ZPTEQR(3F) ZPTEQR(3F)
ZPTEQR - compute all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF and then calling ZBDSQR to compute the singular
values of the bidiagonal factor
SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
CHARACTER COMPZ
INTEGER INFO, LDZ, N
DOUBLE PRECISION D( * ), E( * ), WORK( * )
COMPLEX*16 Z( LDZ, * )
ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF and then calling ZBDSQR to compute the singular
values of the bidiagonal factor.
This routine computes the eigenvalues of the positive definite
tridiagonal matrix to high relative accuracy. This means that if the
eigenvalues range over many orders of magnitude in size, then the small
eigenvalues and corresponding eigenvectors will be computed more
accurately than, for example, with the standard QR method.
The eigenvectors of a full or band positive definite Hermitian matrix can
also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to reduce this
matrix to tridiagonal form. (The reduction to tridiagonal form, however,
may preclude the possibility of obtaining high relative accuracy in the
small eigenvalues of the original matrix, if these eigenvalues range over
many orders of magnitude.)
COMPZ (input) CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvectors of original Hermitian matrix also.
Array Z contains the unitary matrix used to reduce the original
matrix to tridiagonal form. = 'I': Compute eigenvectors of
tridiagonal matrix also.
N (input) INTEGER
The order of the matrix. N >= 0.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix. On
normal exit, D contains the eigenvalues, in descending order.
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ZPTEQR(3F) ZPTEQR(3F)
E (input/output) DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix. On exit, E has been destroyed.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix used in the
reduction to tridiagonal form. On exit, if COMPZ = 'V', the
orthonormal eigenvectors of the original Hermitian matrix; if
COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal
matrix. If INFO > 0 on exit, Z contains the eigenvectors
associated with only the stored eigenvalues. If COMPZ = 'N',
then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if COMPZ =
'V' or 'I', LDZ >= max(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
If COMPZ = 'N', then LWORK = 2*N If COMPZ = 'V' or 'I', then
LWORK = MAX(1,4*N-4)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is: <= N the Cholesky factorization of
the matrix could not be performed because the i-th principal
minor was not positive definite. > N the SVD algorithm failed
to converge; if INFO = N+i, i off-diagonal elements of the
bidiagonal factor did not converge to zero.
ZPTEQR(3F) ZPTEQR(3F)
ZPTEQR - compute all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF and then calling ZBDSQR to compute the singular
values of the bidiagonal factor
SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
CHARACTER COMPZ
INTEGER INFO, LDZ, N
DOUBLE PRECISION D( * ), E( * ), WORK( * )
COMPLEX*16 Z( LDZ, * )
ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF and then calling ZBDSQR to compute the singular
values of the bidiagonal factor.
This routine computes the eigenvalues of the positive definite
tridiagonal matrix to high relative accuracy. This means that if the
eigenvalues range over many orders of magnitude in size, then the small
eigenvalues and corresponding eigenvectors will be computed more
accurately than, for example, with the standard QR method.
The eigenvectors of a full or band positive definite Hermitian matrix can
also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to reduce this
matrix to tridiagonal form. (The reduction to tridiagonal form, however,
may preclude the possibility of obtaining high relative accuracy in the
small eigenvalues of the original matrix, if these eigenvalues range over
many orders of magnitude.)
COMPZ (input) CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvectors of original Hermitian matrix also.
Array Z contains the unitary matrix used to reduce the original
matrix to tridiagonal form. = 'I': Compute eigenvectors of
tridiagonal matrix also.
N (input) INTEGER
The order of the matrix. N >= 0.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix. On
normal exit, D contains the eigenvalues, in descending order.
Page 1
ZPTEQR(3F) ZPTEQR(3F)
E (input/output) DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix. On exit, E has been destroyed.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix used in the
reduction to tridiagonal form. On exit, if COMPZ = 'V', the
orthonormal eigenvectors of the original Hermitian matrix; if
COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal
matrix. If INFO > 0 on exit, Z contains the eigenvectors
associated with only the stored eigenvalues. If COMPZ = 'N',
then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if COMPZ =
'V' or 'I', LDZ >= max(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
If COMPZ = 'N', then LWORK = 2*N If COMPZ = 'V' or 'I', then
LWORK = MAX(1,4*N-4)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is: <= N the Cholesky factorization of
the matrix could not be performed because the i-th principal
minor was not positive definite. > N the SVD algorithm failed
to converge; if INFO = N+i, i off-diagonal elements of the
bidiagonal factor did not converge to zero.
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