1SSPEVX(1) LAPACK driver routine (version 3.1) SSPEVX(1)
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6 SSPEVX - selected eigenvalues and, optionally, eigenvectors of a real
7 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.
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31 JOBZ (input) CHARACTER*1
32 = 'N': Compute eigenvalues only;
33 = 'V': Compute eigenvalues and eigenvectors.
34 RANGE (input) CHARACTER*1
36 = 'A': all eigenvalues will be found;
37 = 'V': all eigenvalues in the half-open interval (VL,VU] will
38 be found; = 'I': the IL-th through IU-th eigenvalues will be
39 found.
40 UPLO (input) CHARACTER*1
42 = 'U': Upper triangle of A is stored;
43 = 'L': Lower triangle of A is stored.
44 N (input) INTEGER
46 The order of the matrix A. N >= 0.
47 AP (input/output) REAL array, dimension (N*(N+1)/2)
49 On entry, the upper or lower triangle of the symmetric matrix
50 A, packed columnwise in a linear array. The j-th column of A
51 is stored in the array AP as follows: if UPLO = 'U', AP(i +
52 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
53 (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
54 On exit, AP is overwritten by values generated during the
56 reduction to tridiagonal form. If UPLO = 'U', the diagonal and
57 first superdiagonal of the tridiagonal matrix T overwrite the
58 corresponding elements of A, and if UPLO = 'L', the diagonal
59 and first subdiagonal of T overwrite the corresponding elements
60 of A.
61 VL (input) REAL
63 VU (input) REAL If RANGE='V', the lower and upper bounds
64 of the interval to be searched for eigenvalues. VL < VU. Not
65 referenced if RANGE = 'A' or 'I'.
66 IL (input) INTEGER
68 IU (input) INTEGER If RANGE='I', the indices (in ascending
69 order) of the smallest and largest eigenvalues to be returned.
70 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
71 referenced if RANGE = 'A' or 'V'.
72 ABSTOL (input) REAL
74 The absolute error tolerance for the eigenvalues. An approxi‐
75 mate eigenvalue is accepted as converged when it is determined
76 to lie in an interval [a,b] of width less than or equal to
77 ABSTOL + EPS * max( |a|,|b| ) ,
79 where EPS is the machine precision. If ABSTOL is less than or
81 equal to zero, then EPS*|T| will be used in its place, where
82 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
83 ing AP to tridiagonal form.
84 Eigenvalues will be computed most accurately when ABSTOL is set
86 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
87 this routine returns with INFO>0, indicating that some eigen‐
88 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
89 See "Computing Small Singular Values of Bidiagonal Matrices
91 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
92 LAPACK Working Note #3.
93 M (output) INTEGER
95 The total number of eigenvalues found. 0 <= M <= N. If RANGE
96 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
97 W (output) REAL array, dimension (N)
99 If INFO = 0, the selected eigenvalues 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) REAL array, dimension (8*N)
118 IWORK (workspace) INTEGER array, dimension (5*N)
120 IFAIL (output) INTEGER array, dimension (N)
122 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
123 are zero. If INFO > 0, then IFAIL contains the indices of the
124 eigenvectors that failed to converge. If JOBZ = 'N', then
125 IFAIL is not referenced.
126 INFO (output) INTEGER
128 = 0: successful exit
129 < 0: if INFO = -i, the i-th argument had an illegal value
130 > 0: if INFO = i, then i eigenvectors failed to converge.
131 Their indices are stored in array IFAIL.
132 LAPACK driver routine (version 3.N1o)vember 2006 SSPEVX(1)