1ZGEHD2(1) LAPACK routine (version 3.2) ZGEHD2(1)
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6 ZGEHD2 - reduces a complex general matrix A to upper Hessenberg form H
7 by a unitary similarity transformation
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10 SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
11 INTEGER IHI, ILO, INFO, LDA, N
13 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
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17 ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H by
18 a unitary similarity transformation: Q' * A * Q = H .
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21 N (input) INTEGER
22 The order of the matrix A. N >= 0.
23 ILO (input) INTEGER
25 IHI (input) INTEGER It is assumed that A is already upper
26 triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI
27 are normally set by a previous call to ZGEBAL; otherwise they
28 should be set to 1 and N respectively. See Further Details.
29 A (input/output) COMPLEX*16 array, dimension (LDA,N)
31 On entry, the n by n general matrix to be reduced. On exit,
32 the upper triangle and the first subdiagonal of A are overwrit‐
33 ten with the upper Hessenberg matrix H, and the elements below
34 the first subdiagonal, with the array TAU, represent the uni‐
35 tary matrix Q as a product of elementary reflectors. See Fur‐
36 ther Details. LDA (input) INTEGER The leading dimension of
37 the array A. LDA >= max(1,N).
38 TAU (output) COMPLEX*16 array, dimension (N-1)
40 The scalar factors of the elementary reflectors (see Further
41 Details).
42 WORK (workspace) COMPLEX*16 array, dimension (N)
44 INFO (output) INTEGER
46 = 0: successful exit
47 < 0: if INFO = -i, the i-th argument had an illegal value.
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50 The matrix Q is represented as a product of (ihi-ilo) elementary
51 reflectors
52 Q = H(ilo) H(ilo+1) . . . H(ihi-1).
53 Each H(i) has the form
54 H(i) = I - tau * v * v'
55 where tau is a complex scalar, and v is a complex vector with v(1:i) =
56 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in
57 A(i+2:ihi,i), and tau in TAU(i).
58 The contents of A are illustrated by the following example, with n = 7,
59 ilo = 2 and ihi = 6:
60 on entry, on exit,
61 ( a a a a a a a ) ( a a h h h h a ) ( a
62 a a a a a ) ( a h h h h a ) ( a a a
63 a a a ) ( h h h h h h ) ( a a a a a
64 a ) ( v2 h h h h h ) ( a a a a a a )
65 ( v2 v3 h h h h ) ( a a a a a a ) (
66 v2 v3 v4 h h h ) ( a ) (
67 a ) where a denotes an element of the original matrix A, h denotes a
68 modified element of the upper Hessenberg matrix H, and vi denotes an
69 element of the vector defining H(i).
70 LAPACK routine (version 3.2) November 2008 ZGEHD2(1)