fast stable computation of the eigenvalues of companion
play

Fast Stable Computation of the Eigenvalues of Companion Matrices - PowerPoint PPT Presentation

Mar 13, 2023 •445 likes •1.57k views

Fast Stable Computation of the Eigenvalues of Companion Matrices David S. Watkins Department of Mathematics Washington State University Spa, Belgium, 2014 David S. Watkins Eigenvalues of Companion Matrices Collaborators This is joint work

  1. The Unitary Part  x x x x   x x   1   1  1 x x x x x x x x          =         x x x 1 x x x x        x x 1 1 x x � � � Q = � � � O ( n ) storage David S. Watkins Eigenvalues of Companion Matrices

  2. The Unitary Part  x x x x   x x   1   1  1 x x x x x x x x          =         x x x 1 x x x x        x x 1 1 x x � � � Q = � � � O ( n ) storage David S. Watkins Eigenvalues of Companion Matrices

  3. The Upper Triangular Part R = U + xy T unitary-plus-rank-one, so R has quasiseparable rank 2.   x · · · x x · · · x . . . ... . . .  . . .      x x x · · ·   R =   x x · · ·    .  ... .   .   x quasiseparable generator representation ( O ( n ) storage) Chandrasekaran et. al. exploit this structure. We do it differently. David S. Watkins Eigenvalues of Companion Matrices

  4. The Upper Triangular Part R = U + xy T unitary-plus-rank-one, so R has quasiseparable rank 2.   x · · · x x · · · x . . . ... . . .  . . .      x x x · · ·   R =   x x · · ·    .  ... .   .   x quasiseparable generator representation ( O ( n ) storage) Chandrasekaran et. al. exploit this structure. We do it differently. David S. Watkins Eigenvalues of Companion Matrices

  5. The Upper Triangular Part R = U + xy T unitary-plus-rank-one, so R has quasiseparable rank 2.   x · · · x x · · · x . . . ... . . .  . . .      x x x · · ·   R =   x x · · ·    .  ... .   .   x quasiseparable generator representation ( O ( n ) storage) Chandrasekaran et. al. exploit this structure. We do it differently. David S. Watkins Eigenvalues of Companion Matrices

  6. The Upper Triangular Part R = U + xy T unitary-plus-rank-one, so R has quasiseparable rank 2.   x · · · x x · · · x . . . ... . . .  . . .      x x x · · ·   R =   x x · · ·    .  ... .   .   x quasiseparable generator representation ( O ( n ) storage) Chandrasekaran et. al. exploit this structure. We do it differently. David S. Watkins Eigenvalues of Companion Matrices

  7. The Upper Triangular Part R = U + xy T unitary-plus-rank-one, so R has quasiseparable rank 2.   x · · · x x · · · x . . . ... . . .  . . .      x x x · · ·   R =   x x · · ·    .  ... .   .   x quasiseparable generator representation ( O ( n ) storage) Chandrasekaran et. al. exploit this structure. We do it differently. David S. Watkins Eigenvalues of Companion Matrices

  8. Our Representation Add a row/column for extra wiggle room  0 1  − a 0 1 − a 1 0   . .  ...  . . A =   . .     1 0 − a n − 1   0 0 Extra zero root can be deflated immediately. A = QR , where  0 ± 1 0   1 0  − a 1 . . ... 1 0 0 . .    . .  . .  ...    . . Q = R =     . . 1 − a n − 1 0         1 0 0 ∓ 1 ± a 0     0 1 0 0 David S. Watkins Eigenvalues of Companion Matrices

  9. Our Representation Add a row/column for extra wiggle room  0 1  − a 0 1 − a 1 0   . .  ...  . . A =   . .     1 0 − a n − 1   0 0 Extra zero root can be deflated immediately. A = QR , where  0 ± 1 0   1 0  − a 1 . . ... 1 0 0 . .    . .  . .  ...    . . Q = R =     . . 1 − a n − 1 0         1 0 0 ∓ 1 ± a 0     0 1 0 0 David S. Watkins Eigenvalues of Companion Matrices

  10. Our Representation  0 ± 1 0  1 0 0   . .  ...  . . Q =   . .     1 0 0   0 1 Q is stored in factored form � � � Q = � � � Q = Q 1 Q 2 · · · Q n − 1 David S. Watkins Eigenvalues of Companion Matrices

