1ZGGHRD(1) LAPACK routine (version 3.1) ZGGHRD(1)
2
6 ZGGHRD - a pair of complex matrices (A,B) to generalized upper Hessen‐
7 berg form using unitary transformations, where A is a general matrix
8 and B is upper triangular
9
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, * )
19
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:
30 Q**H*A*Z = H
31 and transforms B to another upper triangular matrix T:
32 Q**H*B*Z = T
33 in order to reduce the problem to its standard form
34 H*y = lambda*T*y
35 where y = Z**H*x.
36 The unitary matrices Q and Z are determined as products of Givens rota‐
38 tions. They may either be formed explicitly, or they may be postmulti‐
39 plied into input matrices Q1 and Z1, so that
40 Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
41 Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
42 If Q1 is the unitary matrix from the QR factorization of B in the orig‐
43 inal equation A*x = lambda*B*x, then ZGGHRD reduces the original prob‐
44 lem to generalized Hessenberg form.
45
48 COMPQ (input) CHARACTER*1
49 = 'N': do not compute Q;
50 = 'I': Q is initialized to the unit matrix, and the unitary
51 matrix Q is returned; = 'V': Q must contain a unitary matrix Q1
52 on entry, and the product Q1*Q is returned.
53 COMPZ (input) CHARACTER*1
55 = 'N': do not compute Q;
56 = 'I': Q is initialized to the unit matrix, and the unitary
57 matrix Q is returned; = 'V': Q must contain a unitary matrix Q1
58 on entry, and the product Q1*Q is returned.
59 N (input) INTEGER
61 The order of the matrices A and B. N >= 0.
62 ILO (input) INTEGER
64 IHI (input) INTEGER ILO and IHI mark the rows and columns
65 of A which are to be reduced. It is assumed that A is already
66 upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO
67 and IHI are normally set by a previous call to ZGGBAL; other‐
68 wise they should be set to 1 and N respectively. 1 <= ILO <=
69 IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
70 A (input/output) COMPLEX*16 array, dimension (LDA, N)
72 On entry, the N-by-N general matrix to be reduced. On exit,
73 the upper triangle and the first subdiagonal of A are overwrit‐
74 ten with the upper Hessenberg matrix H, and the rest is set to
75 zero.
76 LDA (input) INTEGER
78 The leading dimension of the array A. LDA >= max(1,N).
79 B (input/output) COMPLEX*16 array, dimension (LDB, N)
81 On entry, the N-by-N upper triangular matrix B. On exit, the
82 upper triangular matrix T = Q**H B Z. The elements below the
83 diagonal are set to zero.
84 LDB (input) INTEGER
86 The leading dimension of the array B. LDB >= max(1,N).
87 Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
89 On entry, if COMPQ = 'V', the unitary matrix Q1, typically from
90 the QR factorization of B. On exit, if COMPQ='I', the unitary
91 matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced
92 if COMPQ='N'.
93 LDQ (input) INTEGER
95 The leading dimension of the array Q. LDQ >= N if COMPQ='V' or
96 'I'; LDQ >= 1 otherwise.
97 Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
99 On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if
100 COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the prod‐
101 uct Z1*Z. Not referenced if COMPZ='N'.
102 LDZ (input) INTEGER
104 The leading dimension of the array Z. LDZ >= N if COMPZ='V' or
105 'I'; LDZ >= 1 otherwise.
106 INFO (output) INTEGER
108 = 0: successful exit.
109 < 0: if INFO = -i, the i-th argument had an illegal value.
110
112 This routine reduces A to Hessenberg and B to triangular form by an
113 unblocked reduction, as described in _Matrix_Computations_, by Golub
114 and van Loan (Johns Hopkins Press).
115 LAPACK routine (version 3.1) November 2006 ZGGHRD(1)