1SSBGVD(1) LAPACK driver routine (version 3.2) SSBGVD(1)
2
6 SSBGVD - computes all the eigenvalues, and optionally, the eigenvectors
7 of a real generalized symmetric-definite banded eigenproblem, of the
8 form A*x=(lambda)*B*x
9
11 SUBROUTINE SSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z,
12 LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
13 CHARACTER JOBZ, UPLO
15 INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
17 INTEGER IWORK( * )
19 REAL AB( LDAB, * ), BB( LDBB, * ), W( * ), WORK( * ), Z(
21 LDZ, * )
22
24 SSBGVD computes all the eigenvalues, and optionally, the eigenvectors
25 of a real generalized symmetric-definite banded eigenproblem, of the
26 form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and
27 banded, and B is also positive definite. If eigenvectors are desired,
28 it uses a divide and conquer algorithm.
29 The divide and conquer algorithm makes very mild assumptions about
30 floating point arithmetic. It will work on machines with a guard digit
31 in add/subtract, or on those binary machines without guard digits which
32 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
33 conceivably fail on hexadecimal or decimal machines without guard dig‐
34 its, but we know of none.
35
37 JOBZ (input) CHARACTER*1
38 = 'N': Compute eigenvalues only;
39 = 'V': Compute eigenvalues and eigenvectors.
40 UPLO (input) CHARACTER*1
42 = 'U': Upper triangles of A and B are stored;
43 = 'L': Lower triangles of A and B are stored.
44 N (input) INTEGER
46 The order of the matrices A and B. N >= 0.
47 KA (input) INTEGER
49 The number of superdiagonals of the matrix A if UPLO = 'U', or
50 the number of subdiagonals if UPLO = 'L'. KA >= 0.
51 KB (input) INTEGER
53 The number of superdiagonals of the matrix B if UPLO = 'U', or
54 the number of subdiagonals if UPLO = 'L'. KB >= 0.
55 AB (input/output) REAL array, dimension (LDAB, N)
57 On entry, the upper or lower triangle of the symmetric band
58 matrix A, stored in the first ka+1 rows of the array. The j-th
59 column of A is stored in the j-th column of the array AB as
60 follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-
61 ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
62 j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.
63 LDAB (input) INTEGER
65 The leading dimension of the array AB. LDAB >= KA+1.
66 BB (input/output) REAL array, dimension (LDBB, N)
68 On entry, the upper or lower triangle of the symmetric band
69 matrix B, stored in the first kb+1 rows of the array. The j-th
70 column of B is stored in the j-th column of the array BB as
71 follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-
72 kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for
73 j<=i<=min(n,j+kb). On exit, the factor S from the split
74 Cholesky factorization B = S**T*S, as returned by SPBSTF.
75 LDBB (input) INTEGER
77 The leading dimension of the array BB. LDBB >= KB+1.
78 W (output) REAL array, dimension (N)
80 If INFO = 0, the eigenvalues in ascending order.
81 Z (output) REAL array, dimension (LDZ, N)
83 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
84 eigenvectors, with the i-th column of Z holding the eigenvector
85 associated with W(i). The eigenvectors are normalized so
86 Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.
87 LDZ (input) INTEGER
89 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
90 'V', LDZ >= max(1,N).
91 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
93 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
94 LWORK (input) INTEGER
96 The dimension of the array WORK. If N <= 1,
97 LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 3*N. If JOBZ =
98 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2. If LWORK = -1, then
99 a workspace query is assumed; the routine only calculates the
100 optimal sizes of the WORK and IWORK arrays, returns these val‐
101 ues as the first entries of the WORK and IWORK arrays, and no
102 error message related to LWORK or LIWORK is issued by XERBLA.
103 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
105 On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
106 LIWORK (input) INTEGER
108 The dimension of the array IWORK. If JOBZ = 'N' or N <= 1,
109 LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. If
110 LIWORK = -1, then a workspace query is assumed; the routine
111 only calculates the optimal sizes of the WORK and IWORK arrays,
112 returns these values as the first entries of the WORK and IWORK
113 arrays, and no error message related to LWORK or LIWORK is
114 issued by XERBLA.
115 INFO (output) INTEGER
117 = 0: successful exit
118 < 0: if INFO = -i, the i-th argument had an illegal value
119 > 0: if INFO = i, and i is:
120 <= N: the algorithm failed to converge: i off-diagonal ele‐
121 ments of an intermediate tridiagonal form did not converge to
122 zero; > N: if INFO = N + i, for 1 <= i <= N, then SPBSTF
123 returned INFO = i: B is not positive definite. The factoriza‐
124 tion of B could not be completed and no eigenvalues or eigen‐
125 vectors were computed.
126
128 Based on contributions by
129 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
130 LAPACK driver routine (version 3.N2o)vember 2008 SSBGVD(1)