1SSTEVX(1) LAPACK driver routine (version 3.1) SSTEVX(1)
2
6 SSTEVX - selected eigenvalues and, optionally, eigenvectors of a real
7 symmetric tridiagonal matrix A
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10 SUBROUTINE SSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
11 Z, LDZ, WORK, IWORK, IFAIL, INFO )
12 CHARACTER JOBZ, RANGE
14 INTEGER IL, INFO, IU, LDZ, M, N
16 REAL ABSTOL, VL, VU
18 INTEGER IFAIL( * ), IWORK( * )
20 REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
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24 SSTEVX computes selected eigenvalues and, optionally, eigenvectors of a
25 real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can
26 be selected by specifying either a range of values or a range of
27 indices for the desired eigenvalues.
28
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 N (input) INTEGER
42 The order of the matrix. N >= 0.
43 D (input/output) REAL array, dimension (N)
45 On entry, the n diagonal elements of the tridiagonal matrix A.
46 On exit, D may be multiplied by a constant factor chosen to
47 avoid over/underflow in computing the eigenvalues.
48 E (input/output) REAL array, dimension (max(1,N-1))
50 On entry, the (n-1) subdiagonal elements of the tridiagonal
51 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
52 plied by a constant factor chosen to avoid over/underflow in
53 computing the eigenvalues.
54 VL (input) REAL
56 VU (input) REAL If RANGE='V', the lower and upper bounds
57 of the interval to be searched for eigenvalues. VL < VU. Not
58 referenced if RANGE = 'A' or 'I'.
59 IL (input) INTEGER
61 IU (input) INTEGER If RANGE='I', the indices (in ascending
62 order) of the smallest and largest eigenvalues to be returned.
63 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
64 referenced if RANGE = 'A' or 'V'.
65 ABSTOL (input) REAL
67 The absolute error tolerance for the eigenvalues. An approxi‐
68 mate eigenvalue is accepted as converged when it is determined
69 to lie in an interval [a,b] of width less than or equal to
70 ABSTOL + EPS * max( |a|,|b| ) ,
72 where EPS is the machine precision. If ABSTOL is less than or
74 equal to zero, then EPS*|T| will be used in its place, where
75 |T| is the 1-norm of the tridiagonal matrix.
76 Eigenvalues will be computed most accurately when ABSTOL is set
78 to twice the underflow threshold 2*SLAMCH('S'), not zero. If
79 this routine returns with INFO>0, indicating that some eigen‐
80 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
81 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 The first M elements contain the selected eigenvalues in
92 ascending order.
93 Z (output) REAL array, dimension (LDZ, max(1,M) )
95 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
96 tain the orthonormal eigenvectors of the matrix A corresponding
97 to the selected eigenvalues, with the i-th column of Z holding
98 the eigenvector associated with W(i). If an eigenvector fails
99 to converge (INFO > 0), then that column of Z contains the lat‐
100 est approximation to the eigenvector, and the index of the
101 eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not
102 referenced. Note: the user must ensure that at least max(1,M)
103 columns are supplied in the array Z; if RANGE = 'V', the exact
104 value of M is not known in advance and an upper bound must be
105 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) REAL array, dimension (5*N)
112 IWORK (workspace) INTEGER array, dimension (5*N)
114 IFAIL (output) INTEGER array, dimension (N)
116 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
117 are zero. If INFO > 0, then IFAIL contains the indices of the
118 eigenvectors that failed to converge. If JOBZ = 'N', then
119 IFAIL is not referenced.
120 INFO (output) INTEGER
122 = 0: successful exit
123 < 0: if INFO = -i, the i-th argument had an illegal value
124 > 0: if INFO = i, then i eigenvectors failed to converge.
125 Their indices are stored in array IFAIL.
126 LAPACK driver routine (version 3.N1o)vember 2006 SSTEVX(1)