1ZHEEVX(1) LAPACK driver routine (version 3.1) ZHEEVX(1)
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6 ZHEEVX - selected eigenvalues and, optionally, eigenvectors of a com‐
7 plex 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.
31
34 JOBZ (input) CHARACTER*1
35 = 'N': Compute eigenvalues only;
36 = 'V': Compute eigenvalues and eigenvectors.
37 RANGE (input) CHARACTER*1
39 = 'A': all eigenvalues will be found.
40 = 'V': all eigenvalues in the half-open interval (VL,VU] will
41 be found. = 'I': the IL-th through IU-th eigenvalues will be
42 found.
43 UPLO (input) CHARACTER*1
45 = 'U': Upper triangle of A is stored;
46 = 'L': Lower triangle of A is stored.
47 N (input) INTEGER
49 The order of the matrix A. N >= 0.
50 A (input/output) COMPLEX*16 array, dimension (LDA, N)
52 On entry, the Hermitian matrix A. If UPLO = 'U', the leading
53 N-by-N upper triangular part of A contains the upper triangular
54 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
55 triangular part of A contains the lower triangular part of the
56 matrix A. On exit, the lower triangle (if UPLO='L') or the
57 upper triangle (if UPLO='U') of A, including the diagonal, is
58 destroyed.
59 LDA (input) INTEGER
61 The leading dimension of the array A. LDA >= max(1,N).
62 VL (input) DOUBLE PRECISION
64 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
65 upper bounds of the interval to be searched for eigenvalues. VL
66 < VU. Not referenced if RANGE = 'A' or 'I'.
67 IL (input) INTEGER
69 IU (input) INTEGER If RANGE='I', the indices (in ascending
70 order) of the smallest and largest eigenvalues to be returned.
71 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
72 referenced if RANGE = 'A' or 'V'.
73 ABSTOL (input) DOUBLE PRECISION
75 The absolute error tolerance for the eigenvalues. An approxi‐
76 mate eigenvalue is accepted as converged when it is determined
77 to lie in an interval [a,b] of width less than or equal to
78 ABSTOL + EPS * max( |a|,|b| ) ,
80 where EPS is the machine precision. If ABSTOL is less than or
82 equal to zero, then EPS*|T| will be used in its place, where
83 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
84 ing A to tridiagonal form.
85 Eigenvalues will be computed most accurately when ABSTOL is set
87 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
88 this routine returns with INFO>0, indicating that some eigen‐
89 vectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
90 See "Computing Small Singular Values of Bidiagonal Matrices
92 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
93 LAPACK Working Note #3.
94 M (output) INTEGER
96 The total number of eigenvalues found. 0 <= M <= N. If RANGE
97 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
98 W (output) DOUBLE PRECISION array, dimension (N)
100 On normal exit, the first M elements contain the selected ei‐
101 genvalues in ascending order.
102 Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
104 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
105 tain the orthonormal eigenvectors of the matrix A corresponding
106 to the selected eigenvalues, with the i-th column of Z holding
107 the eigenvector associated with W(i). If an eigenvector fails
108 to converge, then that column of Z contains the latest approxi‐
109 mation to the eigenvector, and the index of the eigenvector is
110 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
111 Note: the user must ensure that at least max(1,M) columns are
112 supplied in the array Z; if RANGE = 'V', the exact value of M
113 is not known in advance and an upper bound must be used.
114 LDZ (input) INTEGER
116 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
117 'V', LDZ >= max(1,N).
118 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
120 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
121 LWORK (input) INTEGER
123 The length of the array WORK. LWORK >= 1, when N <= 1; other‐
124 wise 2*N. For optimal efficiency, LWORK >= (NB+1)*N, where NB
125 is the max of the blocksize for ZHETRD and for ZUNMTR as
126 returned by ILAENV.
127 If LWORK = -1, then a workspace query is assumed; the routine
129 only calculates the optimal size of the WORK array, returns
130 this value as the first entry of the WORK array, and no error
131 message related to LWORK is issued by XERBLA.
132 RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
134 IWORK (workspace) INTEGER array, dimension (5*N)
136 IFAIL (output) INTEGER array, dimension (N)
138 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
139 are zero. If INFO > 0, then IFAIL contains the indices of the
140 eigenvectors that failed to converge. If JOBZ = 'N', then
141 IFAIL is not referenced.
142 INFO (output) INTEGER
144 = 0: successful exit
145 < 0: if INFO = -i, the i-th argument had an illegal value
146 > 0: if INFO = i, then i eigenvectors failed to converge.
147 Their indices are stored in array IFAIL.
148 LAPACK driver routine (version 3.N1o)vember 2006 ZHEEVX(1)