1SSBEVX(1) LAPACK driver routine (version 3.2) SSBEVX(1)
2
6 SSBEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a real 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
32 JOBZ (input) CHARACTER*1
33 = 'N': Compute eigenvalues only;
34 = 'V': Compute eigenvalues and eigenvectors.
35 RANGE (input) CHARACTER*1
37 = 'A': all eigenvalues will be found;
38 = 'V': all eigenvalues in the half-open interval (VL,VU] will
39 be found; = 'I': the IL-th through IU-th eigenvalues will be
40 found.
41 UPLO (input) CHARACTER*1
43 = 'U': Upper triangle of A is stored;
44 = 'L': Lower triangle of A is stored.
45 N (input) INTEGER
47 The order of the matrix A. N >= 0.
48 KD (input) INTEGER
50 The number of superdiagonals of the matrix A if UPLO = 'U', or
51 the number of subdiagonals if UPLO = 'L'. KD >= 0.
52 AB (input/output) REAL array, dimension (LDAB, N)
54 On entry, the upper or lower triangle of the symmetric band
55 matrix A, stored in the first KD+1 rows of the array. The j-th
56 column of A is stored in the j-th column of the array AB as
57 follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-
58 kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
59 j<=i<=min(n,j+kd). On exit, AB is overwritten by values gener‐
60 ated during the reduction to tridiagonal form. If UPLO = 'U',
61 the first superdiagonal and the diagonal of the tridiagonal
62 matrix T are returned in rows KD and KD+1 of AB, and if UPLO =
63 'L', the diagonal and first subdiagonal of T are returned in
64 the first two rows of AB.
65 LDAB (input) INTEGER
67 The leading dimension of the array AB. LDAB >= KD + 1.
68 Q (output) REAL array, dimension (LDQ, N)
70 If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduc‐
71 tion to tridiagonal form. If JOBZ = 'N', the array Q is not
72 referenced.
73 LDQ (input) INTEGER
75 The leading dimension of the array Q. If JOBZ = 'V', then LDQ
76 >= max(1,N).
77 VL (input) REAL
79 VU (input) REAL If RANGE='V', the lower and upper bounds
80 of the interval to be searched for eigenvalues. VL < VU. Not
81 referenced if RANGE = 'A' or 'I'.
82 IL (input) INTEGER
84 IU (input) INTEGER If RANGE='I', the indices (in ascending
85 order) of the smallest and largest eigenvalues to be returned.
86 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
87 referenced if RANGE = 'A' or 'V'.
88 ABSTOL (input) REAL
90 The absolute error tolerance for the eigenvalues. An approxi‐
91 mate eigenvalue is accepted as converged when it is determined
92 to lie in an interval [a,b] of width less than or equal to
93 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
94 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
95 will be used in its place, where |T| is the 1-norm of the
96 tridiagonal matrix obtained by reducing AB to tridiagonal form.
97 Eigenvalues will be computed most accurately when ABSTOL is set
98 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
99 this routine returns with INFO>0, indicating that some eigen‐
100 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
101 See "Computing Small Singular Values of Bidiagonal Matrices
102 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
103 LAPACK Working Note #3.
104 M (output) INTEGER
106 The total number of eigenvalues found. 0 <= M <= N. If RANGE
107 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
108 W (output) REAL array, dimension (N)
110 The first M elements contain the selected eigenvalues in
111 ascending order.
112 Z (output) REAL array, dimension (LDZ, max(1,M))
114 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
115 tain the orthonormal eigenvectors of the matrix A corresponding
116 to the selected eigenvalues, with the i-th column of Z holding
117 the eigenvector associated with W(i). If an eigenvector fails
118 to converge, then that column of Z contains the latest approxi‐
119 mation to the eigenvector, and the index of the eigenvector is
120 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
121 Note: the user must ensure that at least max(1,M) columns are
122 supplied in the array Z; if RANGE = 'V', the exact value of M
123 is not known in advance and an upper bound must be used.
124 LDZ (input) INTEGER
126 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
127 'V', LDZ >= max(1,N).
128 WORK (workspace) REAL array, dimension (7*N)
130 IWORK (workspace) INTEGER array, dimension (5*N)
132 IFAIL (output) INTEGER array, dimension (N)
134 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
135 are zero. If INFO > 0, then IFAIL contains the indices of the
136 eigenvectors that failed to converge. If JOBZ = 'N', then
137 IFAIL is not referenced.
138 INFO (output) INTEGER
140 = 0: successful exit.
141 < 0: if INFO = -i, the i-th argument had an illegal value.
142 > 0: if INFO = i, then i eigenvectors failed to converge.
143 Their indices are stored in array IFAIL.
144 LAPACK driver routine (version 3.N2o)vember 2008 SSBEVX(1)