary Part (Matrix Q) Q factored in core transformations 2 664 3 775 = 2 664 1 1 3 775 2 1 664 1 3 775 2 1 664 1 3 775 Q = O(n) storage J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 9/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Upper Triangular Part (Matrix R) R = U + xyT unitary-plus-rank-one, so R has quasiseparable rank 2. R = 2 666666664 . . . ... ... ... . . . ... 3 777777775 quasiseparable generator representation (O(n) storage) Chandrasekaran, Gu, Xia, and Zhu exploit this structure. We do it dierently. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 10/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix A) Add a row/column for extra wiggle room A = 2 666664 0 a0 1 1 a1 0 . . . ... ... 1 an1 0 0 0 3 777775 Extra zero root can be de ated immediately. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 11/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix A) Add a row/column for extra wiggle room A = 2 666664 0 a0 1 1 a1 0 . . . ... ... 1 an1 0 0 0 3 777775 Extra zero root can be de ated immediately. A = QR, where Q = 2 666664 0 1 0 1 0 0 . . . ... ... 1 0 0 0 1 3 777775 R = 2 666664 1 a1 0 . . . ... ... 1 an1 0 a0 1 0 0 3 777775 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 11/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix Q) Q = 2 666664 0 1 0 1 0 0 . . . ... ... 1 0 0 0 1 3 777775 Q is stored in factored form Q = Q = Q1Q2 Qn1 (mark the n 1 instead of n) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 12/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix R) R = 2 666664 1 a1 0 . . . ... ... 1 an1 0 a0 1 0 0 3 777775 R is unitary-plus-rank-one: 2 666664 1 0 0 . . . ... ... 1 0 0 0 1 1 0 3 777775 + 2 666664 0 a1 0 . . . ... ... 0 an1 0 a0 0 1 0 3 777775 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 13/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R R = U + xyT , where xyT = 2 666664 a1 ... an1 a0 1 3 777775 0 0 1 0 Next step: Roll up x. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 14/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 3 64 75 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 3 64 75 0 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 3 64 75 0 0 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 64 0 0 0 3 75 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 64 0 0 0 3 75 C1 Cn1Cnx = e1 (w.l.g. = 1) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of A Altogether we have A = QR = Q C (B + e1yT ) A = Q1 Qn1 C 1 (B1 Bn + e1yT ) n C Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R . J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 16/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Francis's Iterations We have complex single-shift code, real double-shift code. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Francis's Iterations We have complex single-shift code, real double-shift code. We describe single-shift case for simplicity ignoring rank-one part. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Francis's Iterations We have complex single-shift code, real double-shift code. We describe single-shift case for simplicity ignoring rank-one part. Two basic operations: Fusion ) Turnover , J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Done! Iteration complete! Cost: 3n turnovers/iteration, so O(n) ops/iteration Double-shift iteration is similar: chase two core transformations instead of one, 7n turnovers/iteration. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 19/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Speed Comparison, Fortran implementation Contestants LAPACK code xHSEQR (O(n3)) AMVW (our single-shift code) BBEGG (Bini, Boito, et al. 2010) only single-shift code BEGG (Boito, Eidelman, et al. 2012) only double-shift code CGXZ (Chandrasekaran, Gu, et al. 2007) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 20/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments complex single-shift|random coecients 101 102 103 104 BBEGG: 23.297 BEGG: 15.442 AMVW: 4.269 101 102 105 Time in seconds AMVW ZHSEQR BBEGG BEGG 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 21/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments real double-shift|random coecients 101 102 103 104 CGXZ: 9.023 BBEGG: 7.899 AMVW: 3.817 101 102 105 Time in seconds AMVW DHSEQR BBEGG CGXZ 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 22/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift|random aeb, b 2 [R; R], R = 5; : : : ; 25 101 102 103 104 101 102 105 Time in seconds AMVW ZHSEQR BBEGG BEGG 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 