  11. Our Representation  0 ± 1 0  1 0 0   . .  ...  . . Q =   . .     1 0 0   0 1 Q is stored in factored form � � � Q = � � � Q = Q 1 Q 2 · · · Q n − 1 David S. Watkins Eigenvalues of Companion Matrices

  12. Our Representation  1 0  − a 1 . . ... . .  . .    R =   1 − a n − 1 0     ∓ 1 ± a 0   0 0 R is unitary-plus-rank-one:     1 0 0 0 − a 1 0 . . . . ... ... . . . .  . .   . .      +  1 0 0   0 0  − a n − 1         0 ∓ 1 ± a 0 0     ± 1 0 ∓ 1 0 David S. Watkins Eigenvalues of Companion Matrices

  13. Representation of R R = U + xy T , where   − a 1 . .  .    xy T = � � 0 0 1 0 · · ·   − a n − 1     ± a 0   ∓ 1 Next step: Roll up x . David S. Watkins Eigenvalues of Companion Matrices

  14. Representation of R R = U + xy T , where   − a 1 . .  .    xy T = � � 0 0 1 0 · · ·   − a n − 1     ± a 0   ∓ 1 Next step: Roll up x . David S. Watkins Eigenvalues of Companion Matrices

  15. Representation of R R = U + xy T , where   − a 1 . .  .    xy T = � � 0 0 1 0 · · ·   − a n − 1     ± a 0   ∓ 1 Next step: Roll up x . David S. Watkins Eigenvalues of Companion Matrices

  16. Representation of R     x x x x  =     x x    x x David S. Watkins Eigenvalues of Companion Matrices

  17. Representation of R     x x x x  =     x x    � � x 0 David S. Watkins Eigenvalues of Companion Matrices

  18. Representation of R     x x x x �  =     � 0 x    � � x 0 David S. Watkins Eigenvalues of Companion Matrices

  19. Representation of R     x x � � x 0 �  =     � 0 x    � � x 0 David S. Watkins Eigenvalues of Companion Matrices

  20. Representation of R     x x � � 0 x �  =     � x 0    � � 0 x C 1 · · · C n − 1 C n x = α e 1 (w.l.g. α = 1) David S. Watkins Eigenvalues of Companion Matrices

  21. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  22. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  23. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  24. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  25. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  26. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  27. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  28. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  29. Representation of R C 1 · · · C n − 1 C n x = e 1 Cx = e 1 C ∗ e 1 = x R = U + xy T = U + C ∗ e 1 y T = C ∗ ( CU + e 1 y T ) R = C ∗ ( B + e 1 y T ) B is upper Hessenberg (and unitary) so B = B 1 · · · B n . R = C ∗ ( B + e 1 y T ) = C ∗ 1 ( B 1 · · · B n + e 1 y T ) n · · · C ∗ O ( n ) storage Bonus: Redundancy! No need to keep track of y . David S. Watkins Eigenvalues of Companion Matrices

  30. Representation of A Altogether we have A = QR = Q C ∗ ( B + e 1 y T ) 1 ( B 1 · · · B n + e 1 y T ) A = Q 1 · · · Q n − 1 C ∗ n · · · C ∗   � � � � � � � � �   � � � + · · ·   � � �   � � �   � � � � David S. Watkins Eigenvalues of Companion Matrices

  31. Francis Iterations We have complex single-shift code . . . real double-shift code. We describe single-shift case for simplicity. ignoring rank-one part . . . � � � � � � � � � A = � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  32. Francis Iterations We have complex single-shift code . . . real double-shift code. We describe single-shift case for simplicity. ignoring rank-one part . . . � � � � � � � � � A = � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  33. Francis Iterations We have complex single-shift code . . . real double-shift code. We describe single-shift case for simplicity. ignoring rank-one part . . . � � � � � � � � � A = � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  34. Two Basic Operations Two basic operations: Fusion � � � ⇒ � � � Turnover (aka shift through, Givens swap, . . . ) � � � � � � ⇔ � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  35. Two Basic Operations Two basic operations: Fusion � � � ⇒ � � � Turnover (aka shift through, Givens swap, . . . ) � � � � � � ⇔ � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  36. Two Basic Operations Two basic operations: Fusion � � � ⇒ � � � Turnover (aka shift through, Givens swap, . . . ) � � � � � � ⇔ � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  37. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  38. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  39. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  40. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  41. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  42. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  43. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  44. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  45. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  46. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  47. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  48. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  49. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  50. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  51. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  52. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  53. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  54. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  55. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  56. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  57. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  58. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  59. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  60. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  61. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  62. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  63. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  64. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  65. The Bulge Chase � � � � � � � � � � � � � � � � � � � � � � David S. Watkins Eigenvalues of Companion Matrices

