From 9d434badadac1701a48da3841a3cf7710f1bb6aa Mon Sep 17 00:00:00 2001 From: Pierre-Edouard Portier Date: Sun, 26 Sep 2021 22:15:44 +0200 Subject: [PATCH] =?UTF-8?q?Impl=C3=A9mentation=20de=20la=20factorisation?= =?UTF-8?q?=20QR=20par=20Gram-Schmidt=20modifi=C3=A9.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ML1_12_factorisation_qr.R | 28 ++++++++++++++++++++++++++ ML1_12_factorisation_qr.Rmd | 14 +++++++++++++ ML1_12_factorisation_qr.pdf | Bin 208428 -> 235927 bytes ML1_13_puissance_iteree_par_blocs.Rmd | 2 +- ML1_13_puissance_iteree_par_blocs.pdf | Bin 167886 -> 167886 bytes ML1_14_geometrie_ridge_svd.pdf | Bin 167621 -> 167621 bytes 6 files changed, 43 insertions(+), 1 deletion(-) create mode 100644 ML1_12_factorisation_qr.R diff --git a/ML1_12_factorisation_qr.R b/ML1_12_factorisation_qr.R new file mode 100644 index 0000000..20945d4 --- /dev/null +++ b/ML1_12_factorisation_qr.R @@ -0,0 +1,28 @@ +# Norm 2 +Norm <- function(x) { + stopifnot(is.numeric(x)) + sqrt(sum(x^2)) +} + +# Modified Gram-Schmidt +mgs <- function(A, tol = .Machine$double.eps^0.5) { + stopifnot(is.numeric(A), is.matrix(A)) + M <- nrow(A); N <- ncol(A) + if(MHV)|k>p*nrLDCg_R zlrferR+A~bnU^=1@f#+Hdwqiie*WwQJjmqL^$*=DPp?PJVuCbqb0o6?(*)CyYnf3b z00GPPm0_ycRO@{GUBJpnT7eP9hBG_sdoydcT70n5t(tOOu9i-;%YQ5PoEPp{<>txL zMZT`fnSc2gv(%Wisb6DC2+D+XD?MFK7){r;S+MxdP19;Hi)Ek!-ZJJ1t0C(^j?=6i zK|KvV7?B~4y8&08giIvL6w?+pL^6>uFdWa;b3Z5>4AO{k(Nej=m_`UmM5$`W{L>8x ztqKVceL7g`v4bq1P=A%)8}lk=p0F6qqY5zV^z%Fh0|TvRJ%3b0$Pf&J+YRWqE z16d)utu}Scqb1V)x@E*BV5weH9LJ=iNU?TIlpT!(?-sdXEtP6|Z)q&k-Ij(EuVuHZ`CV*?w=^_3(2{^A4Siy$JIWDfojKpT z5g86_8FS-)K!+Fl*M~Wl}_KstW_aJ$bsG=wkw#SaMkxw%aa^X;GyZsy*_()GFQFNio<>OI1t1R z5T3;J2wnw$$MiS`3-UlIL;qJXn-0<1>!{sc_%4E&LYzPJ)G$9Bf>lObSs>Xv>L0-N zD2;nG0e_e7^Oeu)(x%>_ED~^1Khye!zG7H5sx2%qt1F){yfj-@C z1u>;S-VABpkz3C;uaVK6Zpf7}V~Bo;U@zQtWGx7dN(SlOVO#6FsI6-VL$HYlww}dH zhykRBdFs39euN=I`z(RsT{I4v6eU;=$88-x9e=qAyo<0689cc8VALLC5WDgaf;A#G 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décomposition en valeurs propres -Nous avons vu précédemment que la méthode de la puissance itérée permet de calculer la plus grande valeur propre et un vecteur propre associé pour une matrice carrée $\mathbf{A}$. La décomposition QR est au cœur d'une approche élégante pour étendre la méthode de la puissance itérée au calcul des $S$ plus grandes valeurs propres et lerus vecteurs propres associés. +Nous avons vu précédemment que la méthode de la puissance itérée permet de calculer la plus grande valeur propre et un vecteur propre associé pour une matrice carrée $\mathbf{A}$. La décomposition QR est au cœur d'une approche élégante pour étendre la méthode de la puissance itérée au calcul des $S$ plus grandes valeurs propres et leurs vecteurs propres associés. Une itération consiste à multiplier $\mathbf{A}$ par un bloc $\mathbf{V} \in \mathbb{R}^{N \times S}$ de $S$ vecteurs colonnes. 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