1SSBGVX(1) LAPACK driver routine (version 3.2) SSBGVX(1)
2
6 SSBGVX - computes selected eigenvalues, and optionally, eigenvectors of
7 a real 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
35 JOBZ (input) CHARACTER*1
36 = 'N': Compute eigenvalues only;
37 = 'V': Compute eigenvalues and eigenvectors.
38 RANGE (input) CHARACTER*1
40 = 'A': all eigenvalues will be found.
41 = 'V': all eigenvalues in the half-open interval (VL,VU] will
42 be found. = 'I': the IL-th through IU-th eigenvalues will be
43 found.
44 UPLO (input) CHARACTER*1
46 = 'U': Upper triangles of A and B are stored;
47 = 'L': Lower triangles of A and B are stored.
48 N (input) INTEGER
50 The order of the matrices A and B. N >= 0.
51 KA (input) INTEGER
53 The number of superdiagonals of the matrix A if UPLO = 'U', or
54 the number of subdiagonals if UPLO = 'L'. KA >= 0.
55 KB (input) INTEGER
57 The number of superdiagonals of the matrix B if UPLO = 'U', or
58 the number of subdiagonals if UPLO = 'L'. KB >= 0.
59 AB (input/output) REAL array, dimension (LDAB, N)
61 On entry, the upper or lower triangle of the symmetric band
62 matrix A, stored in the first ka+1 rows of the array. The j-th
63 column of A is stored in the j-th column of the array AB as
64 follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-
65 ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
66 j<=i<=min(n,j+ka). 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). On exit, the factor S from the split
78 Cholesky factorization B = S**T*S, as returned by SPBSTF.
79 LDBB (input) INTEGER
81 The leading dimension of the array BB. LDBB >= KB+1.
82 Q (output) REAL array, dimension (LDQ, N)
84 If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x =
85 (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and con‐
86 sequently C to tridiagonal form. If JOBZ = 'N', the array Q is
87 not referenced.
88 LDQ (input) INTEGER
90 The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1.
91 If JOBZ = 'V', LDQ >= max(1,N).
92 VL (input) REAL
94 VU (input) REAL If RANGE='V', the lower and upper bounds
95 of the interval to be searched for eigenvalues. VL < VU. Not
96 referenced if RANGE = 'A' or 'I'.
97 IL (input) INTEGER
99 IU (input) INTEGER If RANGE='I', the indices (in ascending
100 order) of the smallest and largest eigenvalues to be returned.
101 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
102 referenced if RANGE = 'A' or 'V'.
103 ABSTOL (input) REAL
105 The absolute error tolerance for the eigenvalues. An approxi‐
106 mate eigenvalue is accepted as converged when it is determined
107 to lie in an interval [a,b] of width less than or equal to
108 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
109 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
110 will be used in its place, where |T| is the 1-norm of the
111 tridiagonal matrix obtained by reducing A to tridiagonal form.
112 Eigenvalues will be computed most accurately when ABSTOL is set
113 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
114 this routine returns with INFO>0, indicating that some eigen‐
115 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
116 M (output) INTEGER
118 The total number of eigenvalues found. 0 <= M <= N. If RANGE
119 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
120 W (output) REAL array, dimension (N)
122 If INFO = 0, the eigenvalues in ascending order.
123 Z (output) REAL array, dimension (LDZ, N)
125 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
126 eigenvectors, with the i-th column of Z holding the eigenvector
127 associated with W(i). The eigenvectors are normalized so
128 Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.
129 LDZ (input) INTEGER
131 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
132 'V', LDZ >= max(1,N).
133 WORK (workspace/output) REAL array, dimension (7N)
135 IWORK (workspace/output) INTEGER array, dimension (5N)
137 IFAIL (output) INTEGER array, dimension (M)
139 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
140 are zero. If INFO > 0, then IFAIL contains the indices of the
141 eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL
142 is not referenced.
143 INFO (output) INTEGER
145 = 0 : successful exit
146 < 0 : if INFO = -i, the i-th argument had an illegal value
147 <= N: if INFO = i, then i eigenvectors failed to converge.
148 Their indices are stored in IFAIL. > N : SPBSTF returned an
149 error code; i.e., if INFO = N + i, for 1 <= i <= N, then the
150 leading minor of order i of B is not positive definite. The
151 factorization of B could not be completed and no eigenvalues or
152 eigenvectors were computed.
153
155 Based on contributions by
156 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
157 LAPACK driver routine (version 3.N2o)vember 2008 SSBGVX(1)