1SPTEQR(1) LAPACK routine (version 3.1) SPTEQR(1)
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6 SPTEQR - all eigenvalues and, optionally, eigenvectors of a symmetric
7 positive definite tridiagonal matrix by first factoring the matrix
8 using SPTTRF, and then calling SBDSQR to compute the singular values of
9 the bidiagonal factor
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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, * )
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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.
25 This routine computes the eigenvalues of the positive definite tridiag‐
27 onal matrix to high relative accuracy. This means that if the eigen‐
28 values range over many orders of magnitude in size, then the small ei‐
29 genvalues and corresponding eigenvectors will be computed more accu‐
30 rately than, for example, with the standard QR method.
31 The eigenvectors of a full or band symmetric positive definite matrix
33 can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to reduce
34 this matrix to tridiagonal form. (The reduction to tridiagonal form,
35 however, may preclude the possibility of obtaining high relative accu‐
36 racy in the small eigenvalues of the original matrix, if these eigen‐
37 values range over many orders of magnitude.)
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41 COMPZ (input) CHARACTER*1
42 = 'N': Compute eigenvalues only.
43 = 'V': Compute eigenvectors of original symmetric matrix also.
44 Array Z contains the orthogonal matrix used to reduce the orig‐
45 inal matrix to tridiagonal form. = 'I': Compute eigenvectors
46 of tridiagonal matrix also.
47 N (input) INTEGER
49 The order of the matrix. N >= 0.
50 D (input/output) REAL array, dimension (N)
52 On entry, the n diagonal elements of the tridiagonal matrix.
53 On normal exit, D contains the eigenvalues, in descending
54 order.
55 E (input/output) REAL array, dimension (N-1)
57 On entry, the (n-1) subdiagonal elements of the tridiagonal
58 matrix. On exit, E has been destroyed.
59 Z (input/output) REAL array, dimension (LDZ, N)
61 On entry, if COMPZ = 'V', the orthogonal matrix used in the
62 reduction to tridiagonal form. On exit, if COMPZ = 'V', the
63 orthonormal eigenvectors of the original symmetric matrix; if
64 COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal
65 matrix. If INFO > 0 on exit, Z contains the eigenvectors asso‐
66 ciated with only the stored eigenvalues. If COMPZ = 'N', then
67 Z is not referenced.
68 LDZ (input) INTEGER
70 The leading dimension of the array Z. LDZ >= 1, and if COMPZ =
71 'V' or 'I', LDZ >= max(1,N).
72 WORK (workspace) REAL array, dimension (4*N)
74 INFO (output) INTEGER
76 = 0: successful exit.
77 < 0: if INFO = -i, the i-th argument had an illegal value.
78 > 0: if INFO = i, and i is: <= N the Cholesky factorization
79 of the matrix could not be performed because the i-th principal
80 minor was not positive definite. > N the SVD algorithm
81 failed to converge; if INFO = N+i, i off-diagonal elements of
82 the bidiagonal factor did not converge to zero.
83 LAPACK routine (version 3.1) November 2006 SPTEQR(1)