1DSPEVX(1) LAPACK driver routine (version 3.2) DSPEVX(1)
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6 DSPEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a real symmetric matrix A in packed storage
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10 SUBROUTINE DSPEVX( 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 DOUBLE PRECISION ABSTOL, VL, VU
18 INTEGER IFAIL( * ), IWORK( * )
20 DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
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24 DSPEVX 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) DOUBLE PRECISION 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) DOUBLE PRECISION
60 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
61 upper bounds of the interval to be searched for eigenvalues. VL
62 < VU. Not 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) DOUBLE PRECISION
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*DLAMCH('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*DLAMCH('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) DOUBLE PRECISION array, dimension (N)
91 If INFO = 0, the selected eigenvalues in ascending order.
92 Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DSPEVX(1)