Lecture 12

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Unformatted text preview: % PA where P = I ; 2vvT =(vT v) is a Householder % matrix and v is an m-vector. = ;2=(vT w = AT A=A+v v) v wT 7 (3a) (3b) Re ections Function for a post-Householder update function A = col.house(A v) % col.house: Overwrite A 2 <m n with the product % AP where P = I ; 2vvT =(vT v) is a Householder % matrix and v is an n-vector. = ;2=(vT w= A A=A+w v) v vT Example 4. Let A 2 <m n. Compute Q such that B = QA has bkj = 0, k = j + 1 : m { Partition A A = A11 A12 AA 21 22 where A11 2 <(j ;1) (j ;1), A12 2 <(j ;1) (n;j ), A21 2 <(m;j ) (j ;1) , and A22 2 <(m;j ) (n;j ) { Let Q= Ij;1 0 0P vvT =I;2 T vv 8 0 v= v ~ Re ections Multiply A11 A B = QA = PA21 PA12 22 A typical case is to apply the transform to each column with a goal of making B upper triangular { In this case, A21 = 0 The following code segment does the job v(j : m) = house(A(j : m j )) A(j : m j n) = row.house(A(j : m j : n) v(j : m)) A(j + 1 : m j ) = v(j + 1 : m) { The rst line creates the nonzero portion of v that zeros bkj , k = j + 1 : m { The second line applies the transformation Q to A22 and stores the result back in A { The third line stores the nonzero portion of v in the zeroed portion of the j th column of A 9 Re ections Householder transformations are often applied in sequence Q = Q1Q2 Qr where v(j)(v(j))T v(j) = 0 Qj = I ; 2 (v(j))T v(j) ~ v (j ) ~ { The dimension v(j) is m ; j + 1 It is typically not necessary to store Q, but rather store the Householder vectors v(j ), j = 1 : r { Suppose v(j) is stored in the lower triangular part of a matrix A { The following code segment overwrites C with QC for j = 1 : r v(j + 1 : n) = A(j + 1 : n j ) v(j) = 1 C...
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This document was uploaded on 03/16/2014 for the course CSCI 6800 at Rensselaer Polytechnic Institute.

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