1ZHEGVD(1) LAPACK driver routine (version 3.1) ZHEGVD(1)
2
6 ZHEGVD - all the eigenvalues, and optionally, the eigenvectors of a
7 complex generalized Hermitian-definite eigenproblem, of the form
8 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x
9
11 SUBROUTINE ZHEGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
12 LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
13 CHARACTER JOBZ, UPLO
15 INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LRWORK, LWORK, N
17 INTEGER IWORK( * )
19 DOUBLE PRECISION RWORK( * ), W( * )
21 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
23
25 ZHEGVD computes all the eigenvalues, and optionally, the eigenvectors
26 of a complex generalized Hermitian-definite eigenproblem, of the form
27 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B
28 are assumed to be Hermitian and B is also positive definite. If eigen‐
29 vectors are desired, it uses a divide and conquer algorithm.
30 The divide and conquer algorithm makes very mild assumptions about
32 floating point arithmetic. It will work on machines with a guard digit
33 in add/subtract, or on those binary machines without guard digits which
34 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
35 conceivably fail on hexadecimal or decimal machines without guard dig‐
36 its, but we know of none.
37
40 ITYPE (input) INTEGER
41 Specifies the problem type to be solved:
42 = 1: A*x = (lambda)*B*x
43 = 2: A*B*x = (lambda)*x
44 = 3: B*A*x = (lambda)*x
45 JOBZ (input) CHARACTER*1
47 = 'N': Compute eigenvalues only;
48 = 'V': Compute eigenvalues and eigenvectors.
49 UPLO (input) CHARACTER*1
51 = 'U': Upper triangles of A and B are stored;
52 = 'L': Lower triangles of A and B are stored.
53 N (input) INTEGER
55 The order of the matrices A and B. N >= 0.
56 A (input/output) COMPLEX*16 array, dimension (LDA, N)
58 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
59 N-by-N upper triangular part of A contains the upper triangular
60 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
61 triangular part of A contains the lower triangular part of the
62 matrix A.
63 On exit, if JOBZ = 'V', then if INFO = 0, A contains the matrix
65 Z of eigenvectors. The eigenvectors are normalized as follows:
66 if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z =
67 I. If JOBZ = 'N', then on exit the upper triangle (if
68 UPLO='U') or the lower triangle (if UPLO='L') of A, including
69 the diagonal, is destroyed.
70 LDA (input) INTEGER
72 The leading dimension of the array A. LDA >= max(1,N).
73 B (input/output) COMPLEX*16 array, dimension (LDB, N)
75 On entry, the Hermitian matrix B. If UPLO = 'U', the leading
76 N-by-N upper triangular part of B contains the upper triangular
77 part of the matrix B. If UPLO = 'L', the leading N-by-N lower
78 triangular part of B contains the lower triangular part of the
79 matrix B.
80 On exit, if INFO <= N, the part of B containing the matrix is
82 overwritten by the triangular factor U or L from the Cholesky
83 factorization B = U**H*U or B = L*L**H.
84 LDB (input) INTEGER
86 The leading dimension of the array B. LDB >= max(1,N).
87 W (output) DOUBLE PRECISION array, dimension (N)
89 If INFO = 0, the eigenvalues in ascending order.
90 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
92 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
93 LWORK (input) INTEGER
95 The length of the array WORK. If N <= 1, LWORK
96 >= 1. If JOBZ = 'N' and N > 1, LWORK >= N + 1. If JOBZ =
97 'V' and N > 1, LWORK >= 2*N + N**2.
98 If LWORK = -1, then a workspace query is assumed; the routine
100 only calculates the optimal sizes of the WORK, RWORK and IWORK
101 arrays, returns these values as the first entries of the WORK,
102 RWORK and IWORK arrays, and no error message related to LWORK
103 or LRWORK or LIWORK is issued by XERBLA.
104 RWORK (workspace/output) DOUBLE PRECISION array, dimension
106 (MAX(1,LRWORK))
107 On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
108 LRWORK (input) INTEGER
110 The dimension of the array RWORK. If N <= 1,
111 LRWORK >= 1. If JOBZ = 'N' and N > 1, LRWORK >= N. If JOBZ
112 = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
113 If LRWORK = -1, then a workspace query is assumed; the routine
115 only calculates the optimal sizes of the WORK, RWORK and IWORK
116 arrays, returns these values as the first entries of the WORK,
117 RWORK and IWORK arrays, and no error message related to LWORK
118 or LRWORK or LIWORK is issued by XERBLA.
119 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
121 On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
122 LIWORK (input) INTEGER
124 The dimension of the array IWORK. If N <= 1,
125 LIWORK >= 1. If JOBZ = 'N' and N > 1, LIWORK >= 1. If JOBZ
126 = 'V' and N > 1, LIWORK >= 3 + 5*N.
127 If LIWORK = -1, then a workspace query is assumed; the routine
129 only calculates the optimal sizes of the WORK, RWORK and IWORK
130 arrays, returns these values as the first entries of the WORK,
131 RWORK and IWORK arrays, and no error message related to LWORK
132 or LRWORK or LIWORK is issued by XERBLA.
133 INFO (output) INTEGER
135 = 0: successful exit
136 < 0: if INFO = -i, the i-th argument had an illegal value
137 > 0: ZPOTRF or ZHEEVD returned an error code:
138 <= N: if INFO = i and JOBZ = 'N', then the algorithm failed to
139 converge; i off-diagonal elements of an intermediate tridiago‐
140 nal form did not converge to zero; if INFO = i and JOBZ = 'V',
141 then the algorithm failed to compute an eigenvalue while work‐
142 ing on the submatrix lying in rows and columns INFO/(N+1)
143 through mod(INFO,N+1); > N: if INFO = N + i, for 1 <= i <= N,
144 then the leading minor of order i of B is not positive defi‐
145 nite. The factorization of B could not be completed and no ei‐
146 genvalues or eigenvectors were computed.
147
149 Based on contributions by
150 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
151 Modified so that no backsubstitution is performed if ZHEEVD fails to
153 converge (NEIG in old code could be greater than N causing out of
154 bounds reference to A - reported by Ralf Meyer). Also corrected the
155 description of INFO and the test on ITYPE. Sven, 16 Feb 05.
156 LAPACK driver routine (version 3.N1o)vember 2006 ZHEGVD(1)