1DLAHRD(1) LAPACK auxiliary routine (version 3.2) DLAHRD(1)
2
6 DLAHRD - 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
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10 SUBROUTINE DLAHRD( 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 DLAHRD 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 OBSOLETE auxiliary routine.
24 This routine will be 'deprecated' in a future release.
25 Please use the new routine DLAHR2 instead.
26
28 N (input) INTEGER
29 The order of the matrix A.
30 K (input) INTEGER
32 The offset for the reduction. Elements below the k-th subdiago‐
33 nal in the first NB columns are reduced to zero.
34 NB (input) INTEGER
36 The number of columns to be reduced.
37 A (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)
39 On entry, the n-by-(n-k+1) general matrix A. On exit, the ele‐
40 ments on and above the k-th subdiagonal in the first NB columns
41 are overwritten with the corresponding elements of the reduced
42 matrix; the elements below the k-th subdiagonal, with the array
43 TAU, represent the matrix Q as a product of elementary reflec‐
44 tors. The other columns of A are unchanged. See Further
45 Details. LDA (input) INTEGER The leading dimension of the
46 array A. LDA >= max(1,N).
47 TAU (output) DOUBLE PRECISION array, dimension (NB)
49 The scalar factors of the elementary reflectors. See Further
50 Details.
51 T (output) DOUBLE PRECISION array, dimension (LDT,NB)
53 The upper triangular matrix T.
54 LDT (input) INTEGER
56 The leading dimension of the array T. LDT >= NB.
57 Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
59 The n-by-nb matrix Y.
60 LDY (input) INTEGER
62 The leading dimension of the array Y. LDY >= N.
63
65 The matrix Q is represented as a product of nb elementary reflectors
66 Q = H(1) H(2) . . . H(nb).
67 Each H(i) has the form
68 H(i) = I - tau * v * v'
69 where tau is a real scalar, and v is a real vector with
70 v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
71 A(i+k+1:n,i), and tau in TAU(i).
72 The elements of the vectors v together form the (n-k+1)-by-nb matrix V
73 which is needed, with T and Y, to apply the transformation to the unre‐
74 duced part of the matrix, using an update of the form: A := (I -
75 V*T*V') * (A - Y*V').
76 The contents of A on exit are illustrated by the following example with
77 n = 7, k = 3 and nb = 2:
78 ( a h a a a )
79 ( a h a a a )
80 ( a h a a a )
81 ( h h a a a )
82 ( v1 h a a a )
83 ( v1 v2 a a a )
84 ( v1 v2 a a a )
85 where a denotes an element of the original matrix A, h denotes a modi‐
86 fied element of the upper Hessenberg matrix H, and vi denotes an ele‐
87 ment of the vector defining H(i).
88 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLAHRD(1)