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