laed7

slaed7

Functions

void slaed7(
    const INT  icompq,
    const INT  n,
    const INT  qsiz,
    const INT  tlvls,
    const INT  curlvl,
    const INT  curpbm,
          f32* D,
          f32* Q,
    const INT  ldq,
          INT* indxq,
    const f32  rho,
    const INT  cutpnt,
          f32* qstore,
          INT* qptr,
          INT* prmptr,
          INT* perm,
          INT* givptr,
          INT* givcol,
          f32* givnum,
          f32* work,
          INT* iwork,
          INT* info
);

void slaed7(const INT icompq, const INT n, const INT qsiz, const INT tlvls, const INT curlvl, const INT curpbm, f32 *D, f32 *Q, const INT ldq, INT *indxq, const f32 rho, const INT cutpnt, f32 *qstore, INT *qptr, INT *prmptr, INT *perm, INT *givptr, INT *givcol, f32 *givnum, f32 *work, INT *iwork, INT *info)

SLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix.

This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense symmetric matrix that has been reduced to tridiagonal form. SLAED1 handles the case in which all eigenvalues and eigenvectors of a symmetric tridiagonal matrix are desired.

T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

where Z = Q**Tu, u is a vector of length N with ones in the CUTPNT-1 and CUTPNT th elements and zeros elsewhere (0-based).

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLAED8.

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED9). This routine also calculates the eigenvectors of the current problem.

The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

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.

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.

in

tlvls

The total number of merging levels in the overall divide and conquer tree.

in

curlvl

The current level in the overall merge routine, 0 <= curlvl <= tlvls.

in

curpbm

The current problem in the current level in the overall merge routine (counting from upper left to lower right).

inout

D

Double precision array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

inout

Q

Double precision array, dimension (LDQ, N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

in

ldq

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

out

indxq

Integer array, dimension (N). The permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I ) ) will be in ascending order (0-based indices).

in

rho

The subdiagonal element used to create the rank-1 modification.

in

cutpnt

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

inout

qstore

Double precision array, dimension (N**2+1). Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices (0-based offsets).

inout

qptr

Integer array, dimension (N+2). List of indices (0-based) pointing to beginning of submatrices stored in QSTORE.

in

prmptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in PERM a level’s permutation is stored.

in

perm

Integer array, dimension (N lg N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock (0-based indices).

in

givptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in GIVCOL a level’s Givens rotations are stored.

in

givcol

Integer array, dimension (2 * N lg N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation (0-based indices).

in

givnum

Double precision array, dimension (2 * N lg N). Each number indicates the S value to be used in the corresponding Givens rotation.

out

work

Double precision array, dimension (3*N+2*QSIZ*N).

out

iwork

Integer array, dimension (4*N).

out

info

• = 0: successful exit.

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

• > 0: if info = 1, an eigenvalue did not converge.

dlaed7

Functions

void dlaed7(
    const INT  icompq,
    const INT  n,
    const INT  qsiz,
    const INT  tlvls,
    const INT  curlvl,
    const INT  curpbm,
          f64* D,
          f64* Q,
    const INT  ldq,
          INT* indxq,
    const f64  rho,
    const INT  cutpnt,
          f64* qstore,
          INT* qptr,
          INT* prmptr,
          INT* perm,
          INT* givptr,
          INT* givcol,
          f64* givnum,
          f64* work,
          INT* iwork,
          INT* info
);

void dlaed7(const INT icompq, const INT n, const INT qsiz, const INT tlvls, const INT curlvl, const INT curpbm, f64 *D, f64 *Q, const INT ldq, INT *indxq, const f64 rho, const INT cutpnt, f64 *qstore, INT *qptr, INT *prmptr, INT *perm, INT *givptr, INT *givcol, f64 *givnum, f64 *work, INT *iwork, INT *info)

DLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix.

This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense symmetric matrix that has been reduced to tridiagonal form. DLAED1 handles the case in which all eigenvalues and eigenvectors of a symmetric tridiagonal matrix are desired.

T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

where Z = Q**Tu, u is a vector of length N with ones in the CUTPNT-1 and CUTPNT th elements and zeros elsewhere (0-based).

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED8.

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED9). This routine also calculates the eigenvectors of the current problem.

The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

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.

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.

in

tlvls

The total number of merging levels in the overall divide and conquer tree.

in

curlvl

The current level in the overall merge routine, 0 <= curlvl <= tlvls.

in

curpbm

The current problem in the current level in the overall merge routine (counting from upper left to lower right).

inout

D

Double precision array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

inout

Q

Double precision array, dimension (LDQ, N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

in

ldq

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

out

indxq

Integer array, dimension (N). The permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I ) ) will be in ascending order (0-based indices).

in

rho

The subdiagonal element used to create the rank-1 modification.

in

cutpnt

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

inout

qstore

Double precision array, dimension (N**2+1). Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices (0-based offsets).

inout

qptr

Integer array, dimension (N+2). List of indices (0-based) pointing to beginning of submatrices stored in QSTORE.

in

prmptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in PERM a level’s permutation is stored.

in

perm

Integer array, dimension (N lg N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock (0-based indices).

in

givptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in GIVCOL a level’s Givens rotations are stored.

in

givcol

Integer array, dimension (2 * N lg N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation (0-based indices).

in

givnum

Double precision array, dimension (2 * N lg N). Each number indicates the S value to be used in the corresponding Givens rotation.

out

work

Double precision array, dimension (3*N+2*QSIZ*N).

out

iwork

Integer array, dimension (4*N).

out

info

• = 0: successful exit.

