Ubuntu Manpage: laed8 - laed8: D&C step: deflation

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NAME

       laed8 - laed8: D&C step: deflation

SYNOPSIS

   Functions
       subroutine claed8 (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp,
           indx, indxq, perm, givptr, givcol, givnum, info)
           CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the
           original matrix is dense.
       subroutine dlaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2,
           ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)
           DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the
           original matrix is dense.
       subroutine slaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2,
           ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)
           SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the
           original matrix is dense.
       subroutine zlaed8 (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp,
           indx, indxq, perm, givptr, givcol, givnum, info)
           ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the
           original matrix is dense.

Detailed Description

Function Documentation

subroutine claed8 (integer k, integer n, integer qsiz, complex, dimension( ldq, * ) q, integer
ldq, real, dimension( * ) d, real rho, integer cutpnt, real, dimension( * ) z, real,
dimension( * ) dlambda, complex, dimension( ldq2, * ) q2, integer ldq2, real, dimension( *
) w, integer, dimension( * ) indxp, integer, dimension( * ) indx, integer, dimension( * )
indxq, integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol,
real, dimension( 2, * ) givnum, integer info)
CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the
original matrix is dense.

Purpose:

CLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.

Parameters
K

K is INTEGER
Contains the number of non-deflated eigenvalues.
This is the order of the related secular equation.

N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.

QSIZ

QSIZ is INTEGER
The dimension of the unitary matrix used to reduce
the dense or band matrix to tridiagonal form.
QSIZ >= N if ICOMPQ = 1.

Q is COMPLEX array, dimension (LDQ,N)
On entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).

D is REAL array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to
be combined. On exit, D contains the trailing (N-K) updated
eigenvalues (those which were deflated) sorted into increasing
order.

RHO

RHO is REAL
Contains the off diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined. RHO is modified during the computation to
the value required by SLAED3.

CUTPNT

CUTPNT is INTEGER
Contains the location of the last eigenvalue in the leading
sub-matrix. MIN(1,N) <= CUTPNT <= N.

Z is REAL array, dimension (N)
On input this vector contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix). The contents of Z are
destroyed during the updating process.

DLAMBDA

DLAMBDA is REAL array, dimension (N)
Contains a copy of the first K eigenvalues which will be used
by SLAED3 to form the secular equation.

Q2 is COMPLEX array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
Contains a copy of the first K eigenvectors which will be used
by SLAED7 in a matrix multiply (SGEMM) to update the new
eigenvectors.

LDQ2

LDQ2 is INTEGER
The leading dimension of the array Q2. LDQ2 >= max( 1, N ).

W is REAL array, dimension (N)
This will hold the first k values of the final
deflation-altered z-vector and will be passed to SLAED3.

INDXP

INDXP is INTEGER array, dimension (N)
This will contain the permutation used to place deflated
values of D at the end of the array. On output INDXP(1:K)
points to the nondeflated D-values and INDXP(K+1:N)
points to the deflated eigenvalues.

INDX

INDX is INTEGER array, dimension (N)
This will contain the permutation used to sort the contents of
D into ascending order.

INDXQ

INDXQ is INTEGER array, dimension (N)
This contains the permutation which separately sorts the two
sub-problems in D into ascending order. Note that elements in
the second half of this permutation must first have CUTPNT
added to their values in order to be accurate.

PERM

PERM is INTEGER array, dimension (N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.

GIVPTR

GIVPTR is INTEGER
Contains the number of Givens rotations which took place in
this subproblem.

GIVCOL

GIVCOL is INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.

GIVNUM

GIVNUM is REAL array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author
Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dlaed8 (integer icompq, integer k, integer n, integer qsiz, double precision,
dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer,
dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( *
) z, double precision, dimension( * ) dlambda, double precision, dimension( ldq2, * ) q2,
integer ldq2, double precision, dimension( * ) w, integer, dimension( * ) perm, integer
givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum,
integer, dimension( * ) indxp, integer, dimension( * ) indx, integer info)
DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the
original matrix is dense.

Purpose:

DLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.

Parameters
ICOMPQ

ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.

K is INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation.

N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.

QSIZ

QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.

D is DOUBLE PRECISION array, dimension (N)
On entry, the eigenvalues of the two submatrices to be
combined. On exit, the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.

Q is DOUBLE PRECISION array, dimension (LDQ,N)
If ICOMPQ = 0, Q is not referenced. Otherwise,
on entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).

INDXQ

INDXQ is INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems
in D into ascending order. Note that elements in the second
half of this permutation must first have CUTPNT added to
their values in order to be accurate.

RHO

RHO is DOUBLE PRECISION
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
DLAED3.

CUTPNT

CUTPNT is INTEGER
The location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.

Z is DOUBLE PRECISION array, dimension (N)
On entry, Z contains the updating vector (the last row of
the first sub-eigenvector matrix and the first row of the
second sub-eigenvector matrix).
On exit, the contents of Z are destroyed by the updating
process.

DLAMBDA

DLAMBDA is DOUBLE PRECISION array, dimension (N)
A copy of the first K eigenvalues which will be used by
DLAED3 to form the secular equation.

Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
a copy of the first K eigenvectors which will be used by
DLAED7 in a matrix multiply (DGEMM) to update the new
eigenvectors.

LDQ2

LDQ2 is INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).

