1DSBEVX(1) LAPACK driver routine (version 3.2) DSBEVX(1)
2
6 DSBEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a real symmetric band matrix A
8
10 SUBROUTINE DSBEVX( 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 DOUBLE PRECISION ABSTOL, VL, VU
19 INTEGER IFAIL( * ), IWORK( * )
21 DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK(
23 * ), Z( LDZ, * )
24
26 DSBEVX 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
79 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
80 upper bounds of the interval to be searched for eigenvalues. VL
81 < VU. Not 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) DOUBLE PRECISION
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*DLAMCH('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*DLAMCH('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) DOUBLE PRECISION array, dimension (N)
110 The first M elements contain the selected eigenvalues in
111 ascending order.
112 Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DSBEVX(1)