1DLAHR2(1) LAPACK auxiliary routine (version 3.2) DLAHR2(1)
2
6 DLAHR2 - reduces the first NB columns of A real general n-BY-(n-k+1)
7 matrix A so that elements below the k-th subdiagonal are zero
8
10 SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
11 INTEGER K, LDA, LDT, LDY, N, NB
13 DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), Y(
15 LDY, NB )
16
18 DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
19 matrix A so that elements below the k-th subdiagonal are zero. The
20 reduction is performed by an orthogonal similarity transformation Q' *
21 A * Q. The routine returns the matrices V and T which determine Q as a
22 block reflector I - V*T*V', and also the matrix Y = A * V * T. This is
23 an auxiliary routine called by DGEHRD.
24
26 N (input) INTEGER
27 The order of the matrix A.
28 K (input) INTEGER
30 The offset for the reduction. Elements below the k-th subdiago‐
31 nal in the first NB columns are reduced to zero. K < N.
32 NB (input) INTEGER
34 The number of columns to be reduced.
35 A (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)
37 On entry, the n-by-(n-k+1) general matrix A. On exit, the ele‐
38 ments on and above the k-th subdiagonal in the first NB columns
39 are overwritten with the corresponding elements of the reduced
40 matrix; the elements below the k-th subdiagonal, with the array
41 TAU, represent the matrix Q as a product of elementary reflec‐
42 tors. The other columns of A are unchanged. See Further
43 Details. LDA (input) INTEGER The leading dimension of the
44 array A. LDA >= max(1,N).
45 TAU (output) DOUBLE PRECISION array, dimension (NB)
47 The scalar factors of the elementary reflectors. See Further
48 Details.
49 T (output) DOUBLE PRECISION array, dimension (LDT,NB)
51 The upper triangular matrix T.
52 LDT (input) INTEGER
54 The leading dimension of the array T. LDT >= NB.
55 Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
57 The n-by-nb matrix Y.
58 LDY (input) INTEGER
60 The leading dimension of the array Y. LDY >= N.
61
63 The matrix Q is represented as a product of nb elementary reflectors
64 Q = H(1) H(2) . . . H(nb).
65 Each H(i) has the form
66 H(i) = I - tau * v * v'
67 where tau is a real scalar, and v is a real vector with
68 v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
69 A(i+k+1:n,i), and tau in TAU(i).
70 The elements of the vectors v together form the (n-k+1)-by-nb matrix V
71 which is needed, with T and Y, to apply the transformation to the unre‐
72 duced part of the matrix, using an update of the form: A := (I -
73 V*T*V') * (A - Y*V').
74 The contents of A on exit are illustrated by the following example with
75 n = 7, k = 3 and nb = 2:
76 ( a a a a a )
77 ( a a a a a )
78 ( a a a a a )
79 ( h h a a a )
80 ( v1 h a a a )
81 ( v1 v2 a a a )
82 ( v1 v2 a a a )
83 where a denotes an element of the original matrix A, h denotes a modi‐
84 fied element of the upper Hessenberg matrix H, and vi denotes an ele‐
85 ment of the vector defining H(i).
86 This file is a slight modification of LAPACK-3.0's DLAHRD incorporating
87 improvements proposed by Quintana-Orti and Van de Gejin. Note that the
88 entries of A(1:K,2:NB) differ from those returned by the original
89 LAPACK routine. This function is not backward compatible with
90 LAPACK3.0.
91 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLAHR2(1)