1SSBGVD(1) LAPACK driver routine (version 3.1) SSBGVD(1)
2
6 SSBGVD - all the eigenvalues, and optionally, the eigenvectors of a
7 real generalized symmetric-definite banded eigenproblem, of the form
8 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
31 floating point arithmetic. It will work on machines with a guard digit
32 in 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
39 JOBZ (input) CHARACTER*1
40 = 'N': Compute eigenvalues only;
41 = 'V': Compute eigenvalues and eigenvectors.
42 UPLO (input) CHARACTER*1
44 = 'U': Upper triangles of A and B are stored;
45 = 'L': Lower triangles of A and B are stored.
46 N (input) INTEGER
48 The order of the matrices A and B. N >= 0.
49 KA (input) INTEGER
51 The number of superdiagonals of the matrix A if UPLO = 'U', or
52 the number of subdiagonals if UPLO = 'L'. KA >= 0.
53 KB (input) INTEGER
55 The number of superdiagonals of the matrix B if UPLO = 'U', or
56 the number of subdiagonals if UPLO = 'L'. KB >= 0.
57 AB (input/output) REAL array, dimension (LDAB, N)
59 On entry, the upper or lower triangle of the symmetric band
60 matrix A, stored in the first ka+1 rows of the array. The j-th
61 column of A is stored in the j-th column of the array AB as
62 follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-
63 ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
64 j<=i<=min(n,j+ka).
65 On exit, the contents of AB are destroyed.
67 LDAB (input) INTEGER
69 The leading dimension of the array AB. LDAB >= KA+1.
70 BB (input/output) REAL array, dimension (LDBB, N)
72 On entry, the upper or lower triangle of the symmetric band
73 matrix B, stored in the first kb+1 rows of the array. The j-th
74 column of B is stored in the j-th column of the array BB as
75 follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-
76 kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for
77 j<=i<=min(n,j+kb).
78 On exit, the factor S from the split Cholesky factorization B =
80 S**T*S, as returned by SPBSTF.
81 LDBB (input) INTEGER
83 The leading dimension of the array BB. LDBB >= KB+1.
84 W (output) REAL array, dimension (N)
86 If INFO = 0, the eigenvalues in ascending order.
87 Z (output) REAL array, dimension (LDZ, N)
89 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
90 eigenvectors, with the i-th column of Z holding the eigenvector
91 associated with W(i). The eigenvectors are normalized so
92 Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.
93 LDZ (input) INTEGER
95 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
96 'V', LDZ >= max(1,N).
97 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
99 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
100 LWORK (input) INTEGER
102 The dimension of the array WORK. If N <= 1,
103 LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 3*N. If JOBZ =
104 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
105 If LWORK = -1, then a workspace query is assumed; the routine
107 only calculates the optimal sizes of the WORK and IWORK arrays,
108 returns these values as the first entries of the WORK and IWORK
109 arrays, and no error message related to LWORK or LIWORK is
110 issued by XERBLA.
111 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
113 On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
114 LIWORK (input) INTEGER
116 The dimension of the array IWORK. If JOBZ = 'N' or N <= 1,
117 LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
118 If LIWORK = -1, then a workspace query is assumed; the routine
120 only calculates the optimal sizes of the WORK and IWORK arrays,
121 returns these values as the first entries of the WORK and IWORK
122 arrays, and no error message related to LWORK or LIWORK is
123 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: if INFO = i, and i is:
129 <= N: the algorithm failed to converge: i off-diagonal ele‐
130 ments of an intermediate tridiagonal form did not converge to
131 zero; > N: if INFO = N + i, for 1 <= i <= N, then SPBSTF
132 returned INFO = i: B is not positive definite. The factoriza‐
133 tion of B could not be completed and no eigenvalues or eigen‐
134 vectors were computed.
135
137 Based on contributions by
138 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
139 LAPACK driver routine (version 3.N1o)vember 2006 SSBGVD(1)