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Giansalvo EXIN Cirrincione unité #5

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

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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 = ^

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Décomposition en valeurs singulières (SVD) Full SVD Reduced SVD

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Approximation au sens des moindres carrées Example: polynomial data fitting

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

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

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Posons le problème matriciellement Approximation au sens des moindres carrées

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

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Approximation au sens des moindres carrées Équations normales forme quadratique

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

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The system is nonsingular iff A has full rank.

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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

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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

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

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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

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Conditionnement et précision

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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

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

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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

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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

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

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

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

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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

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

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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.

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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.

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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.

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FINE

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