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  1. unité #5 Analyse numérique matricielle Giansalvo EXIN Cirrincione

  2. U 1  0 V    = 0  n   0  0 Full SVD a11  a1n     valeurs singulières   am1  amn Décomposition en valeurs singulières (SVD)

  3. U 1  0 V    = 0  n   0  0 a11 a11   a1n a1n U 1  0 V        ^ =     0  n     am1 am1   amn amn Reduced SVD Décomposition en valeurs singulières (SVD)

  4. Full SVD Reduced SVD Décomposition en valeurs singulières (SVD)

  5. Approximation au sens des moindres carrées Example: polynomial data fitting

  6. f(x) 2 1 0 yi -1 -2 -3 -4 xi 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Approximation au sens des moindres carrées

  7. discrete square wave interpolation least squares m = n = 11 m = 11 , n = 8 Approximation au sens des moindres carrées

  8. Approximation au sens des moindres carrées Posons le problème matriciellement

  9. Matrice de Vandermonde (1735-1796) Approximation au sens des moindres carrées système linéaire de n équations et n inconnues erreur d’approximation

  10. Approximation au sens des moindres carrées forme quadratique Équations normales

  11. r = b - A x b y = A x range(A) = Pb The system is nonsingular iff A has full rank.

  12. The system is nonsingular iff A has full rank.

  13. Solution par les équations normales factorisation de Cholesky AHA est une matrice n x n hermitienne strictement définie positive 1. Form the matrix AHA and the vector AH b 2. Compute the Cholesky factorization AHA = RHR 3. Solve the lower-triangular system RH w = AH b for w 4. Solve the upper-triangular system R x = w for x

  14. reduced QR factorization 1. Compute the reduced QR factorization 2. Compute the vector 3. Solve the upper-triangular system for x Solution par la factorisation QR (Householder)

  15. 1. Compute the reduced SVD 2. Compute the vector 4. Set 3. Solve the diagonal system for w Solution par la SVD

  16. Comparison of algorithms • speed : normal equations • standard : QR factorization • A close to singular : SVD • Drawbacks • normal equations : not always stable in the presence of rounding errors • QR factoriz.: less-than-ideal stability properties if A is close to singular • SVD : expensive for mn

  17. Conditionnement et précision

  18. r = b - A x b  closeness of the fit = Pb y = A x range(A) Conditionnement du problème des moindres carrées Données : A , b Solutions : x , y

  19. Conditionnement du problème des moindres carrées Données : A , b Solutions : x , y 2-norm relative condition numbers exact for certain  b upper bounds

  20.  highly ill-conditioned basis  very close fit Stabilité des méthodes des moindres carrées exemple Least squares fitting of the function exp(sin(4)) on the interval [0,1] by a polynomial of degree 14  x15 = 1

  21. reduced Stabilité des méthodes des moindres carrées exemple factorisation QR (Householder) The rounding errors have been amplified by a factor of order 10 9. This inaccuracy is explained by ill-conditioning, not instability.

  22. Stabilité des méthodes des moindres carrées exemple factorisation QR (Householder) implicit calculation of the product QH b

  23. Stabilité des méthodes des moindres carrées exemple factorisation QR (Householder) implicit calculation of the product QH b

  24. Stabilité des méthodes des moindres carrées exemple factorisation QR (Householder) backward stable

  25. Stabilité des méthodes des moindres carrées exemple SVD It beats Householder triangularization with column pivoting ( MATLAB's \ ) by a factor of about 3 backward stable

  26. Stabilité des méthodes des moindres carrées exemple équations normales factorisation de Cholesky not even a single digit of accuracy unstable

  27. Stabilité des méthodes des moindres carrées BS least squares algorithm The condition number of the LS problem may lie anywhere in the range  to 2 .

  28. Stabilité des méthodes des moindres carrées BS least squares algorithm Cholesky factorization (BS) The normal equations are typically unstable for ill-conditioned problems involving close fits.

  29. Stabilité des méthodes des moindres carrées The solution of the full-rank least squares problem via the normal equations is unstable. Stability can be achieved, however, by restriction to a class of problems in which (A) is uniformly bounded above or (tan)/is uniformly bounded below. The normal equations are typically unstable for ill-conditioned problems involving close fits.

  30. FINE