1ZHEEVX(1) LAPACK driver routine (version 3.2) ZHEEVX(1)
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6 ZHEEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a complex Hermitian matrix A
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10 SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
11 ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK,
12 IFAIL, INFO )
13 CHARACTER JOBZ, RANGE, UPLO
15 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
17 DOUBLE PRECISION ABSTOL, VL, VU
19 INTEGER IFAIL( * ), IWORK( * )
21 DOUBLE PRECISION RWORK( * ), W( * )
23 COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
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27 ZHEEVX computes selected eigenvalues and, optionally, eigenvectors of a
28 complex Hermitian 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.
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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 A (input/output) COMPLEX*16 array, dimension (LDA, N)
51 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
52 N-by-N upper triangular part of A contains the upper triangular
53 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
54 triangular part of A contains the lower triangular part of the
55 matrix A. On exit, the lower triangle (if UPLO='L') or the
56 upper triangle (if UPLO='U') of A, including the diagonal, is
57 destroyed.
58 LDA (input) INTEGER
60 The leading dimension of the array A. LDA >= max(1,N).
61 VL (input) DOUBLE PRECISION
63 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
64 upper bounds of the interval to be searched for eigenvalues. VL
65 < VU. Not referenced if RANGE = 'A' or 'I'.
66 IL (input) INTEGER
68 IU (input) INTEGER If RANGE='I', the indices (in ascending
69 order) of the smallest and largest eigenvalues to be returned.
70 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
71 referenced if RANGE = 'A' or 'V'.
72 ABSTOL (input) DOUBLE PRECISION
74 The absolute error tolerance for the eigenvalues. An approxi‐
75 mate eigenvalue is accepted as converged when it is determined
76 to lie in an interval [a,b] of width less than or equal to
77 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
78 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
79 will be used in its place, where |T| is the 1-norm of the
80 tridiagonal matrix obtained by reducing A to tridiagonal form.
81 Eigenvalues will be computed most accurately when ABSTOL is set
82 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
83 this routine returns with INFO>0, indicating that some eigen‐
84 vectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
85 See "Computing Small Singular Values of Bidiagonal Matrices
86 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
87 LAPACK Working Note #3.
88 M (output) INTEGER
90 The total number of eigenvalues found. 0 <= M <= N. If RANGE
91 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
92 W (output) DOUBLE PRECISION array, dimension (N)
94 On normal exit, the first M elements contain the selected ei‐
95 genvalues in ascending order.
96 Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
98 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
99 tain the orthonormal eigenvectors of the matrix A corresponding
100 to the selected eigenvalues, with the i-th column of Z holding
101 the eigenvector associated with W(i). If an eigenvector fails
102 to converge, then that column of Z contains the latest approxi‐
103 mation to the eigenvector, and the index of the eigenvector is
104 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
105 Note: the user must ensure that at least max(1,M) columns are
106 supplied in the array Z; if RANGE = 'V', the exact value of M
107 is not known in advance and an upper bound must be used.
108 LDZ (input) INTEGER
110 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
111 'V', LDZ >= max(1,N).
112 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
114 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
115 LWORK (input) INTEGER
117 The length of the array WORK. LWORK >= 1, when N <= 1; other‐
118 wise 2*N. For optimal efficiency, LWORK >= (NB+1)*N, where NB
119 is the max of the blocksize for ZHETRD and for ZUNMTR as
120 returned by ILAENV. If LWORK = -1, then a workspace query is
121 assumed; the routine only calculates the optimal size of the
122 WORK array, returns this value as the first entry of the WORK
123 array, and no error message related to LWORK is issued by
124 XERBLA.
125 RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
127 IWORK (workspace) INTEGER array, dimension (5*N)
129 IFAIL (output) INTEGER array, dimension (N)
131 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
132 are zero. If INFO > 0, then IFAIL contains the indices of the
133 eigenvectors that failed to converge. If JOBZ = 'N', then
134 IFAIL is not referenced.
135 INFO (output) INTEGER
137 = 0: successful exit
138 < 0: if INFO = -i, the i-th argument had an illegal value
139 > 0: if INFO = i, then i eigenvectors failed to converge.
140 Their indices are stored in array IFAIL.
141 LAPACK driver routine (version 3.N2o)vember 2008 ZHEEVX(1)