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

• > 0: if info = 1, an eigenvalue did not converge.

claed7

Functions

void claed7(
    const INT  n,
    const INT  cutpnt,
    const INT  qsiz,
    const INT  tlvls,
    const INT  curlvl,
    const INT  curpbm,
          f32* D,
          c64* Q,
    const INT  ldq,
    const f32  rho,
          INT* indxq,
          f32* qstore,
          INT* qptr,
          INT* prmptr,
          INT* perm,
          INT* givptr,
          INT* givcol,
          f32* givnum,
          c64* work,
          f32* rwork,
          INT* iwork,
          INT* info
);

void claed7(const INT n, const INT cutpnt, const INT qsiz, const INT tlvls, const INT curlvl, const INT curpbm, f32 *D, c64 *Q, const INT ldq, const f32 rho, INT *indxq, f32 *qstore, INT *qptr, INT *prmptr, INT *perm, INT *givptr, INT *givcol, f32 *givnum, c64 *work, f32 *rwork, INT *iwork, INT *info)

CLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix.

This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense or banded Hermitian matrix that has been reduced to tridiagonal form.

T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)

where Z = Q**Hu, u is a vector of length N with ones in the CUTPNT-1 and CUTPNT th elements and zeros elsewhere (0-based).

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLAED2.

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED3). This routine also calculates the eigenvectors of the current problem.

The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

Parameters

in

n

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

in

cutpnt

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

in

qsiz

The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N.

in

tlvls

The total number of merging levels in the overall divide and conquer tree.

in

curlvl

The current level in the overall merge routine, 0 <= curlvl <= tlvls.

in

curpbm

The current problem in the current level in the overall merge routine (counting from upper left to lower right).

inout

D

Single precision array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

inout

Q

Complex array, dimension (LDQ, N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

in

ldq

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

in

rho

Contains the subdiagonal element used to create the rank-1 modification.

out

indxq

Integer array, dimension (N). This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I ) ) will be in ascending order (0-based).

inout

qstore

Single precision array, dimension (N**2+1). Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices (0-based offsets).

inout

qptr

Integer array, dimension (N+2). List of indices (0-based) pointing to beginning of submatrices stored in QSTORE.

in

prmptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in PERM a level’s permutation is stored.

in

perm

Integer array, dimension (N lg N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock (0-based indices).

in

givptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in GIVCOL a level’s Givens rotations are stored.

in

givcol

Integer array, dimension (2 * N lg N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation (0-based indices).

in

givnum

Single precision array, dimension (2 * N lg N). Each number indicates the S value to be used in the corresponding Givens rotation.

out

work

Complex array, dimension (QSIZ*N).

out

rwork

Single precision array, dimension (3*N+2*QSIZ*N).

out

iwork

Integer array, dimension (4*N).

out

info

• = 0: successful exit.

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

• > 0: if info = 1, an eigenvalue did not converge.

zlaed7

Functions

void zlaed7(
    const INT   n,
    const INT   cutpnt,
    const INT   qsiz,
    const INT   tlvls,
    const INT   curlvl,
    const INT   curpbm,
          f64*  D,
          c128* Q,
    const INT   ldq,
    const f64   rho,
          INT*  indxq,
          f64*  qstore,
          INT*  qptr,
          INT*  prmptr,
          INT*  perm,
          INT*  givptr,
          INT*  givcol,
          f64*  givnum,
          c128* work,
          f64*  rwork,
          INT*  iwork,
          INT*  info
);

void zlaed7(const INT n, const INT cutpnt, const INT qsiz, const INT tlvls, const INT curlvl, const INT curpbm, f64 *D, c128 *Q, const INT ldq, const f64 rho, INT *indxq, f64 *qstore, INT *qptr, INT *prmptr, INT *perm, INT *givptr, INT *givcol, f64 *givnum, c128 *work, f64 *rwork, INT *iwork, INT *info)

ZLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix.

This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense or banded Hermitian matrix that has been reduced to tridiagonal form.

T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)

where Z = Q**Hu, u is a vector of length N with ones in the CUTPNT-1 and CUTPNT th elements and zeros elsewhere (0-based).

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED2.

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by SLAED3). This routine also calculates the eigenvectors of the current problem.

The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

Parameters

in

n

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

in

cutpnt

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

in

qsiz

The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N.

in

tlvls

The total number of merging levels in the overall divide and conquer tree.

in

curlvl

The current level in the overall merge routine, 0 <= curlvl <= tlvls.

in

curpbm

The current problem in the current level in the overall merge routine (counting from upper left to lower right).

inout

D

Double precision array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

inout

Q

Complex array, dimension (LDQ, N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

in

ldq

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

in

rho

Contains the subdiagonal element used to create the rank-1 modification.

out

indxq

Integer array, dimension (N). This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I ) ) will be in ascending order (0-based).

inout

qstore

Double precision array, dimension (N**2+1). Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices (0-based offsets).

inout

qptr

Integer array, dimension (N+2). List of indices (0-based) pointing to beginning of submatrices stored in QSTORE.

in

prmptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in PERM a level’s permutation is stored.

in

perm

Integer array, dimension (N lg N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock (0-based indices).

in

givptr

Integer array, dimension (N lg N). Contains a list of pointers (0-based) which indicate where in GIVCOL a level’s Givens rotations are stored.

in

givcol

Integer array, dimension (2 * N lg N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation (0-based indices).

in

givnum

Double precision array, dimension (2 * N lg N). Each number indicates the S value to be used in the corresponding Givens rotation.

out

work

Complex array, dimension (QSIZ*N).

out

rwork

Double precision array, dimension (3*N+2*QSIZ*N).

out

iwork

Integer array, dimension (4*N).

out

info

• = 0: successful exit.

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

• > 0: if info = 1, an eigenvalue did not converge.