1ZGGHRD(1) LAPACK routine (version 3.2) ZGGHRD(1)
2
6 ZGGHRD - reduces a pair of complex matrices (A,B) to generalized upper
7 Hessenberg form using unitary transformations, where A is a general
8 matrix and B is upper triangular
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11 SUBROUTINE ZGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ,
12 Z, LDZ, INFO )
13 CHARACTER COMPQ, COMPZ
15 INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
17 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )
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21 ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper
22 Hessenberg form using unitary transformations, where A is a general
23 matrix and B is upper triangular. The form of the generalized eigen‐
24 value problem is
25 A*x = lambda*B*x,
26 and B is typically made upper triangular by computing its QR factoriza‐
27 tion and moving the unitary matrix Q to the left side of the equation.
28 This subroutine simultaneously reduces A to a Hessenberg matrix H:
29 Q**H*A*Z = H
30 and transforms B to another upper triangular matrix T:
31 Q**H*B*Z = T
32 in order to reduce the problem to its standard form
33 H*y = lambda*T*y
34 where y = Z**H*x.
35 The unitary matrices Q and Z are determined as products of Givens rota‐
36 tions. They may either be formed explicitly, or they may be postmulti‐
37 plied into input matrices Q1 and Z1, so that
38 Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
39 Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
40 If Q1 is the unitary matrix from the QR factorization of B in the orig‐
41 inal equation A*x = lambda*B*x, then ZGGHRD reduces the original prob‐
42 lem to generalized Hessenberg form.
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45 COMPQ (input) CHARACTER*1
46 = 'N': do not compute Q;
47 = 'I': Q is initialized to the unit matrix, and the unitary
48 matrix Q is returned; = 'V': Q must contain a unitary matrix Q1
49 on entry, and the product Q1*Q is returned.
50 COMPZ (input) CHARACTER*1
52 = 'N': do not compute Q;
53 = 'I': Q is initialized to the unit matrix, and the unitary
54 matrix Q is returned; = 'V': Q must contain a unitary matrix Q1
55 on entry, and the product Q1*Q is returned.
56 N (input) INTEGER
58 The order of the matrices A and B. N >= 0.
59 ILO (input) INTEGER
61 IHI (input) INTEGER ILO and IHI mark the rows and columns
62 of A which are to be reduced. It is assumed that A is already
63 upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO
64 and IHI are normally set by a previous call to ZGGBAL; other‐
65 wise they should be set to 1 and N respectively. 1 <= ILO <=
66 IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
67 A (input/output) COMPLEX*16 array, dimension (LDA, N)
69 On entry, the N-by-N general matrix to be reduced. On exit,
70 the upper triangle and the first subdiagonal of A are overwrit‐
71 ten with the upper Hessenberg matrix H, and the rest is set to
72 zero.
73 LDA (input) INTEGER
75 The leading dimension of the array A. LDA >= max(1,N).
76 B (input/output) COMPLEX*16 array, dimension (LDB, N)
78 On entry, the N-by-N upper triangular matrix B. On exit, the
79 upper triangular matrix T = Q**H B Z. The elements below the
80 diagonal are set to zero.
81 LDB (input) INTEGER
83 The leading dimension of the array B. LDB >= max(1,N).
84 Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
86 On entry, if COMPQ = 'V', the unitary matrix Q1, typically from
87 the QR factorization of B. On exit, if COMPQ='I', the unitary
88 matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced
89 if COMPQ='N'.
90 LDQ (input) INTEGER
92 The leading dimension of the array Q. LDQ >= N if COMPQ='V' or
93 'I'; LDQ >= 1 otherwise.
94 Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
96 On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if
97 COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the prod‐
98 uct Z1*Z. Not referenced if COMPZ='N'.
99 LDZ (input) INTEGER
101 The leading dimension of the array Z. LDZ >= N if COMPZ='V' or
102 'I'; LDZ >= 1 otherwise.
103 INFO (output) INTEGER
105 = 0: successful exit.
106 < 0: if INFO = -i, the i-th argument had an illegal value.
107
109 This routine reduces A to Hessenberg and B to triangular form by an
110 unblocked reduction, as described in _Matrix_Computations_, by Golub
111 and van Loan (Johns Hopkins Press).
112 LAPACK routine (version 3.2) November 2008 ZGGHRD(1)