1ZHGEQZ(1) LAPACK routine (version 3.1) ZHGEQZ(1)
2
6 ZHGEQZ - the eigenvalues of a complex matrix pair (H,T),
7
9 SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
10 ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK,
11 INFO )
12 CHARACTER COMPQ, COMPZ, JOB
14 INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
16 DOUBLE PRECISION RWORK( * )
18 COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ), Q( LDQ, * ), T(
20 LDT, * ), WORK( * ), Z( LDZ, * )
21
23 ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T), where H
24 is an upper Hessenberg matrix and T is upper triangular, using the sin‐
25 gle-shift QZ method.
26 Matrix pairs of this type are produced by the reduction to generalized
27 upper Hessenberg form of a complex matrix pair (A,B):
28 A = Q1*H*Z1**H, B = Q1*T*Z1**H,
30 as computed by ZGGHRD.
32 If JOB='S', then the Hessenberg-triangular pair (H,T) is
34 also reduced to generalized Schur form,
35 H = Q*S*Z**H, T = Q*P*Z**H,
37 where Q and Z are unitary matrices and S and P are upper triangular.
39 Optionally, the unitary matrix Q from the generalized Schur factoriza‐
41 tion may be postmultiplied into an input matrix Q1, and the unitary
42 matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1
43 are the unitary matrices from ZGGHRD that reduced the matrix pair (A,B)
44 to generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z
45 are the unitary factors from the generalized Schur factorization of
46 (A,B):
47 A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
49 To avoid overflow, eigenvalues of the matrix pair (H,T)
51 (equivalently, of (A,B)) are computed as a pair of complex values
52 (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigen‐
53 value of the generalized nonsymmetric eigenvalue problem (GNEP)
54 A*x = lambda*B*x
55 and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
56 alternate form of the GNEP
57 mu*A*y = B*y.
58 The values of alpha and beta for the i-th eigenvalue can be read
59 directly from the generalized Schur form: alpha = S(i,i), beta =
60 P(i,i).
61 Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
63 Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
64 pp. 241--256.
65
68 JOB (input) CHARACTER*1
69 = 'E': Compute eigenvalues only;
70 = 'S': Computer eigenvalues and the Schur form.
71 COMPQ (input) CHARACTER*1
73 = 'N': Left Schur vectors (Q) are not computed;
74 = 'I': Q is initialized to the unit matrix and the matrix Q of
75 left Schur vectors of (H,T) is returned; = 'V': Q must contain
76 a unitary matrix Q1 on entry and the product Q1*Q is returned.
77 COMPZ (input) CHARACTER*1
79 = 'N': Right Schur vectors (Z) are not computed;
80 = 'I': Q is initialized to the unit matrix and the matrix Z of
81 right Schur vectors of (H,T) is returned; = 'V': Z must contain
82 a unitary matrix Z1 on entry and the product Z1*Z is returned.
83 N (input) INTEGER
85 The order of the matrices H, T, Q, and Z. N >= 0.
86 ILO (input) INTEGER
88 IHI (input) INTEGER ILO and IHI mark the rows and columns
89 of H which are in Hessenberg form. It is assumed that A is
90 already upper triangular in rows and columns 1:ILO-1 and
91 IHI+1:N. If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and
92 IHI=0.
93 H (input/output) COMPLEX*16 array, dimension (LDH, N)
95 On entry, the N-by-N upper Hessenberg matrix H. On exit, if
96 JOB = 'S', H contains the upper triangular matrix S from the
97 generalized Schur factorization. If JOB = 'E', the diagonal of
98 H matches that of S, but the rest of H is unspecified.
99 LDH (input) INTEGER
101 The leading dimension of the array H. LDH >= max( 1, N ).
102 T (input/output) COMPLEX*16 array, dimension (LDT, N)
104 On entry, the N-by-N upper triangular matrix T. On exit, if
105 JOB = 'S', T contains the upper triangular matrix P from the
106 generalized Schur factorization. If JOB = 'E', the diagonal of
107 T matches that of P, but the rest of T is unspecified.
108 LDT (input) INTEGER
110 The leading dimension of the array T. LDT >= max( 1, N ).
111 ALPHA (output) COMPLEX*16 array, dimension (N)
113 The complex scalars alpha that define the eigenvalues of GNEP.
114 ALPHA(i) = S(i,i) in the generalized Schur factorization.
115 BETA (output) COMPLEX*16 array, dimension (N)
117 The real non-negative scalars beta that define the eigenvalues
118 of GNEP. BETA(i) = P(i,i) in the generalized Schur factoriza‐
119 tion.
120 Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
122 represent the j-th eigenvalue of the matrix pair (A,B), in one
123 of the forms lambda = alpha/beta or mu = beta/alpha. Since
124 either lambda or mu may overflow, they should not, in general,
125 be computed.
126 Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
128 On entry, if COMPZ = 'V', the unitary matrix Q1 used in the
129 reduction of (A,B) to generalized Hessenberg form. On exit, if
130 COMPZ = 'I', the unitary matrix of left Schur vectors of (H,T),
131 and if COMPZ = 'V', the unitary matrix of left Schur vectors of
132 (A,B). Not referenced if COMPZ = 'N'.
133 LDQ (input) INTEGER
135 The leading dimension of the array Q. LDQ >= 1. If COMPQ='V'
136 or 'I', then LDQ >= N.
137 Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
139 On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
140 reduction of (A,B) to generalized Hessenberg form. On exit, if
141 COMPZ = 'I', the unitary matrix of right Schur vectors of
142 (H,T), and if COMPZ = 'V', the unitary matrix of right Schur
143 vectors of (A,B). Not referenced if COMPZ = 'N'.
144 LDZ (input) INTEGER
146 The leading dimension of the array Z. LDZ >= 1. If COMPZ='V'
147 or 'I', then LDZ >= N.
148 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
150 On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
151 LWORK (input) INTEGER
153 The dimension of the array WORK. LWORK >= max(1,N).
154 If LWORK = -1, then a workspace query is assumed; the routine
156 only calculates the optimal size of the WORK array, returns
157 this value as the first entry of the WORK array, and no error
158 message related to LWORK is issued by XERBLA.
159 RWORK (workspace) DOUBLE PRECISION array, dimension (N)
161 INFO (output) INTEGER
163 = 0: successful exit
164 < 0: if INFO = -i, the i-th argument had an illegal value
165 = 1,...,N: the QZ iteration did not converge. (H,T) is not in
166 Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be
167 correct. = N+1,...,2*N: the shift calculation failed. (H,T)
168 is not in Schur form, but ALPHA(i) and BETA(i), i=INFO-
169 N+1,...,N should be correct.
170
172 We assume that complex ABS works as long as its value is less than
173 overflow.
174 LAPACK routine (version 3.1) November 2006 ZHGEQZ(1)