1ZHBGVD(1) LAPACK driver routine (version 3.2) ZHBGVD(1)
2
6 ZHBGVD - computes all the eigenvalues, and optionally, the eigenvectors
7 of a complex generalized Hermitian-definite banded eigenproblem, of the
8 form A*x=(lambda)*B*x
9
11 SUBROUTINE ZHBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z,
12 LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO
13 )
14 CHARACTER JOBZ, UPLO
16 INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LRWORK,
18 LWORK, N
19 INTEGER IWORK( * )
21 DOUBLE PRECISION RWORK( * ), W( * )
23 COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), WORK( * ), Z( LDZ, * )
25
27 ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors
28 of a complex generalized Hermitian-definite banded eigenproblem, of the
29 form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and
30 banded, and B is also positive definite. If eigenvectors are desired,
31 it uses a divide and conquer algorithm.
32 The divide and conquer algorithm makes very mild assumptions about
33 floating point arithmetic. It will work on machines with a guard digit
34 in add/subtract, or on those binary machines without guard digits which
35 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
36 conceivably fail on hexadecimal or decimal machines without guard dig‐
37 its, but we know of none.
38
40 JOBZ (input) CHARACTER*1
41 = 'N': Compute eigenvalues only;
42 = 'V': Compute eigenvalues and eigenvectors.
43 UPLO (input) CHARACTER*1
45 = 'U': Upper triangles of A and B are stored;
46 = 'L': Lower triangles of A and B are stored.
47 N (input) INTEGER
49 The order of the matrices A and B. N >= 0.
50 KA (input) INTEGER
52 The number of superdiagonals of the matrix A if UPLO = 'U', or
53 the number of subdiagonals if UPLO = 'L'. KA >= 0.
54 KB (input) INTEGER
56 The number of superdiagonals of the matrix B if UPLO = 'U', or
57 the number of subdiagonals if UPLO = 'L'. KB >= 0.
58 AB (input/output) COMPLEX*16 array, dimension (LDAB, N)
60 On entry, the upper or lower triangle of the Hermitian band
61 matrix A, stored in the first ka+1 rows of the array. The j-th
62 column of A is stored in the j-th column of the array AB as
63 follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-
64 ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
65 j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.
66 LDAB (input) INTEGER
68 The leading dimension of the array AB. LDAB >= KA+1.
69 BB (input/output) COMPLEX*16 array, dimension (LDBB, N)
71 On entry, the upper or lower triangle of the Hermitian band
72 matrix B, stored in the first kb+1 rows of the array. The j-th
73 column of B is stored in the j-th column of the array BB as
74 follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-
75 kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for
76 j<=i<=min(n,j+kb). On exit, the factor S from the split
77 Cholesky factorization B = S**H*S, as returned by ZPBSTF.
78 LDBB (input) INTEGER
80 The leading dimension of the array BB. LDBB >= KB+1.
81 W (output) DOUBLE PRECISION array, dimension (N)
83 If INFO = 0, the eigenvalues in ascending order.
84 Z (output) COMPLEX*16 array, dimension (LDZ, N)
86 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
87 eigenvectors, with the i-th column of Z holding the eigenvector
88 associated with W(i). The eigenvectors are normalized so that
89 Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.
90 LDZ (input) INTEGER
92 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
93 'V', LDZ >= N.
94 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
96 On exit, if INFO=0, WORK(1) returns the optimal LWORK.
97 LWORK (input) INTEGER
99 The dimension of the array WORK. If N <= 1,
100 LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= N. If JOBZ =
101 'V' and N > 1, LWORK >= 2*N**2. If LWORK = -1, then a
102 workspace query is assumed; the routine only calculates the
103 optimal sizes of the WORK, RWORK and IWORK arrays, returns
104 these values as the first entries of the WORK, RWORK and IWORK
105 arrays, and no error message related to LWORK or LRWORK or
106 LIWORK is issued by XERBLA.
107 RWORK (workspace/output) DOUBLE PRECISION array, dimension
109 (MAX(1,LRWORK))
110 On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.
111 LRWORK (input) INTEGER
113 The dimension of array RWORK. If N <= 1, LRWORK
114 >= 1. If JOBZ = 'N' and N > 1, LRWORK >= N. If JOBZ = 'V' and
115 N > 1, LRWORK >= 1 + 5*N + 2*N**2. If LRWORK = -1, then a
116 workspace query is assumed; the routine only calculates the
117 optimal sizes of the WORK, RWORK and IWORK arrays, returns
118 these values as the first entries of the WORK, RWORK and IWORK
119 arrays, and no error message related to LWORK or LRWORK or
120 LIWORK is issued by XERBLA.
121 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
123 On exit, if INFO=0, IWORK(1) returns the optimal LIWORK.
124 LIWORK (input) INTEGER
126 The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK
127 >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. If LIWORK =
128 -1, then a workspace query is assumed; the routine only calcu‐
129 lates the optimal sizes of the WORK, RWORK and IWORK arrays,
130 returns these values as the first entries of the WORK, RWORK
131 and IWORK arrays, and no error message related to LWORK or
132 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: if INFO = i, and i is:
138 <= N: the algorithm failed to converge: i off-diagonal ele‐
139 ments of an intermediate tridiagonal form did not converge to
140 zero; > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF
141 returned INFO = i: B is not positive definite. The factoriza‐
142 tion of B could not be completed and no eigenvalues or eigen‐
143 vectors were computed.
144
146 Based on contributions by
147 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
148 LAPACK driver routine (version 3.N2o)vember 2008 ZHBGVD(1)