1ZHBGVD(1) LAPACK driver routine (version 3.1) ZHBGVD(1)
2
6 ZHBGVD - all the eigenvalues, and optionally, the eigenvectors of a
7 complex generalized Hermitian-definite banded eigenproblem, of the form
8 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
34 floating point arithmetic. It will work on machines with a guard digit
35 in add/subtract, or on those binary machines without guard digits which
36 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
37 conceivably fail on hexadecimal or decimal machines without guard dig‐
38 its, but we know of none.
39
42 JOBZ (input) CHARACTER*1
43 = 'N': Compute eigenvalues only;
44 = 'V': Compute eigenvalues and eigenvectors.
45 UPLO (input) CHARACTER*1
47 = 'U': Upper triangles of A and B are stored;
48 = 'L': Lower triangles of A and B are stored.
49 N (input) INTEGER
51 The order of the matrices A and B. N >= 0.
52 KA (input) INTEGER
54 The number of superdiagonals of the matrix A if UPLO = 'U', or
55 the number of subdiagonals if UPLO = 'L'. KA >= 0.
56 KB (input) INTEGER
58 The number of superdiagonals of the matrix B if UPLO = 'U', or
59 the number of subdiagonals if UPLO = 'L'. KB >= 0.
60 AB (input/output) COMPLEX*16 array, dimension (LDAB, N)
62 On entry, the upper or lower triangle of the Hermitian band
63 matrix A, stored in the first ka+1 rows of the array. The j-th
64 column of A is stored in the j-th column of the array AB as
65 follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-
66 ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
67 j<=i<=min(n,j+ka).
68 On exit, the contents of AB are destroyed.
70 LDAB (input) INTEGER
72 The leading dimension of the array AB. LDAB >= KA+1.
73 BB (input/output) COMPLEX*16 array, dimension (LDBB, N)
75 On entry, the upper or lower triangle of the Hermitian band
76 matrix B, stored in the first kb+1 rows of the array. The j-th
77 column of B is stored in the j-th column of the array BB as
78 follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-
79 kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for
80 j<=i<=min(n,j+kb).
81 On exit, the factor S from the split Cholesky factorization B =
83 S**H*S, as returned by ZPBSTF.
84 LDBB (input) INTEGER
86 The leading dimension of the array BB. LDBB >= KB+1.
87 W (output) DOUBLE PRECISION array, dimension (N)
89 If INFO = 0, the eigenvalues in ascending order.
90 Z (output) COMPLEX*16 array, dimension (LDZ, N)
92 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
93 eigenvectors, with the i-th column of Z holding the eigenvector
94 associated with W(i). The eigenvectors are normalized so that
95 Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.
96 LDZ (input) INTEGER
98 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
99 'V', LDZ >= N.
100 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
102 On exit, if INFO=0, WORK(1) returns the optimal LWORK.
103 LWORK (input) INTEGER
105 The dimension of the array WORK. If N <= 1,
106 LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= N. If JOBZ =
107 'V' and N > 1, LWORK >= 2*N**2.
108 If LWORK = -1, then a workspace query is assumed; the routine
110 only calculates the optimal sizes of the WORK, RWORK and IWORK
111 arrays, returns these values as the first entries of the WORK,
112 RWORK and IWORK arrays, and no error message related to LWORK
113 or LRWORK or LIWORK is issued by XERBLA.
114 RWORK (workspace/output) DOUBLE PRECISION array, dimension
116 (MAX(1,LRWORK))
117 On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.
118 LRWORK (input) INTEGER
120 The dimension of array RWORK. If N <= 1, LRWORK
121 >= 1. If JOBZ = 'N' and N > 1, LRWORK >= N. If JOBZ = 'V' and
122 N > 1, LRWORK >= 1 + 5*N + 2*N**2.
123 If LRWORK = -1, then a workspace query is assumed; the routine
125 only calculates the optimal sizes of the WORK, RWORK and IWORK
126 arrays, returns these values as the first entries of the WORK,
127 RWORK and IWORK arrays, and no error message related to LWORK
128 or LRWORK or LIWORK is issued by XERBLA.
129 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
131 On exit, if INFO=0, IWORK(1) returns the optimal LIWORK.
132 LIWORK (input) INTEGER
134 The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK
135 >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
136 If LIWORK = -1, then a workspace query is assumed; the routine
138 only calculates the optimal sizes of the WORK, RWORK and IWORK
139 arrays, returns these values as the first entries of the WORK,
140 RWORK and IWORK arrays, and no error message related to LWORK
141 or LRWORK or LIWORK is issued by XERBLA.
142 INFO (output) INTEGER
144 = 0: successful exit
145 < 0: if INFO = -i, the i-th argument had an illegal value
146 > 0: if INFO = i, and i is:
147 <= N: the algorithm failed to converge: i off-diagonal ele‐
148 ments of an intermediate tridiagonal form did not converge to
149 zero; > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF
150 returned INFO = i: B is not positive definite. The factoriza‐
151 tion of B could not be completed and no eigenvalues or eigen‐
152 vectors were computed.
153
155 Based on contributions by
156 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
157 LAPACK driver routine (version 3.N1o)vember 2006 ZHBGVD(1)