23/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift|random aeb, b 2 [R; R], R = 5; : : : ; 25 101 102 103 104 101 102 105 Time in seconds AMVW DHSEQR BBEGG CGXZ 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 24/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift|zn 1 101 102 103 104 BBEGG: 27.847 BEGG: 16.933 AMVW: 3.198 101 102 105 Time in seconds AMVW ZHSEQR BBEGG BEGG 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 25/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift|zn 1 101 102 103 104 CGXZ: 8.883 BBEGG: 6.595 AMVW: 3.554 101 102 105 Time in seconds AMVW DHSEQR BBEGG CGXZ 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 26/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials Roots known! No. Description Deg. 1 Wilkinson polynomial 10 2 Wilkinson polynomial 15 3 Wilkinson polynomial 20 4 scaled and shifted Wilkinson poly. 20 5 reverse Wilkinson polynomial 10 6 reverse Wilkinson polynomial 15 7 reverse Wilkinson polynomial 20 8 prescribed roots of varying scale 20 9 prescribed roots of varying scale 3 20 10 Chebyshev polynomial 20 11 z20 + z19 + + z + 1 20 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 27/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials (2) Roots known! No. Description Deg. MPSolve 12 trv m, C. Traverso 24 13 mand31 Mandelbrot example (k = 5) 31 14 mand63 Mandelbrot example (k = 6) 63 Vanni Noferini 15 polynomial from V. Noferini 12 16 polynomial from V. Noferini 35 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 28/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials (3) Roots known! No. Description Deg. Jenkins and Traub 17 p1(z) with a =1 e8 3 18 p1(z) with a =1 e15 3 19 p1(z) with a =1 e+8 3 20 p1(z) with a =1 e+15 3 21 p3(z) under ow test 10 22 p3(z) under ow test 20 23 p10(z) de ation test a = 10 e+3 3 24 p10(z) de ation test a = 10 e+6 3 25 p10(z) de ation test a = 10 e+9 3 26 p11(z) de ation test m = 15 60 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 29/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials (4) Roots unknown! No. Description Deg. 27 Bernoulli polynomial (k = 20) 20 28 truncated exponential (k = 20) 20 used in Bevilacqua, Del Corso, and Gemignani 29{33 p1(z) with m = 10, 20, 30, 256, 512 2m 34{38 p2(z) with m = 10, 20, 30, 256, 512 l 2m 39{43 p3(z) with m + 1 = 20, : : : , 1024, = 0:9 m + 1 44{48 p3(z) with m + 1 = 20, : : : , 1024, = 0:999 m + 1 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 30/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error No AMVW ZHSEQR ZGEEV BBEGG BEGG 1 15 16 16 01 15 2 15 15 15 01 15 3 14 15 14 01 01 4 15 15 15 01 14 5 16 15 15 06 15 6 15 15 15 06 14 7 15 15 15 04 14 8 15 16 15 01 14 9 14 15 15 01 01 10 15 15 15 01 14 11 15 15 15 01 14 12 15 13 15 01 06 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 31/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error (2) No AMVW ZHSEQR ZGEEV BBEGG BEGG 13 15 15 15 01 14 14 15 15 15 01 14 15 15 15 12 01 01 16 15 15 10 01 01 17 17 16 22 01 17 18 17 16 30 01 16 19 16 01 24 +07 01 20 17 +13 17 +14 01 21 16 16 18 03 15 22 16 15 17 02 13 23 16 16 15 +03 16 24 16 16 16 +06 15 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 32/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error (3) No AMVW ZHSEQR ZGEEV BBEGG BEGG 25 26 16 16 +09 11 26 15 15 14 01 13 27 15 15 15 01 14 28 12 14 15 01 03 29 15 15 15 01 14 30 15 15 14 01 13 31 15 14 14 02 13 32 13 13 13 01 11 33 13 13 13 01 10 34 15 15 15 01 14 35 15 15 15 02 14 36 15 14 15 01 14 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 33/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error (4) No AMVW ZHSEQR ZGEEV BBEGG BEGG 37 14 14 14 02 12 38 14 13 14 02 11 39 15 15 15 01 14 40 15 14 14 01 13 41 15 14 14 01 13 42 13 13 13 +00 11 43 13 13 13 01 10 44 15 14 14 +00 13 45 15 14 14 +00 13 46 15 14 14 01 13 47 13 13 13 +00 11 48 12 12 13 +00 10 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 34/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error No AMVW DHSEQR DGEEV BBEGG CGXZ 1 15 15 15 09 11 2 15 15 15 01 06 3 14 15 15 01 01 4 15 15 15 13 14 5 16 16 16 15 15 6 15 15 16 02 14 7 15 15 15 02 15 8 15 15 15 01 14 9 15 15 15 01 04 10 15 15 15 14 14 11 15 15 15 15 15 12 06 12 15 01 01 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 35/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error (2) No AMVW DHSEQR DGEEV BBEGG CGXZ 13 15 15 16 02 14 14 15 15 15 04 13 15 15 15 10 01 10 16 15 15 10 01 05 17 17 16 24 16 16 18 18 17 +00 16 16 19 02 +00 16 01 01 20 01 17 16 01 01 21 17 16 17 15 05 22 16 16 17 02 01 23 