# zlaed8.f - Man Page

SRC/zlaed8.f

## Synopsis

### Functions/Subroutines

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.

## Function/Subroutine Documentation

### 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*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.

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

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zlaed8(3) is an alias of zlaed8.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK