laed8#

Functions

void slaed8(
    const INT  icompq,
          INT* K,
    const INT  n,
    const INT  qsiz,
          f32* D,
          f32* Q,
    const INT  ldq,
          INT* indxq,
          f32* rho,
    const INT  cutpnt,
          f32* Z,
          f32* dlambda,
          f32* Q2,
    const INT  ldq2,
          f32* W,
          INT* perm,
          INT* givptr,
          INT* givcol,
          f32* givnum,
          INT* indxp,
          INT* indx,
          INT* info
);
void slaed8(const INT icompq, INT *K, const INT n, const INT qsiz, f32 *D, f32 *Q, const INT ldq, INT *indxq, f32 *rho, const INT cutpnt, f32 *Z, f32 *dlambda, f32 *Q2, const INT ldq2, f32 *W, INT *perm, INT *givptr, INT *givcol, f32 *givnum, INT *indxp, INT *indx, INT *info)#

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

in
icompq

= 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.

out
K

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

in
n

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

in
qsiz

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

inout
D

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.

inout
Q

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.

in
ldq

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

in
indxq

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.

inout
rho

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.

in
cutpnt

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

inout
Z

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.

out
dlambda

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

out
Q2

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 SLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors.

in
ldq2

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

out
W

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

out
perm

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

out
givptr

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

out
givcol

Integer array, dimension (2 * N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Stored column-major with leading dimension 2.

out
givnum

Double precision array, dimension (2 * N). Each number indicates the C and S values to be used in the corresponding Givens rotation. Stored column-major with leading dimension 2.

out
indxp

Integer array, dimension (N). The permutation used to place deflated values of D at the end of the array. INDXP(0:K-1) points to the nondeflated D-values and INDXP(K:N-1) points to the deflated eigenvalues.

out
indx

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

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

Functions

void dlaed8(
    const INT  icompq,
          INT* K,
    const INT  n,
    const INT  qsiz,
          f64* D,
          f64* Q,
    const INT  ldq,
          INT* indxq,
          f64* rho,
    const INT  cutpnt,
          f64* Z,
          f64* dlambda,
          f64* Q2,
    const INT  ldq2,
          f64* W,
          INT* perm,
          INT* givptr,
          INT* givcol,
          f64* givnum,
          INT* indxp,
          INT* indx,
          INT* info
);
void dlaed8(const INT icompq, INT *K, const INT n, const INT qsiz, f64 *D, f64 *Q, const INT ldq, INT *indxq, f64 *rho, const INT cutpnt, f64 *Z, f64 *dlambda, f64 *Q2, const INT ldq2, f64 *W, INT *perm, INT *givptr, INT *givcol, f64 *givnum, INT *indxp, INT *indx, INT *info)#

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

in
icompq

= 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.

out
K

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

in
n

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

in
qsiz

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

inout
D

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.

inout
Q

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.

in
ldq

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

in
indxq

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.

inout
rho

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.

in
cutpnt

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

inout
Z

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.

out
dlambda

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

out
Q2

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.

in
ldq2

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

out
W

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

out
perm

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

out
givptr

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

out
givcol

Integer array, dimension (2 * N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Stored column-major with leading dimension 2.

out
givnum

Double precision array, dimension (2 * N). Each number indicates the C and S values to be used in the corresponding Givens rotation. Stored column-major with leading dimension 2.

out
indxp

Integer array, dimension (N). The permutation used to place deflated values of D at the end of the array. INDXP(0:K-1) points to the nondeflated D-values and INDXP(K:N-1) points to the deflated eigenvalues.

out
indx

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

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

Functions

void claed8(
          INT* K,
    const INT  n,
    const INT  qsiz,
          c64* Q,
    const INT  ldq,
          f32* D,
          f32* rho,
    const INT  cutpnt,
          f32* Z,
          f32* dlambda,
          c64* Q2,
    const INT  ldq2,
          f32* W,
          INT* indxp,
          INT* indx,
          INT* indxq,
          INT* perm,
          INT* givptr,
          INT* givcol,
          f32* givnum,
          INT* info
);
void claed8(INT *K, const INT n, const INT qsiz, c64 *Q, const INT ldq, f32 *D, f32 *rho, const INT cutpnt, f32 *Z, f32 *dlambda, c64 *Q2, const INT ldq2, f32 *W, INT *indxp, INT *indx, INT *indxq, INT *perm, INT *givptr, INT *givcol, f32 *givnum, INT *info)#

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

out
K

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

in
n

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

in
qsiz

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

inout
Q

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.

in
ldq

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

inout
D

Single 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.

inout
rho

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.

in
cutpnt

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

inout
Z

Single 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.

out
dlambda

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

out
Q2

Complex array, dimension (LDQ2, N). A copy of the first K eigenvectors which will be used by SLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors.

in
ldq2

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

out
W

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

out
indxp

Integer array, dimension (N). The permutation used to place deflated values of D at the end of the array. INDXP(0:K-1) points to the nondeflated D-values and INDXP(K:N-1) points to the deflated eigenvalues.

out
indx

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

in
indxq

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.

out
perm

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

out
givptr

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

out
givcol

Integer array, dimension (2 * N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Stored column-major with leading dimension 2.

out
givnum

Single precision array, dimension (2 * N). Each number indicates the C and S values to be used in the corresponding Givens rotation. Stored column-major with leading dimension 2.

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

Functions

void zlaed8(
          INT*  K,
    const INT   n,
    const INT   qsiz,
          c128* Q,
    const INT   ldq,
          f64*  D,
          f64*  rho,
    const INT   cutpnt,
          f64*  Z,
          f64*  dlambda,
          c128* Q2,
    const INT   ldq2,
          f64*  W,
          INT*  indxp,
          INT*  indx,
          INT*  indxq,
          INT*  perm,
          INT*  givptr,
          INT*  givcol,
          f64*  givnum,
          INT*  info
);
void zlaed8(INT *K, const INT n, const INT qsiz, c128 *Q, const INT ldq, f64 *D, f64 *rho, const INT cutpnt, f64 *Z, f64 *dlambda, c128 *Q2, const INT ldq2, f64 *W, INT *indxp, INT *indx, INT *indxq, INT *perm, INT *givptr, INT *givcol, f64 *givnum, INT *info)#

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

out
K

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

in
n

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

in
qsiz

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

inout
Q

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.

in
ldq

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

inout
D

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.

inout
rho

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.

in
cutpnt

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

inout
Z

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.

out
dlambda

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

out
Q2

Complex array, dimension (LDQ2, N). A copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors.

in
ldq2

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

out
W

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

out
indxp

Integer array, dimension (N). The permutation used to place deflated values of D at the end of the array. INDXP(0:K-1) points to the nondeflated D-values and INDXP(K:N-1) points to the deflated eigenvalues.

out
indx

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

in
indxq

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.

out
perm

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

out
givptr

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

out
givcol

Integer array, dimension (2 * N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Stored column-major with leading dimension 2.

out
givnum

Double precision array, dimension (2 * N). Each number indicates the C and S values to be used in the corresponding Givens rotation. Stored column-major with leading dimension 2.

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

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