16 16 16 16 16 24 14 16 15 15 15 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 36/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error (3) No AMVW DHSEQR DGEEV BBEGG CGXZ 25 16 16 24 16 11 26 15 14 15 14 14 27 15 15 15 11 14 28 14 14 15 01 01 29 15 15 15 15 15 30 15 14 14 15 14 31 15 14 14 15 14 32 14 14 13 14 11 33 13 13 13 13 09 34 15 15 15 15 15 35 15 15 15 14 15 36 15 14 15 15 15 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 37/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error (4) No AMVW DHSEQR DGEEV BBEGG CGXZ 37 14 14 14 12 12 38 14 13 14 11 +07 39 15 14 15 15 15 40 15 14 14 13 14 41 15 14 14 13 14 42 14 13 13 11 13 43 13 12 12 10 12 44 15 14 14 13 14 45 15 14 14 13 13 46 14 14 14 13 13 47 14 13 13 12 12 48 13 12 12 10 12 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 38/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Backward stability We have a proof for U(A + A)U = ^A; where kAk2 kcoecients of p(x)k22 O(m): J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Backward stability We have a proof for U(A + A)U = ^A; where kAk2 kcoecients of p(x)k22 O(m): The square is annoying, and in the analysis also a dependency of ja0j. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Backward stability We have a proof for U(A + A)U = ^A; where kAk2 kcoecients of p(x)k22 O(m): The square is annoying, and in the analysis also a dependency of ja0j. But in the numerical experiments Tests with small ja0j no problem: as accurate as LAPACK. Tests with growing kxk2 no problem: as accurate as LAPACK. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Conclusion We have a new fast companion eigenvalue method: about as accurate as LAPACK, and more accurate than the other fast methods, does OK on harder problems J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Conclusion We have a new fast companion eigenvalue method: about as accurate as LAPACK, and more accurate than the other fast methods, does OK on harder problems faster than LAPACK from size 16 on, faster than all other fast methods, J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments Conclusion We have a new fast companion eigenvalue method: about as accurate as LAPACK, and more accurate than the other fast methods, does OK on harder problems faster than LAPACK from size 16 on, faster than all other fast methods, almost backward stable (see preprint). J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/ homepage/software/companion_qr.php. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/ homepage/software/companion_qr.php. Package by Andreas Noack in julia (http://julialang.org/): Pkg.clone(https://github.com/andreasnoackjensen/AMVW.jl) Pkg.build(AMVW). J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/ homepage/software/companion_qr.php. Package by Andreas Noack in julia (http://julialang.org/): Pkg.clone(https://github.com/andreasnoackjensen/AMVW.jl) Pkg.build(AMVW). Thank you for your attention. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41

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Fast and backward stable computation of roots of polynomials

  • 1. 1 . . . 1 a0 a1 ... an1 Fast and stable computation of the roots of polynomials Jared L. Aurentz1, Thomas Mach2, Raf Vandebril2, and David S. Watkins1 1Dept. of Mathematics, Washington State University 2Dept. Computer Science, KU Leuven Structured Numerical Linear and Multilinear Algebra: Analysis, Algorithms and Applications 11 September 2014 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 1/41
  • 2. Introduction Factorization Facts Francis's Algorithm Numerical experiments Collaborators This is joint work with David S. Watkins (WSU) Jared L. Aurentz (WSU) Raf Vandebril (KU Leuven) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 2/41
  • 3. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Problem p(x) = xn + an1xn1 + an2xn2 + + a0 = 0 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 3/41
  • 4. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Problem p(x) = xn + an1xn1 + an2xn2 + + a0 = 0 Companion matrix A = 2 66666664 a0 1 a1 1 a2 . . . ... 1 an2 1 an1 3 77777775 Get the zeros of p by computing the eigenvalues of A. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 3/41
  • 5. Introduction Factorization Facts Francis's Algorithm Numerical experiments Cost If structure not exploited: If structure exploited: J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 4/41
