# laed8 - Man Page

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*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*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*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*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*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*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.

Definition at line **225** of file **claed8.f**.

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

*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.

*D*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*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*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*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*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

Definition at line **240** of file **dlaed8.f**.

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

*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.

*D*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*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*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 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*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*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*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*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

Definition at line **240** of file **slaed8.f**.

### 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*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*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*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*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*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*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*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**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.

Definition at line **225** of file **zlaed8.f**.

## Author

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