1CHBEVX(1) LAPACK driver routine (version 3.2) CHBEVX(1)
2
6 CHBEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a complex Hermitian band matrix A
8
10 SUBROUTINE CHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU,
11 IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,
12 IFAIL, 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 RWORK( * ), W( * )
23 COMPLEX AB( LDAB, * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
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27 CHBEVX computes selected eigenvalues and, optionally, eigenvectors of a
28 complex Hermitian band matrix A. Eigenvalues and eigenvectors can be
29 selected by specifying either a range of values or a range of indices
30 for the desired eigenvalues.
31
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) COMPLEX array, dimension (LDAB, N)
55 On entry, the upper or lower triangle of the Hermitian 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). On exit, AB is overwritten by values gener‐
61 ated during the reduction to tridiagonal form.
62 LDAB (input) INTEGER
64 The leading dimension of the array AB. LDAB >= KD + 1.
65 Q (output) COMPLEX array, dimension (LDQ, N)
67 If JOBZ = 'V', the N-by-N unitary matrix used in the reduction
68 to tridiagonal form. If JOBZ = 'N', the array Q is not refer‐
69 enced.
70 LDQ (input) INTEGER
72 The leading dimension of the array Q. If JOBZ = 'V', then LDQ
73 >= max(1,N).
74 VL (input) REAL
76 VU (input) REAL If RANGE='V', the lower and upper bounds
77 of the interval to be searched for eigenvalues. VL < VU. Not
78 referenced if RANGE = 'A' or 'I'.
79 IL (input) INTEGER
81 IU (input) INTEGER If RANGE='I', the indices (in ascending
82 order) of the smallest and largest eigenvalues to be returned.
83 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
84 referenced if RANGE = 'A' or 'V'.
85 ABSTOL (input) REAL
87 The absolute error tolerance for the eigenvalues. An approxi‐
88 mate eigenvalue is accepted as converged when it is determined
89 to lie in an interval [a,b] of width less than or equal to
90 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
91 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
92 will be used in its place, where |T| is the 1-norm of the
93 tridiagonal matrix obtained by reducing AB to tridiagonal form.
94 Eigenvalues will be computed most accurately when ABSTOL is set
95 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
96 this routine returns with INFO>0, indicating that some eigen‐
97 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
98 See "Computing Small Singular Values of Bidiagonal Matrices
99 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
100 LAPACK Working Note #3.
101 M (output) INTEGER
103 The total number of eigenvalues found. 0 <= M <= N. If RANGE
104 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
105 W (output) REAL array, dimension (N)
107 The first M elements contain the selected eigenvalues in
108 ascending order.
109 Z (output) COMPLEX array, dimension (LDZ, max(1,M))
111 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
112 tain the orthonormal eigenvectors of the matrix A corresponding
113 to the selected eigenvalues, with the i-th column of Z holding
114 the eigenvector associated with W(i). If an eigenvector fails
115 to converge, then that column of Z contains the latest approxi‐
116 mation to the eigenvector, and the index of the eigenvector is
117 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
118 Note: the user must ensure that at least max(1,M) columns are
119 supplied in the array Z; if RANGE = 'V', the exact value of M
120 is not known in advance and an upper bound must be used.
121 LDZ (input) INTEGER
123 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
124 'V', LDZ >= max(1,N).
125 WORK (workspace) COMPLEX array, dimension (N)
127 RWORK (workspace) REAL array, dimension (7*N)
129 IWORK (workspace) INTEGER array, dimension (5*N)
131 IFAIL (output) INTEGER array, dimension (N)
133 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
134 are zero. If INFO > 0, then IFAIL contains the indices of the
135 eigenvectors that failed to converge. If JOBZ = 'N', then
136 IFAIL is not referenced.
137 INFO (output) INTEGER
139 = 0: successful exit
140 < 0: if INFO = -i, the i-th argument had an illegal value
141 > 0: if INFO = i, then i eigenvectors failed to converge.
142 Their indices are stored in array IFAIL.
143 LAPACK driver routine (version 3.N2o)vember 2008 CHBEVX(1)