  • 6. Introduction Factorization Facts Francis's Algorithm Numerical experiments Cost If structure not exploited: O(n2) storage O(n3) ops Francis's implicitly-shifted QR algorithm Matlab's roots command. If structure exploited: O(n) storage O(n2) ops data-sparse representation + Francis's algorithm several methods proposed including some by my coauthors (fast but potentially unstable) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 4/41
  • 7. Introduction Factorization Facts Francis's Algorithm Numerical experiments Some of the Competitors Chandrasekaran, Gu, Xia, Zhu (2007) Bini, Boito, Eidelman, Gemignani, Gohberg (2010) Boito, Eidelman, Gemignani, Gohberg (2012) Fortran codes available Quasiseparable generator representation we use essentially 2 2 unitary matrices results on backward stability J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 5/41
  • 8. Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Contribution We present yet another O(n) representation Francis's algorithm in O(n) ops/iteration Fortran codes (we're faster) normwise almost backward stable (We can prove it.) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 6/41
  • 9. Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure companion matrix is unitary-plus-rank-one A = 2 0 0 ei 1 0 6664 . . . ... 1 0 3 7775+ 2 6664 0 0 ei a0 0 0 a1 ... ... ... 0 0 an1 3 7775 preserved by unitary similarities J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
  • 10. Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure companion matrix is unitary-plus-rank-one A = 2 0 0 ei 1 0 6664 . . . ... 1 0 3 7775+ 2 6664 0 0 ei a0 0 0 a1 ... ... ... 0 0 an1 3 7775 preserved by unitary similarities companion matrix is also upper Hessenberg. preserved by Francis's algorithm J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
  • 11. Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure companion matrix is unitary-plus-rank-one A = 2 0 0 ei 1 0 6664 . . . ... 1 0 3 7775+ 2 6664 0 0 ei a0 0 0 a1 ... ... ... 0 0 an1 3 7775 preserved by unitary similarities companion matrix is also upper Hessenberg. preserved by Francis's algorithm We exploit this structure. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
  • 12. Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure Chandrasekaran, Gu, Xia, Zhu (2007) Compute the companion's QR factorization A = QR Q is upper Hessenberg and unitary. R is upper triangular and unitary-plus-rank-one. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 8/41
  • 13. Introduction Factorization Facts Francis's Algorithm Numerical experiments Structure Chandrasekaran, Gu, Xia, Zhu (2007) Compute the companion's QR factorization A = QR Q is upper Hessenberg and unitary. R is upper triangular and unitary-plus-rank-one. We do this too. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 8/41
  • 14. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Unitary Part (Matrix Q) Q factored in core transformations 2 664 3 775 = 2 664 1 1 3 775 2 1 664 1 3 775 2 1 664 1 3 775 Q = O(n) storage J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 9/41
  • 15. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Upper Triangular Part (Matrix R) R = U + xyT unitary-plus-rank-one, so R has quasiseparable rank 2. R = 2 666666664 . . . ... ... ... . . . ... 3 777777775 quasiseparable generator representation (O(n) storage) Chandrasekaran, Gu, Xia, and Zhu exploit this structure. We do it dierently. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 10/41
  • 16. Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix A) Add a row/column for extra wiggle room A = 2 666664 0 a0 1 1 a1 0 . . . ... ... 1 an1 0 0 0 3 777775 Extra zero root can be de ated immediately. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 11/41
  • 17. Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix A) Add a row/column for extra wiggle room A = 2 666664 0 a0 1 1 a1 0 . . . ... ... 1 an1 0 0 0 3 777775 Extra zero root can be de ated immediately. A = QR, where Q = 2 666664 0 1 0 1 0 0 . . . ... ... 1 0 0 0 1 3 777775 R = 2 666664 1 a1 0 . . . ... ... 1 an1 0 a0 1 0 0 3 777775 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 11/41
  • 18. Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix Q) Q = 2 666664 0 1 0 1 0 0 . . . ... ... 1 0 0 0 1 3 777775 Q is stored in factored form Q = Q = Q1Q2 Qn1 (mark the n 1 instead of n) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 12/41
  • 19. Introduction Factorization Facts Francis's Algorithm Numerical experiments Our Representation (Matrix R) R = 2 666664 1 a1 0 . . . ... ... 1 an1 0 a0 1 0 0 3 777775 R is unitary-plus-rank-one: 2 666664 1 0 0 . . . ... ... 1 0 0 0 1 1 0 3 777775 + 2 666664 0 a1 0 . . . ... ... 0 an1 0 a0 0 1 0 3 777775 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 13/41