  66. Done! iteration complete! Cost: 3 n turnovers/iteration, so O ( n ) flops/iteration Double-shift iteration is similar. (Chase two core transformations instead of one.) David S. Watkins Eigenvalues of Companion Matrices

  67. Done! iteration complete! Cost: 3 n turnovers/iteration, so O ( n ) flops/iteration Double-shift iteration is similar. (Chase two core transformations instead of one.) David S. Watkins Eigenvalues of Companion Matrices

  68. Done! iteration complete! Cost: 3 n turnovers/iteration, so O ( n ) flops/iteration Double-shift iteration is similar. (Chase two core transformations instead of one.) David S. Watkins Eigenvalues of Companion Matrices

  69. Speed Comparison, Complex Case Contestants ( O ( n 3 )) LAPACK code ZHSEQR BEGG (Boito et. al. 2012) AMVW (our single-shift code) David S. Watkins Eigenvalues of Companion Matrices

  70. Speed Comparison, Complex Case 3 10 LAPACK BEGG 2 10 AMVW 1 10 0 10 time (seconds) −1 10 −2 10 −3 10 −4 10 −5 10 0 1 2 3 4 5 10 10 10 10 10 10 degree David S. Watkins Eigenvalues of Companion Matrices

  71. Speed Comparison, Complex Case At degree 1024: method time LAPACK 7.14 BEGG 1.02 AMVW 0.18 David S. Watkins Eigenvalues of Companion Matrices

  72. Speed Comparison, Real Case Results similar for double-shift code . . . . . . but we’re only about twice as fast as the competition. (compared with Chandrasekaran et. al.) David S. Watkins Eigenvalues of Companion Matrices

Recommend

Fast computation of eigenvalues of companion, comrade, and related matrices David S. Watkins

Fast computation of eigenvalues of companion, comrade, and related matrices David S. Watkins

Fast computation of eigenvalues of companion, comrade, and related matrices David S. Watkins Department of Mathematics Washington State University Providence, June 2013 David S. Watkins companion, comrade, and related matrices Collaborators

1.4k views • 137 slides
Computing the eigenvalues of a Joint work with Raf Vandebril and Paul Van Dooren companion

Computing the eigenvalues of a Joint work with Raf Vandebril and Paul Van Dooren companion

Computing the eigenvalues of a companion matrix Marc Van Barel Computing the eigenvalues of a Joint work with Raf Vandebril and Paul Van Dooren companion matrix Structured Linear Algebra Problems: Analysis, Algorithms, and Applications

1.12k views • 64 slides
( A I ) = 0 However, there is no exact formula for polynomials of degree five

( A I ) = 0 However, there is no exact formula for polynomials of degree five

Lesson 21 C OMPUTATION OF EIGENVALUES We consider the problem of computation of eigenvalues To make life (much!) simpler, let's assume we have a symmetric matrix A R n n , which has decomposition A = Q Q Unlike previous

503 views • 26 slides
Chapter 12: Iterative Methods ES 240: Scientific and Engineering Computation. Iterative Methods

Chapter 12: Iterative Methods ES 240: Scientific and Engineering Computation. Iterative Methods

ES 240: Scientific and Engineering Computation. Iterative Methods and Eigenvalues Uchechukwu Ofoegbu Temple University Chapter 12: Iterative Methods ES 240: Scientific and Engineering Computation. Iterative Methods and Eigenvalues Gauss-

261 views • 13 slides
OPERATOR SPLITTING METHODS FOR COMPUTATION OF EIGENVALUES OF REGULAR STURM-LIOUVILLE PROBLEMS

OPERATOR SPLITTING METHODS FOR COMPUTATION OF EIGENVALUES OF REGULAR STURM-LIOUVILLE PROBLEMS

OPERATOR SPLITTING METHODS FOR COMPUTATION OF EIGENVALUES OF REGULAR STURM-LIOUVILLE PROBLEMS Ismail G UZEL ismailgzel@gmail.com Dokuz Eyl ul University IZM IR 13/06/2016 Liouville J. Sturm J.C.F OPERATOR SPLITTING METHOD

