1ZHPEVX(1) LAPACK driver routine (version 3.1) ZHPEVX(1)
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6 ZHPEVX - selected eigenvalues and, optionally, eigenvectors of a com‐
7 plex Hermitian matrix A in packed storage
8
10 SUBROUTINE ZHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
11 W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
12 CHARACTER JOBZ, RANGE, UPLO
14 INTEGER IL, INFO, IU, LDZ, M, N
16 DOUBLE PRECISION ABSTOL, VL, VU
18 INTEGER IFAIL( * ), IWORK( * )
20 DOUBLE PRECISION RWORK( * ), W( * )
22 COMPLEX*16 AP( * ), WORK( * ), Z( LDZ, * )
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26 ZHPEVX computes selected eigenvalues and, optionally, eigenvectors of a
27 complex Hermitian matrix A in packed storage. Eigenvalues/vectors can
28 be selected by specifying either a range of values or a range of
29 indices 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 AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
51 On entry, the upper or lower triangle of the Hermitian matrix
52 A, packed columnwise in a linear array. The j-th column of A
53 is stored in the array AP as follows: if UPLO = 'U', AP(i +
54 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
55 (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
56 On exit, AP is overwritten by values generated during the
58 reduction to tridiagonal form. If UPLO = 'U', the diagonal and
59 first superdiagonal of the tridiagonal matrix T overwrite the
60 corresponding elements of A, and if UPLO = 'L', the diagonal
61 and first subdiagonal of T overwrite the corresponding elements
62 of A.
63 VL (input) DOUBLE PRECISION
65 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
66 upper bounds of the interval to be searched for eigenvalues. VL
67 < VU. Not referenced if RANGE = 'A' or 'I'.
68 IL (input) INTEGER
70 IU (input) INTEGER If RANGE='I', the indices (in ascending
71 order) of the smallest and largest eigenvalues to be returned.
72 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
73 referenced if RANGE = 'A' or 'V'.
74 ABSTOL (input) DOUBLE PRECISION
76 The absolute error tolerance for the eigenvalues. An approxi‐
77 mate eigenvalue is accepted as converged when it is determined
78 to lie in an interval [a,b] of width less than or equal to
79 ABSTOL + EPS * max( |a|,|b| ) ,
81 where EPS is the machine precision. If ABSTOL is less than or
83 equal to zero, then EPS*|T| will be used in its place, where
84 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
85 ing AP to tridiagonal form.
86 Eigenvalues will be computed most accurately when ABSTOL is set
88 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
89 this routine returns with INFO>0, indicating that some eigen‐
90 vectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
91 See "Computing Small Singular Values of Bidiagonal Matrices
93 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
94 LAPACK Working Note #3.
95 M (output) INTEGER
97 The total number of eigenvalues found. 0 <= M <= N. If RANGE
98 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
99 W (output) DOUBLE PRECISION array, dimension (N)
101 If INFO = 0, the selected eigenvalues 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) COMPLEX*16 array, dimension (2*N)
120 RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
122 IWORK (workspace) INTEGER array, dimension (5*N)
124 IFAIL (output) INTEGER array, dimension (N)
126 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
127 are zero. If INFO > 0, then IFAIL contains the indices of the
128 eigenvectors that failed to converge. If JOBZ = 'N', then
129 IFAIL is not referenced.
130 INFO (output) INTEGER
132 = 0: successful exit
133 < 0: if INFO = -i, the i-th argument had an illegal value
134 > 0: if INFO = i, then i eigenvectors failed to converge.
135 Their indices are stored in array IFAIL.
136 LAPACK driver routine (version 3.N1o)vember 2006 ZHPEVX(1)