  • 20. Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R R = U + xyT , where xyT = 2 666664 a1 ... an1 a0 1 3 777775 0 0 1 0 Next step: Roll up x. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 14/41
  • 21. Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 3 64 75 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
  • 22. Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 3 64 75 0 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
  • 23. Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 3 64 75 0 0 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
  • 24. Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 64 0 0 0 3 75 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
  • 25. Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of R 2 64 3 75 = 2 64 0 0 0 3 75 C1 Cn1Cnx = e1 (w.l.g. = 1) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
  • 26. Introduction Factorization Facts Francis's Algorithm Numerical experiments Representation of A Altogether we have A = QR = Q C (B + e1yT ) A = Q1 Qn1 C 1 (B1 Bn + e1yT ) n C Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R . J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 16/41
  • 27. Introduction Factorization Facts Francis's Algorithm Numerical experiments Francis's Iterations We have complex single-shift code, real double-shift code. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
  • 28. Introduction Factorization Facts Francis's Algorithm Numerical experiments Francis's Iterations We have complex single-shift code, real double-shift code. We describe single-shift case for simplicity ignoring rank-one part. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
  • 29. Introduction Factorization Facts Francis's Algorithm Numerical experiments Francis's Iterations We have complex single-shift code, real double-shift code. We describe single-shift case for simplicity ignoring rank-one part. Two basic operations: Fusion ) Turnover , J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
  • 30. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 31. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 32. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 33. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 34. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 35. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 36. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 37. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 38. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 39. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 40. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 41. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 42. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 43. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 44. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 45. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 46. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 47. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 48. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 49. Introduction Factorization Facts Francis's Algorithm Numerical experiments The Bulge Chase Q1 Qn1 C n C 1 B = B1 Bn + 1 0 0 0 0 e1 yT R J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
  • 50. Introduction Factorization Facts Francis's Algorithm Numerical experiments Done! Iteration complete! Cost: 3n turnovers/iteration, so O(n) ops/iteration Double-shift iteration is similar: chase two core transformations instead of one, 7n turnovers/iteration. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 19/41
  • 51. Introduction Factorization Facts Francis's Algorithm Numerical experiments Speed Comparison, Fortran implementation Contestants LAPACK code xHSEQR (O(n3)) AMVW (our single-shift code) BBEGG (Bini, Boito, et al. 2010) only single-shift code BEGG (Boito, Eidelman, et al. 2012) only double-shift code CGXZ (Chandrasekaran, Gu, et al. 2007) J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 20/41
  • 52. Introduction Factorization Facts Francis's Algorithm Numerical experiments complex single-shift|random coecients 101 102 103 104 BBEGG: 23.297 BEGG: 15.442 AMVW: 4.269 101 102 105 Time in seconds AMVW ZHSEQR BBEGG BEGG 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 21/41
  • 53. Introduction Factorization Facts Francis's Algorithm Numerical experiments real double-shift|random coecients 101 102 103 104 CGXZ: 9.023 BBEGG: 7.899 AMVW: 3.817 101 102 105 Time in seconds AMVW DHSEQR BBEGG CGXZ 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 22/41
