# laed7 - Man Page

laed7: D&C step: merge subproblems

## Synopsis

### Functions

subroutine **claed7** (n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info)**CLAED7** used by CSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

subroutine **dlaed7** (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info)**DLAED7** used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

subroutine **slaed7** (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info)**SLAED7** used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

subroutine **zlaed7** (n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info)**ZLAED7** used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

## Detailed Description

## Function Documentation

### subroutine claed7 (integer n, integer cutpnt, integer qsiz, integer tlvls, integer curlvl, integer curpbm, real, dimension( * ) d, complex, dimension( ldq, * ) q, integer ldq, real rho, integer, dimension( * ) indxq, real, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

**CLAED7** used by CSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

**Purpose:**

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 and CUTPNT + 1 th elements and zeros elsewhere. 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***N*N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.

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

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

*TLVLS*TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.

*CURLVL*CURLVL is INTEGER The current level in the overall merge routine, 0 <= curlvl <= tlvls.

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

*D*D is REAL array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

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

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

*RHO*RHO is REAL Contains the subdiagonal element used to create the rank-1 modification.

*INDXQ*INDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, ie. D( INDXQ( I = 1, N ) ) will be in ascending order.

*IWORK*IWORK is INTEGER array, dimension (4*N)

*RWORK*RWORK is REAL array, dimension (3*N+2*QSIZ*N)

*WORK*WORK is COMPLEX array, dimension (QSIZ*N)

*QSTORE*QSTORE is REAL array, dimension (N**2+1) Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices.

*QPTR*QPTR is INTEGER array, dimension (N+2) List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.

*PRMPTR*PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.

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

*GIVPTR*GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.

*GIVCOL*GIVCOL is INTEGER array, dimension (2, N lg 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 lg 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. > 0: if INFO = 1, an eigenvalue did not converge

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **245** of file **claed7.f**.

### subroutine dlaed7 (integer icompq, integer n, integer qsiz, integer tlvls, integer curlvl, integer curpbm, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

**DLAED7** used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

**Purpose:**

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 and CUTPNT + 1 th elements and zeros elsewhere. 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***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.

*N*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.

*TLVLS*TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.

*CURLVL*CURLVL is INTEGER The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.

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

*D*D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

*Q*Q is 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.

*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 will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in ascending order.

*RHO*RHO is DOUBLE PRECISION The subdiagonal element used to create the rank-1 modification.

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

*QSTORE*QSTORE is 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.

*QPTR*QPTR is INTEGER array, dimension (N+2) List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.

*PRMPTR*PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.

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

*GIVPTR*GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.

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

*WORK*WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)

*IWORK*IWORK is INTEGER 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 = 1, an eigenvalue did not converge

**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

Definition at line **256** of file **dlaed7.f**.

### subroutine slaed7 (integer icompq, integer n, integer qsiz, integer tlvls, integer curlvl, integer curpbm, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

**SLAED7** used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

**Purpose:**

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 and CUTPNT + 1 th elements and zeros elsewhere. 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***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.

*N*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.

*TLVLS*TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.

*CURLVL*CURLVL is INTEGER The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.

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

*D*D is REAL array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

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

*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 will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in ascending order.

*RHO*RHO is REAL The subdiagonal element used to create the rank-1 modification.

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

*QSTORE*QSTORE is REAL array, dimension (N**2+1) Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices.

*QPTR*QPTR is INTEGER array, dimension (N+2) List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.

*PRMPTR*PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.

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

*GIVPTR*GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.

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

*WORK*WORK is REAL array, dimension (3*N+2*QSIZ*N)

*IWORK*IWORK is INTEGER 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 = 1, an eigenvalue did not converge

**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

Definition at line **256** of file **slaed7.f**.

### subroutine zlaed7 (integer n, integer cutpnt, integer qsiz, integer tlvls, integer curlvl, integer curpbm, double precision, dimension( * ) d, complex*16, dimension( ldq, * ) q, integer ldq, double precision rho, integer, dimension( * ) indxq, double precision, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

**ZLAED7** used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

**Purpose:**

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 and CUTPNT + 1 th elements and zeros elsewhere. 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***N*N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.

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

*TLVLS*TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.

*CURLVL*CURLVL is INTEGER The current level in the overall merge routine, 0 <= curlvl <= tlvls.

*CURPBM**D*D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

*Q*Q is COMPLEX*16 array, dimension (LDQ,N) On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

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

*RHO*RHO is DOUBLE PRECISION Contains the subdiagonal element used to create the rank-1 modification.

*INDXQ*INDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, ie. D( INDXQ( I = 1, N ) ) will be in ascending order.

*IWORK*IWORK is INTEGER array, dimension (4*N)

*RWORK*RWORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)

*WORK*WORK is COMPLEX*16 array, dimension (QSIZ*N)

*QSTORE*QSTORE is 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.

*QPTR**PRMPTR**PERM**GIVPTR**GIVCOL**GIVNUM*GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation.

*INFO***Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **245** of file **zlaed7.f**.

## Author

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