1ZLAHR2 ‐ the first NB columns of A complex general n‐BY‐(n‐k+1)
2matrix A so that elements below the k‐th subdiagonal are zero
3SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
4 INTEGER K, LDA, LDT, LDY, N, NB
5 COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )
6ZLAHR2 reduces the first NB columns of A complex 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 unitary similarity transfor‐
9mation Q' * A * Q. The routine returns the matrices V and T which
10determine Q as a block reflector I ‐ V*T*V', and also the matrix
11Y = A * V * T.
12This is an auxiliary routine called by ZGEHRD.
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) COMPLEX*16 array, dimension (LDA,N‐K+1) On
20entry, the n‐by‐(n‐k+1) general matrix A. On exit, the elements
21on and above the k‐th subdiagonal in the first NB columns are
22overwritten with the corresponding elements of the reduced ma‐
23trix; the elements below the k‐th subdiagonal, with the array
24TAU, represent the matrix Q as a product of elementary reflec‐
25tors. The other columns of A are unchanged. See Further Details.
26LDA (input) INTEGER The leading dimension of the array A.
27LDA >= max(1,N). TAU (output) COMPLEX*16 array, dimension
28(NB) The scalar factors of the elementary reflectors. See Further
29Details. T (output) COMPLEX*16 array, dimension (LDT,NB)
30The upper triangular matrix T. LDT (input) INTEGER The lead‐
31ing dimension of the array T. LDT >= NB. Y (output) COM‐
32PLEX*16 array, dimension (LDY,NB) The n‐by‐nb matrix Y. LDY
33(input) INTEGER The leading dimension of the array Y. LDY >= N.
34The matrix Q is represented as a product of nb elementary reflec‐
35tors
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 complex scalar, and v is a complex 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 ZLAHRD 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