  • 54. Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift|random aeb, b 2 [R; R], R = 5; : : : ; 25 101 102 103 104 101 102 105 Time in seconds AMVW ZHSEQR BBEGG BEGG 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 23/41
  • 55. Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift|random aeb, b 2 [R; R], R = 5; : : : ; 25 101 102 103 104 101 102 105 Time in seconds AMVW DHSEQR BBEGG CGXZ 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 24/41
  • 56. Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift|zn 1 101 102 103 104 BBEGG: 27.847 BEGG: 16.933 AMVW: 3.198 101 102 105 Time in seconds AMVW ZHSEQR BBEGG BEGG 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 25/41
  • 57. Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift|zn 1 101 102 103 104 CGXZ: 8.883 BBEGG: 6.595 AMVW: 3.554 101 102 105 Time in seconds AMVW DHSEQR BBEGG CGXZ 108 Accuracy 1012 101 102 103 104 1016 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 26/41
  • 58. Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials Roots known! No. Description Deg. 1 Wilkinson polynomial 10 2 Wilkinson polynomial 15 3 Wilkinson polynomial 20 4 scaled and shifted Wilkinson poly. 20 5 reverse Wilkinson polynomial 10 6 reverse Wilkinson polynomial 15 7 reverse Wilkinson polynomial 20 8 prescribed roots of varying scale 20 9 prescribed roots of varying scale 3 20 10 Chebyshev polynomial 20 11 z20 + z19 + + z + 1 20 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 27/41
  • 59. Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials (2) Roots known! No. Description Deg. MPSolve 12 trv m, C. Traverso 24 13 mand31 Mandelbrot example (k = 5) 31 14 mand63 Mandelbrot example (k = 6) 63 Vanni Noferini 15 polynomial from V. Noferini 12 16 polynomial from V. Noferini 35 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 28/41
  • 60. Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials (3) Roots known! No. Description Deg. Jenkins and Traub 17 p1(z) with a =1 e8 3 18 p1(z) with a =1 e15 3 19 p1(z) with a =1 e+8 3 20 p1(z) with a =1 e+15 3 21 p3(z) under ow test 10 22 p3(z) under ow test 20 23 p10(z) de ation test a = 10 e+3 3 24 p10(z) de ation test a = 10 e+6 3 25 p10(z) de ation test a = 10 e+9 3 26 p11(z) de ation test m = 15 60 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 29/41
  • 61. Introduction Factorization Facts Francis's Algorithm Numerical experiments Special Polynomials (4) Roots unknown! No. Description Deg. 27 Bernoulli polynomial (k = 20) 20 28 truncated exponential (k = 20) 20 used in Bevilacqua, Del Corso, and Gemignani 29{33 p1(z) with m = 10, 20, 30, 256, 512 2m 34{38 p2(z) with m = 10, 20, 30, 256, 512 l 2m 39{43 p3(z) with m + 1 = 20, : : : , 1024, = 0:9 m + 1 44{48 p3(z) with m + 1 = 20, : : : , 1024, = 0:999 m + 1 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 30/41
  • 62. Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error No AMVW ZHSEQR ZGEEV BBEGG BEGG 1 15 16 16 01 15 2 15 15 15 01 15 3 14 15 14 01 01 4 15 15 15 01 14 5 16 15 15 06 15 6 15 15 15 06 14 7 15 15 15 04 14 8 15 16 15 01 14 9 14 15 15 01 01 10 15 15 15 01 14 11 15 15 15 01 14 12 15 13 15 01 06 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 31/41
  • 63. Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error (2) No AMVW ZHSEQR ZGEEV BBEGG BEGG 13 15 15 15 01 14 14 15 15 15 01 14 15 15 15 12 01 01 16 15 15 10 01 01 17 17 16 22 01 17 18 17 16 30 01 16 19 16 01 24 +07 01 20 17 +13 17 +14 01 21 16 16 18 03 15 22 16 15 17 02 13 23 16 16 15 +03 16 24 16 16 16 +06 15 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 32/41
  • 64. Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error (3) No AMVW ZHSEQR ZGEEV BBEGG BEGG 25 26 16 16 +09 11 26 15 15 14 01 13 27 15 15 15 01 14 28 12 14 15 01 03 29 15 15 15 01 14 30 15 15 14 01 13 31 15 14 14 02 13 32 13 13 13 01 11 33 13 13 13 01 10 34 15 15 15 01 14 35 15 15 15 02 14 36 15 14 15 01 14 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 33/41
  • 65. Introduction Factorization Facts Francis's Algorithm Numerical experiments single-shift, unbalanced, relative backward error (4) No AMVW ZHSEQR ZGEEV BBEGG BEGG 37 14 14 14 02 12 38 14 13 14 02 11 39 15 15 15 01 14 40 15 14 14 01 13 41 15 14 14 01 13 42 13 13 13 +00 11 43 13 13 13 01 10 44 15 14 14 +00 13 45 15 14 14 +00 13 46 15 14 14 01 13 47 13 13 13 +00 11 48 12 12 13 +00 10 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 34/41
