1SPTEQR(1) LAPACK routine (version 3.2) SPTEQR(1)
2
6 SPTEQR - computes all eigenvalues and, optionally, eigenvectors of a
7 symmetric positive definite tridiagonal matrix by first factoring the
8 matrix using SPTTRF, and then calling SBDSQR to compute the singular
9 values of the bidiagonal factor
10
12 SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
13 CHARACTER COMPZ
15 INTEGER INFO, LDZ, N
17 REAL D( * ), E( * ), WORK( * ), Z( LDZ, * )
19
21 SPTEQR computes all eigenvalues and, optionally, eigenvectors of a sym‐
22 metric positive definite tridiagonal matrix by first factoring the
23 matrix using SPTTRF, and then calling SBDSQR to compute the singular
24 values of the bidiagonal factor. This routine computes the eigenvalues
25 of the positive definite tridiagonal matrix to high relative accuracy.
26 This means that if the eigenvalues range over many orders of magnitude
27 in size, then the small eigenvalues and corresponding eigenvectors will
28 be computed more accurately than, for example, with the standard QR
29 method. The eigenvectors of a full or band symmetric positive definite
30 matrix can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
31 reduce this matrix to tridiagonal form. (The reduction to tridiagonal
32 form, however, may preclude the possibility of obtaining high relative
33 accuracy in the small eigenvalues of the original matrix, if these ei‐
34 genvalues range over many orders of magnitude.)
35
37 COMPZ (input) CHARACTER*1
38 = 'N': Compute eigenvalues only.
39 = 'V': Compute eigenvectors of original symmetric matrix also.
40 Array Z contains the orthogonal matrix used to reduce the orig‐
41 inal matrix to tridiagonal form. = 'I': Compute eigenvectors
42 of tridiagonal matrix also.
43 N (input) INTEGER
45 The order of the matrix. N >= 0.
46 D (input/output) REAL array, dimension (N)
48 On entry, the n diagonal elements of the tridiagonal matrix.
49 On normal exit, D contains the eigenvalues, in descending
50 order.
51 E (input/output) REAL array, dimension (N-1)
53 On entry, the (n-1) subdiagonal elements of the tridiagonal
54 matrix. On exit, E has been destroyed.
55 Z (input/output) REAL array, dimension (LDZ, N)
57 On entry, if COMPZ = 'V', the orthogonal matrix used in the
58 reduction to tridiagonal form. On exit, if COMPZ = 'V', the
59 orthonormal eigenvectors of the original symmetric matrix; if
60 COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal
61 matrix. If INFO > 0 on exit, Z contains the eigenvectors asso‐
62 ciated with only the stored eigenvalues. If COMPZ = 'N', then
63 Z is not referenced.
64 LDZ (input) INTEGER
66 The leading dimension of the array Z. LDZ >= 1, and if COMPZ =
67 'V' or 'I', LDZ >= max(1,N).
68 WORK (workspace) REAL array, dimension (4*N)
70 INFO (output) INTEGER
72 = 0: successful exit.
73 < 0: if INFO = -i, the i-th argument had an illegal value.
74 > 0: if INFO = i, and i is: <= N the Cholesky factorization
75 of the matrix could not be performed because the i-th principal
76 minor was not positive definite. > N the SVD algorithm
77 failed to converge; if INFO = N+i, i off-diagonal elements of
78 the bidiagonal factor did not converge to zero.
79 LAPACK routine (version 3.2) November 2008 SPTEQR(1)