1SSYEVX(1) LAPACK driver routine (version 3.1) SSYEVX(1)
2
6 SSYEVX - selected eigenvalues and, optionally, eigenvectors of a real
7 symmetric matrix A
8
10 SUBROUTINE SSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
11 ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL,
12 INFO )
13 CHARACTER JOBZ, RANGE, UPLO
15 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
17 REAL ABSTOL, VL, VU
19 INTEGER IFAIL( * ), IWORK( * )
21 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
23
25 SSYEVX computes selected eigenvalues and, optionally, eigenvectors of a
26 real symmetric matrix A. Eigenvalues and eigenvectors can be selected
27 by specifying either a range of values or a range of indices for the
28 desired eigenvalues.
29
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 A (input/output) REAL array, dimension (LDA, N)
50 On entry, the symmetric matrix A. If UPLO = 'U', the leading
51 N-by-N upper triangular part of A contains the upper triangular
52 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
53 triangular part of A contains the lower triangular part of the
54 matrix A. On exit, the lower triangle (if UPLO='L') or the
55 upper triangle (if UPLO='U') of A, including the diagonal, is
56 destroyed.
57 LDA (input) INTEGER
59 The leading dimension of the array A. LDA >= max(1,N).
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| ) ,
78 where EPS is the machine precision. If ABSTOL is less than or
80 equal to zero, then EPS*|T| will be used in its place, where
81 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
82 ing A to tridiagonal form.
83 Eigenvalues will be computed most accurately when ABSTOL is set
85 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
86 this routine returns with INFO>0, indicating that some eigen‐
87 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
88 See "Computing Small Singular Values of Bidiagonal Matrices
90 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
91 LAPACK Working Note #3.
92 M (output) INTEGER
94 The total number of eigenvalues found. 0 <= M <= N. If RANGE
95 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
96 W (output) REAL array, dimension (N)
98 On normal exit, the first M elements contain the selected ei‐
99 genvalues in ascending order.
100 Z (output) REAL array, dimension (LDZ, max(1,M))
102 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
103 tain the orthonormal eigenvectors of the matrix A corresponding
104 to the selected eigenvalues, with the i-th column of Z holding
105 the eigenvector associated with W(i). If an eigenvector fails
106 to converge, then that column of Z contains the latest approxi‐
107 mation to the eigenvector, and the index of the eigenvector is
108 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
109 Note: the user must ensure that at least max(1,M) columns are
110 supplied in the array Z; if RANGE = 'V', the exact value of M
111 is not known in advance and an upper bound must be used.
112 LDZ (input) INTEGER
114 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
115 'V', LDZ >= max(1,N).
116 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
118 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
119 LWORK (input) INTEGER
121 The length of the array WORK. LWORK >= 1, when N <= 1; other‐
122 wise 8*N. For optimal efficiency, LWORK >= (NB+3)*N, where NB
123 is the max of the blocksize for SSYTRD and SORMTR returned by
124 ILAENV.
125 If LWORK = -1, then a workspace query is assumed; the routine
127 only calculates the optimal size of the WORK array, returns
128 this value as the first entry of the WORK array, and no error
129 message related to LWORK is issued by XERBLA.
130 IWORK (workspace) INTEGER array, dimension (5*N)
132 IFAIL (output) INTEGER array, dimension (N)
134 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
135 are zero. If INFO > 0, then IFAIL contains the indices of the
136 eigenvectors that failed to converge. If JOBZ = 'N', then
137 IFAIL is not referenced.
138 INFO (output) INTEGER
140 = 0: successful exit
141 < 0: if INFO = -i, the i-th argument had an illegal value
142 > 0: if INFO = i, then i eigenvectors failed to converge.
143 Their indices are stored in array IFAIL.
144 LAPACK driver routine (version 3.N1o)vember 2006 SSYEVX(1)