  • 66. Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error No AMVW DHSEQR DGEEV BBEGG CGXZ 1 15 15 15 09 11 2 15 15 15 01 06 3 14 15 15 01 01 4 15 15 15 13 14 5 16 16 16 15 15 6 15 15 16 02 14 7 15 15 15 02 15 8 15 15 15 01 14 9 15 15 15 01 04 10 15 15 15 14 14 11 15 15 15 15 15 12 06 12 15 01 01 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 35/41
  • 67. Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error (2) No AMVW DHSEQR DGEEV BBEGG CGXZ 13 15 15 16 02 14 14 15 15 15 04 13 15 15 15 10 01 10 16 15 15 10 01 05 17 17 16 24 16 16 18 18 17 +00 16 16 19 02 +00 16 01 01 20 01 17 16 01 01 21 17 16 17 15 05 22 16 16 17 02 01 23 16 16 16 16 16 24 14 16 15 15 15 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 36/41
  • 68. Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error (3) No AMVW DHSEQR DGEEV BBEGG CGXZ 25 16 16 24 16 11 26 15 14 15 14 14 27 15 15 15 11 14 28 14 14 15 01 01 29 15 15 15 15 15 30 15 14 14 15 14 31 15 14 14 15 14 32 14 14 13 14 11 33 13 13 13 13 09 34 15 15 15 15 15 35 15 15 15 14 15 36 15 14 15 15 15 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 37/41
  • 69. Introduction Factorization Facts Francis's Algorithm Numerical experiments double-shift, unbalanced, relative backward error (4) No AMVW DHSEQR DGEEV BBEGG CGXZ 37 14 14 14 12 12 38 14 13 14 11 +07 39 15 14 15 15 15 40 15 14 14 13 14 41 15 14 14 13 14 42 14 13 13 11 13 43 13 12 12 10 12 44 15 14 14 13 14 45 15 14 14 13 13 46 14 14 14 13 13 47 14 13 13 12 12 48 13 12 12 10 12 J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 38/41
  • 70. Introduction Factorization Facts Francis's Algorithm Numerical experiments Backward stability We have a proof for U(A + A)U = ^A; where kAk2 kcoecients of p(x)k22 O(m): J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
  • 71. Introduction Factorization Facts Francis's Algorithm Numerical experiments Backward stability We have a proof for U(A + A)U = ^A; where kAk2 kcoecients of p(x)k22 O(m): The square is annoying, and in the analysis also a dependency of ja0j. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
  • 72. Introduction Factorization Facts Francis's Algorithm Numerical experiments Backward stability We have a proof for U(A + A)U = ^A; where kAk2 kcoecients of p(x)k22 O(m): The square is annoying, and in the analysis also a dependency of ja0j. But in the numerical experiments Tests with small ja0j no problem: as accurate as LAPACK. Tests with growing kxk2 no problem: as accurate as LAPACK. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
  • 73. Introduction Factorization Facts Francis's Algorithm Numerical experiments Conclusion We have a new fast companion eigenvalue method: about as accurate as LAPACK, and more accurate than the other fast methods, does OK on harder problems J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
  • 74. Introduction Factorization Facts Francis's Algorithm Numerical experiments Conclusion We have a new fast companion eigenvalue method: about as accurate as LAPACK, and more accurate than the other fast methods, does OK on harder problems faster than LAPACK from size 16 on, faster than all other fast methods, J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
  • 75. Introduction Factorization Facts Francis's Algorithm Numerical experiments Conclusion We have a new fast companion eigenvalue method: about as accurate as LAPACK, and more accurate than the other fast methods, does OK on harder problems faster than LAPACK from size 16 on, faster than all other fast methods, almost backward stable (see preprint). J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
  • 76. Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
  • 77. Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/ homepage/software/companion_qr.php. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
  • 78. Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/ homepage/software/companion_qr.php. Package by Andreas Noack in julia (http://julialang.org/): Pkg.clone(https://github.com/andreasnoackjensen/AMVW.jl) Pkg.build(AMVW). J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
  • 79. Introduction Factorization Facts Francis's Algorithm Numerical experiments More ... Preprint: http://www.cs.kuleuven.be/publicaties/ rapporten/tw/TW654.abs.html. Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/ homepage/software/companion_qr.php. Package by Andreas Noack in julia (http://julialang.org/): Pkg.clone(https://github.com/andreasnoackjensen/AMVW.jl) Pkg.build(AMVW). Thank you for your attention. J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41