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Giansalvo EXIN Cirrincione unité #5 Décomposition en valeurs singulières (SVD) valeurs singulières a m1 a mn a 11 a 1n U 0 0 0 n n 10V = Full SVD.

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Presentation on theme: "Giansalvo EXIN Cirrincione unité #5 Décomposition en valeurs singulières (SVD) valeurs singulières a m1 a mn a 11 a 1n U 0 0 0 n n 10V = Full SVD."— Presentation transcript:

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2 Giansalvo EXIN Cirrincione unité #5

3 Décomposition en valeurs singulières (SVD) valeurs singulières a m1 a mn a 11 a 1n U n n 10V = Full SVD

4 Décomposition en valeurs singulières (SVD) a m1 a mn a 11 a 1n U n n 10V = Reduced SVD a m1 a mn a 11 a 1n U 0 n n 10V = ^

5 Décomposition en valeurs singulières (SVD) Full SVD Reduced SVD

6 Approximation au sens des moindres carrées Example: polynomial data fitting

7 f(x)f(x) xixi yiyi Approximation au sens des moindres carrées

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

9 Posons le problème matriciellement Approximation au sens des moindres carrées

10 système linéaire de n équations et n inconnues Approximation au sens des moindres carrées erreurdapproximation Matrice de Vandermonde ( )

11 Approximation au sens des moindres carrées Équations normales forme quadratique

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

14 The system is nonsingular iff A has full rank.

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

16 Solution par la factorisation QR (Householder)reduced QR factorization 1. Compute the reduced QR factorization2. Compute the vector3. Solve the upper-triangular system for x

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

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

19 Conditionnement et précision

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

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

22 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 x 15 = 1 highly ill-conditioned basis very close fit

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

24 Stabilité des méthodes des moindres carrées exemple factorisation QR (Householder) implicit calculation of the product Q H b

25 Stabilité des méthodes des moindres carrées exemple factorisation QR (Householder) implicit calculation of the product Q H b

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

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

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

29 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.

30 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.

31 Stabilité des méthodes des moindres carrées The normal equations are typically unstable for ill-conditioned problems involving close fits. 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.

32 FINE


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