pteqr(3)

NAME
pteqr - pteqr: eig, positive definite tridiagonal

SYNOPSIS

  Functions

subroutine cpteqr (compz, n, d, e, z, ldz, work, info)
CPTEQR
subroutine dpteqr (compz, n, d, e, z, ldz, work, info)
DPTEQR
subroutine spteqr (compz, n, d, e, z, ldz, work, info)
SPTEQR
subroutine zpteqr (compz, n, d, e, z, ldz, work, info)
ZPTEQR

Detailed Description

Function Documentation

  subroutine cpteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

CPTEQR

Purpose:

    !>
    !> CPTEQR computes all eigenvalues and, optionally, eigenvectors of a
    !> symmetric positive definite tridiagonal matrix by first factoring the
    !> matrix using SPTTRF and then calling CBDSQR 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 CHETRD, CHPTRD, or CHBTRD 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.)
    !>

Parameters
    COMPZ

    !>          COMPZ is 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

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          On entry, the n diagonal elements of the tridiagonal matrix.
    !>          On normal exit, D contains the eigenvalues, in descending
    !>          order.
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          On entry, the (n-1) subdiagonal elements of the tridiagonal
    !>          matrix.
    !>          On exit, E has been destroyed.
    !>

    Z

    !>          Z is COMPLEX 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          COMPZ = 'V' or 'I', LDZ >= max(1,N).
    !>

    WORK

    !>          WORK is REAL array, dimension (4*N)
    !>

    INFO

    !>          INFO is 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 leading principal
    !>                      minor of order i was not positive.
    !>                > N   the SVD algorithm failed to converge;
    !>                      if INFO = N+i, i off-diagonal elements of the
    !>                      bidiagonal factor did not converge to zero.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dpteqr (character compz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer info)

DPTEQR

Purpose:

    !>
    !> DPTEQR computes all eigenvalues and, optionally, eigenvectors of a
    !> symmetric positive definite tridiagonal matrix by first factoring the
    !> matrix using DPTTRF, and then calling DBDSQR 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 symmetric positive definite matrix
    !> can also be found if DSYTRD, DSPTRD, or DSBTRD 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.)
    !>

Parameters
    COMPZ

    !>          COMPZ is CHARACTER*1
    !>          = 'N':  Compute eigenvalues only.
    !>          = 'V':  Compute eigenvectors of original symmetric
    !>                  matrix also.  Array Z contains the orthogonal
    !>                  matrix used to reduce the original matrix to
    !>                  tridiagonal form.
    !>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    D

    !>          D is 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.
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          On entry, the (n-1) subdiagonal elements of the tridiagonal
    !>          matrix.
    !>          On exit, E has been destroyed.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
    !>          On entry, if COMPZ = 'V', the orthogonal matrix used in the
    !>          reduction to tridiagonal form.
    !>          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
    !>          original symmetric 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          COMPZ = 'V' or 'I', LDZ >= max(1,N).
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (4*N)
    !>

    INFO

    !>          INFO is 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 leading principal
    !>                      minor of order i was not positive.
    !>                > N   the SVD algorithm failed to converge;
    !>                      if INFO = N+i, i off-diagonal elements of the
    !>                      bidiagonal factor did not converge to zero.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine spteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

SPTEQR

Purpose:

    !>
    !> SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
    !> symmetric positive definite tridiagonal matrix by first factoring the
    !> matrix using SPTTRF, and then calling SBDSQR 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 symmetric positive definite matrix
    !> can also be found if SSYTRD, SSPTRD, or SSBTRD 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.)
    !>

Parameters
    COMPZ

    !>          COMPZ is CHARACTER*1
    !>          = 'N':  Compute eigenvalues only.
    !>          = 'V':  Compute eigenvectors of original symmetric
    !>                  matrix also.  Array Z contains the orthogonal
    !>                  matrix used to reduce the original matrix to
    !>                  tridiagonal form.
    !>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          On entry, the n diagonal elements of the tridiagonal
    !>          matrix.
    !>          On normal exit, D contains the eigenvalues, in descending
    !>          order.
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          On entry, the (n-1) subdiagonal elements of the tridiagonal
    !>          matrix.
    !>          On exit, E has been destroyed.
    !>

    Z

    !>          Z is REAL array, dimension (LDZ, N)
    !>          On entry, if COMPZ = 'V', the orthogonal matrix used in the
    !>          reduction to tridiagonal form.
    !>          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
    !>          original symmetric 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          COMPZ = 'V' or 'I', LDZ >= max(1,N).
    !>

    WORK

    !>          WORK is REAL array, dimension (4*N)
    !>

    INFO

    !>          INFO is 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 leading principal
    !>                      minor of order i was not positive.
    !>                > N   the SVD algorithm failed to converge;
    !>                      if INFO = N+i, i off-diagonal elements of the
    !>                      bidiagonal factor did not converge to zero.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zpteqr (character compz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer info)

ZPTEQR

Purpose:

    !>
    !> 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.)
    !>

Parameters
    COMPZ

    !>          COMPZ is 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

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    D

    !>          D is 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.
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          On entry, the (n-1) subdiagonal elements of the tridiagonal
    !>          matrix.
    !>          On exit, E has been destroyed.
    !>

    Z

    !>          Z is 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          COMPZ = 'V' or 'I', LDZ >= max(1,N).
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (4*N)
    !>

    INFO

    !>          INFO is 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 leading principal
    !>                      minor of order i was not positive.
    !>                > N   the SVD algorithm failed to converge;
    !>                      if INFO = N+i, i off-diagonal elements of the
    !>                      bidiagonal factor did not converge to zero.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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