1SLAHR2 ‐ the first NB columns of A real general n‐BY‐(n‐k+1) ma‐
2trix A so that elements below the k‐th subdiagonal are zero SUB‐
3ROUTINE SLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
4 INTEGER K, LDA, LDT, LDY, N, NB
5 REAL A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )
6SLAHR2 reduces the first NB columns of A real general n‐BY‐(n‐
7k+1) matrix A so that elements below the k‐th subdiagonal are ze‐
8ro. The reduction is performed by an orthogonal similarity trans‐
9formation Q' * A * Q. The routine returns the matrices V and T
10which determine Q as a block reflector I ‐ V*T*V', and also the
11matrix Y = A * V * T.
12This is an auxiliary routine called by SGEHRD.
14N (input) INTEGER The order of the matrix A. K (in‐
16put) INTEGER The offset for the reduction. Elements below the k‐
17th subdiagonal in the first NB columns are reduced to zero. K <
18N. NB (input) INTEGER The number of columns to be reduced.
19A (input/output) REAL array, dimension (LDA,N‐K+1) On en‐
20try, the n‐by‐(n‐k+1) general matrix A. On exit, the elements on
21and above the k‐th subdiagonal in the first NB columns are over‐
22written with the corresponding elements of the reduced matrix;
23the elements below the k‐th subdiagonal, with the array TAU, rep‐
24resent the matrix Q as a product of elementary reflectors. The
25other columns of A are unchanged. See Further Details. LDA
26(input) INTEGER The leading dimension of the array A. LDA >=
27max(1,N). TAU (output) REAL array, dimension (NB) The scalar
28factors of the elementary reflectors. See Further Details. T
29(output) REAL array, dimension (LDT,NB) The upper triangular ma‐
30trix T. LDT (input) INTEGER The leading dimension of the ar‐
31ray T. LDT >= NB. Y (output) REAL array, dimension
32(LDY,NB) The n‐by‐nb matrix Y. LDY (input) INTEGER The lead‐
33ing dimension of the array Y. LDY >= N. The matrix Q is repre‐
34sented as a product of nb elementary reflectors
35 Q = H(1) H(2) . . . H(nb).
37Each H(i) has the form
39 H(i) = I ‐ tau * v * v'
41where tau is a real scalar, and v is a real vector with
43v(1:i+k‐1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
44A(i+k+1:n,i), and tau in TAU(i).
45The elements of the vectors v together form the (n‐k+1)‐by‐nb ma‐
47trix V which is needed, with T and Y, to apply the transformation
48to the unreduced part of the matrix, using an update of the form:
49A := (I ‐ V*T*V') * (A ‐ Y*V').
50The contents of A on exit are illustrated by the following exam‐
52ple with n = 7, k = 3 and nb = 2:
53 ( a a a a a )
55 ( a a a a a )
56 ( a a a a a )
57 ( h h a a a )
58 ( v1 h a a a )
59 ( v1 v2 a a a )
60 ( v1 v2 a a a )
61where a denotes an element of the original matrix A, h denotes a
63modified element of the upper Hessenberg matrix H, and vi denotes
64an element of the vector defining H(i).
65This file is a slight modification of LAPACK‐3.0's SLAHRD incor‐
67porating improvements proposed by Quintana‐Orti and Van de Gejin.
68Note that the entries of A(1:K,2:NB) differ from those returned
69by the original LAPACK routine. This function is not backward
70compatible with LAPACK3.0.
71