W is DOUBLE PRECISION array, dimension (N)
The first k values of the final deflation-altered z-vector and
will be passed to DLAED3.

PERM

PERM is INTEGER array, dimension (N)
The permutations (from deflation and sorting) to be applied
to each eigenblock.

GIVPTR

GIVPTR is INTEGER
The number of Givens rotations which took place in this
subproblem.

GIVCOL

GIVCOL is INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.

GIVNUM

GIVNUM is DOUBLE PRECISION array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.

INDXP

INDXP is INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end
of the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.

INDX

INDX is INTEGER array, dimension (N)
The permutation used to sort the contents of D into ascending
order.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author
Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

subroutine slaed8 (integer icompq, integer k, integer n, integer qsiz, real, dimension( * ) d,
real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer
cutpnt, real, dimension( * ) z, real, dimension( * ) dlambda, real, dimension( ldq2, * )
q2, integer ldq2, real, dimension( * ) w, integer, dimension( * ) perm, integer givptr,
integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, integer, dimension( * )
indxp, integer, dimension( * ) indx, integer info)
SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the
original matrix is dense.

Purpose:

SLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.

Parameters
ICOMPQ

K is INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation.

N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.

QSIZ

QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.

D is REAL array, dimension (N)
On entry, the eigenvalues of the two submatrices to be
combined. On exit, the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.

Q is REAL array, dimension (LDQ,N)
If ICOMPQ = 0, Q is not referenced. Otherwise,
on entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).

INDXQ

RHO

RHO is REAL
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
SLAED3.

CUTPNT

CUTPNT is INTEGER
The location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.

Z is REAL array, dimension (N)
On entry, Z contains the updating vector (the last row of
the first sub-eigenvector matrix and the first row of the
second sub-eigenvector matrix).
On exit, the contents of Z are destroyed by the updating
process.

DLAMBDA

DLAMBDA is REAL array, dimension (N)
A copy of the first K eigenvalues which will be used by
SLAED3 to form the secular equation.

Q2 is REAL array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
a copy of the first K eigenvectors which will be used by
SLAED7 in a matrix multiply (SGEMM) to update the new
eigenvectors.

LDQ2

LDQ2 is INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).

W is REAL array, dimension (N)
The first k values of the final deflation-altered z-vector and
will be passed to SLAED3.

PERM

PERM is INTEGER array, dimension (N)
The permutations (from deflation and sorting) to be applied
to each eigenblock.

GIVPTR

GIVPTR is INTEGER
The number of Givens rotations which took place in this
subproblem.

GIVCOL

GIVCOL is INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.

GIVNUM

GIVNUM is REAL array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.

INDXP

INDX

INDX is INTEGER array, dimension (N)
The permutation used to sort the contents of D into ascending
order.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author
Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

subroutine zlaed8 (integer k, integer n, integer qsiz, complex*16, dimension( ldq, * ) q,
integer ldq, double precision, dimension( * ) d, double precision rho, integer cutpnt,
double precision, dimension( * ) z, double precision, dimension( * ) dlambda, complex*16,
dimension( ldq2, * ) q2, integer ldq2, double precision, dimension( * ) w, integer,
dimension( * ) indxp, integer, dimension( * ) indx, integer, dimension( * ) indxq,
integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol, double
precision, dimension( 2, * ) givnum, integer info)
ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the
original matrix is dense.

Purpose:

ZLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.

Parameters
K

K is INTEGER
Contains the number of non-deflated eigenvalues.
This is the order of the related secular equation.

N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.

QSIZ

QSIZ is INTEGER
The dimension of the unitary matrix used to reduce
the dense or band matrix to tridiagonal form.
QSIZ >= N if ICOMPQ = 1.

Q is COMPLEX*16 array, dimension (LDQ,N)
On entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).

D is DOUBLE PRECISION array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to
be combined. On exit, D contains the trailing (N-K) updated
eigenvalues (those which were deflated) sorted into increasing
order.

RHO

RHO is DOUBLE PRECISION
Contains the off diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined. RHO is modified during the computation to
the value required by DLAED3.

CUTPNT

CUTPNT is INTEGER
Contains the location of the last eigenvalue in the leading
sub-matrix. MIN(1,N) <= CUTPNT <= N.

Z is DOUBLE PRECISION array, dimension (N)
On input this vector contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix). The contents of Z are
destroyed during the updating process.

DLAMBDA

DLAMBDA is DOUBLE PRECISION array, dimension (N)
Contains a copy of the first K eigenvalues which will be used
by DLAED3 to form the secular equation.

Q2 is COMPLEX*16 array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
Contains a copy of the first K eigenvectors which will be used
by DLAED7 in a matrix multiply (DGEMM) to update the new
eigenvectors.

LDQ2

LDQ2 is INTEGER
The leading dimension of the array Q2. LDQ2 >= max( 1, N ).

W is DOUBLE PRECISION array, dimension (N)
This will hold the first k values of the final
deflation-altered z-vector and will be passed to DLAED3.

INDXP

INDX

INDX is INTEGER array, dimension (N)
This will contain the permutation used to sort the contents of
D into ascending order.

INDXQ

PERM

PERM is INTEGER array, dimension (N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.

GIVPTR

GIVPTR is INTEGER
Contains the number of Givens rotations which took place in
this subproblem.

GIVCOL

GIVCOL is INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.

GIVNUM

GIVNUM is DOUBLE PRECISION array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author
Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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