607 views • 59 slides
Companion Planting: Plants Need Friends, Too! What is Companion Planting? Planting two or

Companion Planting: Plants Need Friends, Too! What is Companion Planting? Planting two or

Companion Planting: Plants Need Friends, Too! What is Companion Planting? Planting two or more plant species in close proximity to culturally benefit each other by attracting or deterring pests, keeping the soil healthy, and improving

858 views • 22 slides
Graph Computation on Computer Cluster? Steep learning curve Cost Overkill for smaller

Graph Computation on Computer Cluster? Steep learning curve Cost Overkill for smaller

MMap Fast Billion-Scale Graph Computation on a PC via Memory Mapping Lead by Zhiyuan (Jerry) Lin Georgia Tech CS Undergrad Now: Stanford 1st year PhD student MMap: Fast Billion-Scale Graph Computation on a PC via Memory Mapping .

130 views • 11 slides
Fast Constant-Time GCD Computation and Modular Inversion Daniel J. Bernstein 1,2 Bo-Yin Yang 3 1

Fast Constant-Time GCD Computation and Modular Inversion Daniel J. Bernstein 1,2 Bo-Yin Yang 3 1

Fast Constant-Time GCD Computation and Modular Inversion Daniel J. Bernstein 1,2 Bo-Yin Yang 3 1 University of Illinois at Chicago 2 Ruhr Universt at Bochum 3 Academia Sinica Monday, August 26, 2019 Summary: Fast, Safe GCD and Inversions

431 views • 23 slides
Sol Fast Distributed Computation Over Slow Networks Fan Lai , Jie You, Xiangfeng Zhu Harsha V.

Sol Fast Distributed Computation Over Slow Networks Fan Lai , Jie You, Xiangfeng Zhu Harsha V.

Sol Fast Distributed Computation Over Slow Networks Fan Lai , Jie You, Xiangfeng Zhu Harsha V. Madhyastha, Mosharaf Chowdhury 1 Distributed Data Processing is Ubiquitous Distributed computation in Local-Area Networks (LAN) To accelerate

1.92k views • 77 slides
Multiple Instance Learning for Fast, Stable and Early RNN Predictions Don Dennis , Microsoft

Multiple Instance Learning for Fast, Stable and Early RNN Predictions Don Dennis , Microsoft

The Edge of Machine Learning Multiple Instance Learning for Fast, Stable and Early RNN Predictions Don Dennis , Microsoft Research India, Joint work with Chirag P., Harsha and Prateek Accepted to NIPS 18 1 Algorithms for the IDE - EdgeML

977 views • 70 slides
Numerically Stable Parallel Computation of (Co-)Variance Erich Schubert, Michael Gertz 30th Int.

Numerically Stable Parallel Computation of (Co-)Variance Erich Schubert, Michael Gertz 30th Int.

Numerically Stable Parallel Computation of (Co-)Variance Erich Schubert, Michael Gertz 30th Int. Conf. on Scientific and Statistical Database Management (SSDBM 18) July 911, 2018, Bozen-Bolzano, Italy Ruprecht-Karls-Universitt

356 views • 32 slides
Fast computation of power series solutions of systems of differential equations Alin Bostan

Fast computation of power series solutions of systems of differential equations Alin Bostan

Fast computation of power series solutions of systems of differential equations Alin Bostan ALGO, INRIA Rocquencourt joint work with F. Chyzak , F. Ollivier , B. Salvy , E. Schost , A. Sedoglavic Context find fast algorithms for solving (in

911 views • 22 slides
Chapter 5 Eigenvalues and Eigenvectors Section 5.1 Eigenvectors and Eigenvalues Motivation:

Chapter 5 Eigenvalues and Eigenvectors Section 5.1 Eigenvectors and Eigenvalues Motivation:

Chapter 5 Eigenvalues and Eigenvectors Section 5.1 Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics : 1. half of the newborn rabbits survive their first

353 views • 22 slides
Companion House March 2014 Making a referral to Companion House This presentation is designed

Companion House March 2014 Making a referral to Companion House This presentation is designed

Making referrals to Companion House March 2014 Making a referral to Companion House This presentation is designed to provide information to service providers about how to refer a client to Companion House Aims of the presentation To support

750 views • 36 slides
Fast and stable rational RBF-based Partition of Unity interpolation 1 Stefano De Marchi

Fast and stable rational RBF-based Partition of Unity interpolation 1 Stefano De Marchi

Fast and stable rational RBF-based Partition of Unity interpolation 1 Stefano De Marchi Department of Mathematics Tullio Levi-Civita University of Padova SMART 2017 - Gaeta September 18, 2017 1 joint work with A. Martinez (Padova) and E.

