--- name: "zlaed8.f" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/zlaed8.f" --- # zlaed8.f(3) ## NAME 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.