1ZHEGVD(1) LAPACK driver routine (version 3.2) ZHEGVD(1)
2
6 ZHEGVD - computes all the eigenvalues, and optionally, the eigenvectors
7 of a 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. The
30 divide and conquer algorithm makes very mild assumptions about floating
31 point arithmetic. It will work on machines with a guard digit in
32 add/subtract, or on those binary machines without guard digits which
33 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
34 conceivably fail on hexadecimal or decimal machines without guard dig‐
35 its, but we know of none.
36
38 ITYPE (input) INTEGER
39 Specifies the problem type to be solved:
40 = 1: A*x = (lambda)*B*x
41 = 2: A*B*x = (lambda)*x
42 = 3: B*A*x = (lambda)*x
43 JOBZ (input) CHARACTER*1
45 = 'N': Compute eigenvalues only;
46 = 'V': Compute eigenvalues and eigenvectors.
47 UPLO (input) CHARACTER*1
49 = 'U': Upper triangles of A and B are stored;
50 = 'L': Lower triangles of A and B are stored.
51 N (input) INTEGER
53 The order of the matrices A and B. N >= 0.
54 A (input/output) COMPLEX*16 array, dimension (LDA, N)
56 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
57 N-by-N upper triangular part of A contains the upper triangular
58 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
59 triangular part of A contains the lower triangular part of the
60 matrix A. On exit, if JOBZ = 'V', then if INFO = 0, A contains
61 the matrix Z of eigenvectors. The eigenvectors are normalized
62 as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3,
63 Z**H*inv(B)*Z = I. If JOBZ = 'N', then on exit the upper tri‐
64 angle (if UPLO='U') or the lower triangle (if UPLO='L') of A,
65 including the diagonal, is destroyed.
66 LDA (input) INTEGER
68 The leading dimension of the array A. LDA >= max(1,N).
69 B (input/output) COMPLEX*16 array, dimension (LDB, N)
71 On entry, the Hermitian matrix B. If UPLO = 'U', the leading
72 N-by-N upper triangular part of B contains the upper triangular
73 part of the matrix B. If UPLO = 'L', the leading N-by-N lower
74 triangular part of B contains the lower triangular part of the
75 matrix B. On exit, if INFO <= N, the part of B containing the
76 matrix is overwritten by the triangular factor U or L from the
77 Cholesky factorization B = U**H*U or B = L*L**H.
78 LDB (input) INTEGER
80 The leading dimension of the array B. LDB >= max(1,N).
81 W (output) DOUBLE PRECISION array, dimension (N)
83 If INFO = 0, the eigenvalues in ascending order.
84 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
86 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
87 LWORK (input) INTEGER
89 The length of the array WORK. If N <= 1, LWORK
90 >= 1. If JOBZ = 'N' and N > 1, LWORK >= N + 1. If JOBZ =
91 'V' and N > 1, LWORK >= 2*N + N**2. If LWORK = -1, then a
92 workspace query is assumed; the routine only calculates the
93 optimal sizes of the WORK, RWORK and IWORK arrays, returns
94 these values as the first entries of the WORK, RWORK and IWORK
95 arrays, and no error message related to LWORK or LRWORK or
96 LIWORK is issued by XERBLA.
97 RWORK (workspace/output) DOUBLE PRECISION array, dimension
99 (MAX(1,LRWORK))
100 On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
101 LRWORK (input) INTEGER
103 The dimension of the array RWORK. If N <= 1,
104 LRWORK >= 1. If JOBZ = 'N' and N > 1, LRWORK >= N. If JOBZ
105 = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2. If LRWORK = -1,
106 then a workspace query is assumed; the routine only calculates
107 the optimal sizes of the WORK, RWORK and IWORK arrays, returns
108 these values as the first entries of the WORK, RWORK and IWORK
109 arrays, and no error message related to LWORK or LRWORK or
110 LIWORK is issued by XERBLA.
111 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
113 On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
114 LIWORK (input) INTEGER
116 The dimension of the array IWORK. If N <= 1,
117 LIWORK >= 1. If JOBZ = 'N' and N > 1, LIWORK >= 1. If JOBZ
118 = 'V' and N > 1, LIWORK >= 3 + 5*N. If LIWORK = -1, then a
119 workspace query is assumed; the routine only calculates the
120 optimal sizes of the WORK, RWORK and IWORK arrays, returns
121 these values as the first entries of the WORK, RWORK and IWORK
122 arrays, and no error message related to LWORK or LRWORK or
123 LIWORK is issued by XERBLA.
124 INFO (output) INTEGER
126 = 0: successful exit
127 < 0: if INFO = -i, the i-th argument had an illegal value
128 > 0: ZPOTRF or ZHEEVD returned an error code:
129 <= N: if INFO = i and JOBZ = 'N', then the algorithm failed to
130 converge; i off-diagonal elements of an intermediate tridiago‐
131 nal form did not converge to zero; if INFO = i and JOBZ = 'V',
132 then the algorithm failed to compute an eigenvalue while work‐
133 ing on the submatrix lying in rows and columns INFO/(N+1)
134 through mod(INFO,N+1); > N: if INFO = N + i, for 1 <= i <= N,
135 then the leading minor of order i of B is not positive defi‐
136 nite. The factorization of B could not be completed and no ei‐
137 genvalues or eigenvectors were computed.
138
140 Based on contributions by
141 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA Modi‐
142 fied so that no backsubstitution is performed if ZHEEVD fails to con‐
143 verge (NEIG in old code could be greater than N causing out of bounds
144 reference to A - reported by Ralf Meyer). Also corrected the descrip‐
145 tion of INFO and the test on ITYPE. Sven, 16 Feb 05.
146 LAPACK driver routine (version 3.N2o)vember 2008 ZHEGVD(1)