622 views • 37 slides
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they are intimately connected with the

377 views • 25 slides
Bio Detectors Accurate and precise Stable system Fast and visible response Versatile

Bio Detectors Accurate and precise Stable system Fast and visible response Versatile

Introduction Bio Detectors Accurate and precise Stable system Fast and visible response Versatile Our Idea Our Idea Fast Visible Output Accurate Detection

692 views • 39 slides
Section 5.5 Complex Eigenvalues (Part II) Motivation: Complex Versus Two Real Eigenvalues

Section 5.5 Complex Eigenvalues (Part II) Motivation: Complex Versus Two Real Eigenvalues

Section 5.5 Complex Eigenvalues (Part II) Motivation: Complex Versus Two Real Eigenvalues Todays decomposition is very analogous to diagonalization. Theorem Let A be a 2 2 matrix with linearly independent eigenvectors v 1 , v 2 and

700 views • 16 slides
Humanoid Companion Embedded GPUs can make your robotic companion more alive Hi Alex!

Humanoid Companion Embedded GPUs can make your robotic companion more alive Hi Alex!

Smarter Humanoid Companion Embedded GPUs can make your robotic companion more alive Hi Alex! Alexandre Mazel Innovation Software Director Oct 2017 Agenda Overview Innovation Team Presentation Mummer Research Project

399 views • 29 slides
March, 2012 Mahindra & Mahindra 2 India - Fast, Sustained and Stable Growth Story GDP

March, 2012 Mahindra & Mahindra 2 India - Fast, Sustained and Stable Growth Story GDP

March, 2012 Mahindra & Mahindra 2 India - Fast, Sustained and Stable Growth Story GDP growth F10 8.4% 8.4% F11 - 8.4 % F12E 6.9 % <GOIs first GDP estimate for FY12> An exponential increase in the consumer class

1.01k views • 65 slides
Stable Distributions for Stream Computation: Its as easy as 0,1,2 Graham Cormode

Stable Distributions for Stream Computation: Its as easy as 0,1,2 Graham Cormode

Stable Distributions for Stream Computation: Its as easy as 0,1,2 Graham Cormode graham@dimacs.rutgers.edu dimacs.rutgers.edu/ ~ graham "I had a really good night last night. I found I can count up to 1023 on my fingers. Chris

349 views • 11 slides
Fast and Stable Maximum Likelihood Estimation for Incomplete Multinomial Models Chenyang Zhang,

Fast and Stable Maximum Likelihood Estimation for Incomplete Multinomial Models Chenyang Zhang,

Fast and Stable Maximum Likelihood Estimation for Incomplete Multinomial Models Chenyang Zhang, Guosheng Yin Department of Statistics and Actuarial Science, The University of Hong Kong June 13, 2019 (HKU SAAS) ICML 2019 June 13, 2019 1 / 9

135 views • 10 slides
For matrix A ( p p ) with real eigenvalues, define F A , the empirical distribution function of

For matrix A ( p p ) with real eigenvalues, define F A , the empirical distribution function of

For matrix A ( p p ) with real eigenvalues, define F A , the empirical distribution function of the eigenvalues of A , to be F A ( x ) (1 /p ) (number of eigenvalues of A x ) . For and p.d.f. G the Stieltjes transform of G is defined

447 views • 19 slides
For matrix A ( p p ) with real eigenvalues, define F A , the empirical distribution function of

For matrix A ( p p ) with real eigenvalues, define F A , the empirical distribution function of

For matrix A ( p p ) with real eigenvalues, define F A , the empirical distribution function of the eigenvalues of A , to be F A ( x ) (1 /p ) (number of eigenvalues of A x ) . For and p.d.f. G the Stieltjes transform of G is defined

451 views • 24 slides

More recommend