1SSPEVX(1) LAPACK driver routine (version 3.2) SSPEVX(1)
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6 SSPEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a real symmetric matrix A in packed storage
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10 SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
11 W, Z, LDZ, WORK, 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 AP( * ), W( * ), WORK( * ), Z( LDZ, * )
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24 SSPEVX computes selected eigenvalues and, optionally, eigenvectors of a
25 real symmetric matrix A in packed storage. Eigenvalues/vectors can be
26 selected by specifying either a range of values or a range of indices
27 for the desired eigenvalues.
28
30 JOBZ (input) CHARACTER*1
31 = 'N': Compute eigenvalues only;
32 = 'V': Compute eigenvalues and eigenvectors.
33 RANGE (input) CHARACTER*1
35 = 'A': all eigenvalues will be found;
36 = 'V': all eigenvalues in the half-open interval (VL,VU] will
37 be found; = 'I': the IL-th through IU-th eigenvalues will be
38 found.
39 UPLO (input) CHARACTER*1
41 = 'U': Upper triangle of A is stored;
42 = 'L': Lower triangle of A is stored.
43 N (input) INTEGER
45 The order of the matrix A. N >= 0.
46 AP (input/output) REAL array, dimension (N*(N+1)/2)
48 On entry, the upper or lower triangle of the symmetric matrix
49 A, packed columnwise in a linear array. The j-th column of A
50 is stored in the array AP as follows: if UPLO = 'U', AP(i +
51 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
52 (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is over‐
53 written by values generated during the reduction to tridiagonal
54 form. If UPLO = 'U', the diagonal and first superdiagonal of
55 the tridiagonal matrix T overwrite the corresponding elements
56 of A, and if UPLO = 'L', the diagonal and first subdiagonal of
57 T overwrite the corresponding elements of A.
58 VL (input) REAL
60 VU (input) REAL If RANGE='V', the lower and upper bounds
61 of the interval to be searched for eigenvalues. VL < VU. Not
62 referenced if RANGE = 'A' or 'I'.
63 IL (input) INTEGER
65 IU (input) INTEGER If RANGE='I', the indices (in ascending
66 order) of the smallest and largest eigenvalues to be returned.
67 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
68 referenced if RANGE = 'A' or 'V'.
69 ABSTOL (input) REAL
71 The absolute error tolerance for the eigenvalues. An approxi‐
72 mate eigenvalue is accepted as converged when it is determined
73 to lie in an interval [a,b] of width less than or equal to
74 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
75 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
76 will be used in its place, where |T| is the 1-norm of the
77 tridiagonal matrix obtained by reducing AP to tridiagonal form.
78 Eigenvalues will be computed most accurately when ABSTOL is set
79 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
80 this routine returns with INFO>0, indicating that some eigen‐
81 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
82 See "Computing Small Singular Values of Bidiagonal Matrices
83 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
84 LAPACK Working Note #3.
85 M (output) INTEGER
87 The total number of eigenvalues found. 0 <= M <= N. If RANGE
88 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
89 W (output) REAL array, dimension (N)
91 If INFO = 0, the selected eigenvalues in ascending order.
92 Z (output) REAL array, dimension (LDZ, max(1,M))
94 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
95 tain the orthonormal eigenvectors of the matrix A corresponding
96 to the selected eigenvalues, with the i-th column of Z holding
97 the eigenvector associated with W(i). If an eigenvector fails
98 to converge, then that column of Z contains the latest approxi‐
99 mation to the eigenvector, and the index of the eigenvector is
100 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
101 Note: the user must ensure that at least max(1,M) columns are
102 supplied in the array Z; if RANGE = 'V', the exact value of M
103 is not known in advance and an upper bound must be used.
104 LDZ (input) INTEGER
106 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
107 'V', LDZ >= max(1,N).
108 WORK (workspace) REAL array, dimension (8*N)
110 IWORK (workspace) INTEGER array, dimension (5*N)
112 IFAIL (output) INTEGER array, dimension (N)
114 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
115 are zero. If INFO > 0, then IFAIL contains the indices of the
116 eigenvectors that failed to converge. If JOBZ = 'N', then
117 IFAIL is not referenced.
118 INFO (output) INTEGER
120 = 0: successful exit
121 < 0: if INFO = -i, the i-th argument had an illegal value
122 > 0: if INFO = i, then i eigenvectors failed to converge.
123 Their indices are stored in array IFAIL.
124 LAPACK driver routine (version 3.N2o)vember 2008 SSPEVX(1)