1SSBEVX(1) LAPACK driver routine (version 3.1) SSBEVX(1)
2
6 SSBEVX - selected eigenvalues and, optionally, eigenvectors of a real
7 symmetric band matrix A
8
10 SUBROUTINE SSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU,
11 IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
12 INFO )
13 CHARACTER JOBZ, RANGE, UPLO
15 INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
17 REAL ABSTOL, VL, VU
19 INTEGER IFAIL( * ), IWORK( * )
21 REAL AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ), Z(
23 LDZ, * )
24
26 SSBEVX computes selected eigenvalues and, optionally, eigenvectors of a
27 real symmetric band matrix A. Eigenvalues and eigenvectors can be
28 selected by specifying either a range of values or a range of indices
29 for the desired eigenvalues.
30
33 JOBZ (input) CHARACTER*1
34 = 'N': Compute eigenvalues only;
35 = 'V': Compute eigenvalues and eigenvectors.
36 RANGE (input) CHARACTER*1
38 = 'A': all eigenvalues will be found;
39 = 'V': all eigenvalues in the half-open interval (VL,VU] will
40 be found; = 'I': the IL-th through IU-th eigenvalues will be
41 found.
42 UPLO (input) CHARACTER*1
44 = 'U': Upper triangle of A is stored;
45 = 'L': Lower triangle of A is stored.
46 N (input) INTEGER
48 The order of the matrix A. N >= 0.
49 KD (input) INTEGER
51 The number of superdiagonals of the matrix A if UPLO = 'U', or
52 the number of subdiagonals if UPLO = 'L'. KD >= 0.
53 AB (input/output) REAL array, dimension (LDAB, N)
55 On entry, the upper or lower triangle of the symmetric band
56 matrix A, stored in the first KD+1 rows of the array. The j-th
57 column of A is stored in the j-th column of the array AB as
58 follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-
59 kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
60 j<=i<=min(n,j+kd).
61 On exit, AB is overwritten by values generated during the
63 reduction to tridiagonal form. If UPLO = 'U', the first super‐
64 diagonal and the diagonal of the tridiagonal matrix T are
65 returned in rows KD and KD+1 of AB, and if UPLO = 'L', the
66 diagonal and first subdiagonal of T are returned in the first
67 two rows of AB.
68 LDAB (input) INTEGER
70 The leading dimension of the array AB. LDAB >= KD + 1.
71 Q (output) REAL array, dimension (LDQ, N)
73 If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduc‐
74 tion to tridiagonal form. If JOBZ = 'N', the array Q is not
75 referenced.
76 LDQ (input) INTEGER
78 The leading dimension of the array Q. If JOBZ = 'V', then LDQ
79 >= max(1,N).
80 VL (input) REAL
82 VU (input) REAL If RANGE='V', the lower and upper bounds
83 of the interval to be searched for eigenvalues. VL < VU. Not
84 referenced if RANGE = 'A' or 'I'.
85 IL (input) INTEGER
87 IU (input) INTEGER If RANGE='I', the indices (in ascending
88 order) of the smallest and largest eigenvalues to be returned.
89 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
90 referenced if RANGE = 'A' or 'V'.
91 ABSTOL (input) REAL
93 The absolute error tolerance for the eigenvalues. An approxi‐
94 mate eigenvalue is accepted as converged when it is determined
95 to lie in an interval [a,b] of width less than or equal to
96 ABSTOL + EPS * max( |a|,|b| ) ,
98 where EPS is the machine precision. If ABSTOL is less than or
100 equal to zero, then EPS*|T| will be used in its place, where
101 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
102 ing AB to tridiagonal form.
103 Eigenvalues will be computed most accurately when ABSTOL is set
105 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
106 this routine returns with INFO>0, indicating that some eigen‐
107 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
108 See "Computing Small Singular Values of Bidiagonal Matrices
110 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
111 LAPACK Working Note #3.
112 M (output) INTEGER
114 The total number of eigenvalues found. 0 <= M <= N. If RANGE
115 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
116 W (output) REAL array, dimension (N)
118 The first M elements contain the selected eigenvalues in
119 ascending order.
120 Z (output) REAL array, dimension (LDZ, max(1,M))
122 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
123 tain the orthonormal eigenvectors of the matrix A corresponding
124 to the selected eigenvalues, with the i-th column of Z holding
125 the eigenvector associated with W(i). If an eigenvector fails
126 to converge, then that column of Z contains the latest approxi‐
127 mation to the eigenvector, and the index of the eigenvector is
128 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
129 Note: the user must ensure that at least max(1,M) columns are
130 supplied in the array Z; if RANGE = 'V', the exact value of M
131 is not known in advance and an upper bound must be used.
132 LDZ (input) INTEGER
134 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
135 'V', LDZ >= max(1,N).
136 WORK (workspace) REAL array, dimension (7*N)
138 IWORK (workspace) INTEGER array, dimension (5*N)
140 IFAIL (output) INTEGER array, dimension (N)
142 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
143 are zero. If INFO > 0, then IFAIL contains the indices of the
144 eigenvectors that failed to converge. If JOBZ = 'N', then
145 IFAIL is not referenced.
146 INFO (output) INTEGER
148 = 0: successful exit.
149 < 0: if INFO = -i, the i-th argument had an illegal value.
150 > 0: if INFO = i, then i eigenvectors failed to converge.
151 Their indices are stored in array IFAIL.
152 LAPACK driver routine (version 3.N1o)vember 2006 SSBEVX(1)