1CHPEVX(1) LAPACK driver routine (version 3.2) CHPEVX(1)
2
6 CHPEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a complex Hermitian matrix A in packed storage
8
10 SUBROUTINE CHPEVX( 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 REAL ABSTOL, VL, VU
18 INTEGER IFAIL( * ), IWORK( * )
20 REAL RWORK( * ), W( * )
22 COMPLEX AP( * ), WORK( * ), Z( LDZ, * )
24
26 CHPEVX 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.
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 AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
50 On entry, the upper or lower triangle of the Hermitian matrix
51 A, packed columnwise in a linear array. The j-th column of A
52 is stored in the array AP as follows: if UPLO = 'U', AP(i +
53 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
54 (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is over‐
55 written by values generated during the reduction to tridiagonal
56 form. If UPLO = 'U', the diagonal and first superdiagonal of
57 the tridiagonal matrix T overwrite the corresponding elements
58 of A, and if UPLO = 'L', the diagonal and first subdiagonal of
59 T overwrite the corresponding elements of A.
60 VL (input) REAL
62 VU (input) REAL If RANGE='V', the lower and upper bounds
63 of the interval to be searched for eigenvalues. VL < VU. Not
64 referenced if RANGE = 'A' or 'I'.
65 IL (input) INTEGER
67 IU (input) INTEGER If RANGE='I', the indices (in ascending
68 order) of the smallest and largest eigenvalues to be returned.
69 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
70 referenced if RANGE = 'A' or 'V'.
71 ABSTOL (input) REAL
73 The absolute error tolerance for the eigenvalues. An approxi‐
74 mate eigenvalue is accepted as converged when it is determined
75 to lie in an interval [a,b] of width less than or equal to
76 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
77 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
78 will be used in its place, where |T| is the 1-norm of the
79 tridiagonal matrix obtained by reducing AP to tridiagonal form.
80 Eigenvalues will be computed most accurately when ABSTOL is set
81 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
82 this routine returns with INFO>0, indicating that some eigen‐
83 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
84 See "Computing Small Singular Values of Bidiagonal Matrices
85 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
86 LAPACK Working Note #3.
87 M (output) INTEGER
89 The total number of eigenvalues found. 0 <= M <= N. If RANGE
90 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
91 W (output) REAL array, dimension (N)
93 If INFO = 0, the selected eigenvalues in ascending order.
94 Z (output) COMPLEX array, dimension (LDZ, max(1,M))
96 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
97 tain the orthonormal eigenvectors of the matrix A corresponding
98 to the selected eigenvalues, with the i-th column of Z holding
99 the eigenvector associated with W(i). If an eigenvector fails
100 to converge, then that column of Z contains the latest approxi‐
101 mation to the eigenvector, and the index of the eigenvector is
102 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
103 Note: the user must ensure that at least max(1,M) columns are
104 supplied in the array Z; if RANGE = 'V', the exact value of M
105 is not known in advance and an upper bound must be used.
106 LDZ (input) INTEGER
108 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
109 'V', LDZ >= max(1,N).
110 WORK (workspace) COMPLEX array, dimension (2*N)
112 RWORK (workspace) REAL array, dimension (7*N)
114 IWORK (workspace) INTEGER array, dimension (5*N)
116 IFAIL (output) INTEGER array, dimension (N)
118 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
119 are zero. If INFO > 0, then IFAIL contains the indices of the
120 eigenvectors that failed to converge. If JOBZ = 'N', then
121 IFAIL is not referenced.
122 INFO (output) INTEGER
124 = 0: successful exit
125 < 0: if INFO = -i, the i-th argument had an illegal value
126 > 0: if INFO = i, then i eigenvectors failed to converge.
127 Their indices are stored in array IFAIL.
128 LAPACK driver routine (version 3.N2o)vember 2008 CHPEVX(1)