1DLAHR2 ‐ 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 DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
4 INTEGER K, LDA, LDT, LDY, N, NB
5 DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), Y(
6LDY, NB ) DLAHR2 reduces the first NB columns of A real general
7n‐BY‐(n‐k+1) matrix A so that elements below the k‐th subdiagonal
8are zero. The reduction is performed by an orthogonal similarity
9transformation Q' * A * Q. The routine returns the matrices V and
10T which 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 DGEHRD.
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) DOUBLE PRECISION array, dimension (LDA,N‐
20K+1) On entry, the n‐by‐(n‐k+1) general matrix A. On exit, the
21elements on and above the k‐th subdiagonal in the first NB col‐
22umns are overwritten with the corresponding elements of the re‐
23duced matrix; the elements below the k‐th subdiagonal, with the
24array TAU, represent the matrix Q as a product of elementary re‐
25flectors. The other columns of A are unchanged. See Further De‐
26tails. LDA (input) INTEGER The leading dimension of the ar‐
27ray A. LDA >= max(1,N). TAU (output) DOUBLE PRECISION ar‐
28ray, dimension (NB) The scalar factors of the elementary reflec‐
29tors. See Further Details. T (output) DOUBLE PRECISION ar‐
30ray, dimension (LDT,NB) The upper triangular matrix T. LDT
31(input) INTEGER The leading dimension of the array T. LDT >= NB.
32Y (output) DOUBLE PRECISION array, dimension (LDY,NB) The
33n‐by‐nb matrix Y. LDY (input) INTEGER The leading dimension
34of the array Y. LDY >= N. The matrix Q is represented as a prod‐
35uct of nb elementary reflectors
36 Q = H(1) H(2) . . . H(nb).
38Each H(i) has the form
40 H(i) = I ‐ tau * v * v'
42where tau is a real scalar, and v is a real vector with
44v(1:i+k‐1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
45A(i+k+1:n,i), and tau in TAU(i).
46The elements of the vectors v together form the (n‐k+1)‐by‐nb ma‐
48trix V which is needed, with T and Y, to apply the transformation
49to the unreduced part of the matrix, using an update of the form:
50A := (I ‐ V*T*V') * (A ‐ Y*V').
51The contents of A on exit are illustrated by the following exam‐
53ple with n = 7, k = 3 and nb = 2:
54 ( a a a a a )
56 ( a a a a a )
57 ( a a a a a )
58 ( h h a a a )
59 ( v1 h a a a )
60 ( v1 v2 a a a )
61 ( v1 v2 a a a )
62where a denotes an element of the original matrix A, h denotes a
64modified element of the upper Hessenberg matrix H, and vi denotes
65an element of the vector defining H(i).
66This file is a slight modification of LAPACK‐3.0's DLAHRD incor‐
68porating improvements proposed by Quintana‐Orti and Van de Gejin.
69Note that the entries of A(1:K,2:NB) differ from those returned
70by the original LAPACK routine. This function is not backward
71compatible with LAPACK3.0.
72