1SSBGVX(1) LAPACK driver routine (version 3.1) SSBGVX(1)
2
6 SSBGVX - selected eigenvalues, and optionally, eigenvectors of a real
7 generalized symmetric-definite banded eigenproblem, of the form
8 A*x=(lambda)*B*x
9
11 SUBROUTINE SSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q,
12 LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
13 IWORK, IFAIL, INFO )
14 CHARACTER JOBZ, RANGE, UPLO
16 INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N
18 REAL ABSTOL, VL, VU
20 INTEGER IFAIL( * ), IWORK( * )
22 REAL AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), W( * ),
24 WORK( * ), Z( LDZ, * )
25
27 SSBGVX computes selected eigenvalues, and optionally, eigenvectors of a
28 real generalized symmetric-definite banded eigenproblem, of the form
29 A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded,
30 and B is also positive definite. Eigenvalues and eigenvectors can be
31 selected by specifying either all eigenvalues, a range of values or a
32 range of indices for the desired eigenvalues.
33
36 JOBZ (input) CHARACTER*1
37 = 'N': Compute eigenvalues only;
38 = 'V': Compute eigenvalues and eigenvectors.
39 RANGE (input) CHARACTER*1
41 = 'A': all eigenvalues will be found.
42 = 'V': all eigenvalues in the half-open interval (VL,VU] will
43 be found. = 'I': the IL-th through IU-th eigenvalues will be
44 found.
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) REAL array, dimension (LDAB, N)
62 On entry, the upper or lower triangle of the symmetric 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) REAL array, dimension (LDBB, N)
75 On entry, the upper or lower triangle of the symmetric 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(ka+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**T*S, as returned by SPBSTF.
84 LDBB (input) INTEGER
86 The leading dimension of the array BB. LDBB >= KB+1.
87 Q (output) REAL array, dimension (LDQ, N)
89 If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x =
90 (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and con‐
91 sequently C to tridiagonal form. If JOBZ = 'N', the array Q is
92 not referenced.
93 LDQ (input) INTEGER
95 The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1.
96 If JOBZ = 'V', LDQ >= max(1,N).
97 VL (input) REAL
99 VU (input) REAL If RANGE='V', the lower and upper bounds
100 of the interval to be searched for eigenvalues. VL < VU. Not
101 referenced if RANGE = 'A' or 'I'.
102 IL (input) INTEGER
104 IU (input) INTEGER If RANGE='I', the indices (in ascending
105 order) of the smallest and largest eigenvalues to be returned.
106 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
107 referenced if RANGE = 'A' or 'V'.
108 ABSTOL (input) REAL
110 The absolute error tolerance for the eigenvalues. An approxi‐
111 mate eigenvalue is accepted as converged when it is determined
112 to lie in an interval [a,b] of width less than or equal to
113 ABSTOL + EPS * max( |a|,|b| ) ,
115 where EPS is the machine precision. If ABSTOL is less than or
117 equal to zero, then EPS*|T| will be used in its place, where
118 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
119 ing A to tridiagonal form.
120 Eigenvalues will be computed most accurately when ABSTOL is set
122 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
123 this routine returns with INFO>0, indicating that some eigen‐
124 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
125 M (output) INTEGER
127 The total number of eigenvalues found. 0 <= M <= N. If RANGE
128 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
129 W (output) REAL array, dimension (N)
131 If INFO = 0, the eigenvalues in ascending order.
132 Z (output) REAL array, dimension (LDZ, N)
134 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
135 eigenvectors, with the i-th column of Z holding the eigenvector
136 associated with W(i). The eigenvectors are normalized so
137 Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.
138 LDZ (input) INTEGER
140 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
141 'V', LDZ >= max(1,N).
142 WORK (workspace/output) REAL array, dimension (7N)
144 IWORK (workspace/output) INTEGER array, dimension (5N)
146 IFAIL (output) INTEGER array, dimension (M)
148 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
149 are zero. If INFO > 0, then IFAIL contains the indices of the
150 eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL
151 is not referenced.
152 INFO (output) INTEGER
154 = 0 : successful exit
155 < 0 : if INFO = -i, the i-th argument had an illegal value
156 <= N: if INFO = i, then i eigenvectors failed to converge.
157 Their indices are stored in IFAIL. > N : SPBSTF returned an
158 error code; i.e., if INFO = N + i, for 1 <= i <= N, then the
159 leading minor of order i of B is not positive definite. The
160 factorization of B could not be completed and no eigenvalues or
161 eigenvectors were computed.
162
164 Based on contributions by
165 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
166 LAPACK driver routine (version 3.N1o)vember 2006 SSBGVX(1)