?laed7

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?laed7

Used by ?stedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Syntax

call slaed7( icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info )

call dlaed7( icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info )

call claed7( n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info )

call zlaed7( n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info )

Include Files

mkl.fi

Description

The routine ?laed7 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/Hermitian matrix that has been reduced to tridiagonal form. For real flavors, slaed1/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) for real flavors, or

T = Q(in)*(D(in)+rho*Z*Z^H)*Q^H(in) = Q(out)*D(out)*Q^H(out) for complex flavors

where Z = Q^T*u for real flavors and Z = Q^H*u for complex flavors, 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/dlaed8 (for real flavors) or by the routine slaed2/dlaed2 (for complex flavors).

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine ?laed4 (as called by ?laed9 or ?laed3). 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.

Input Parameters

icompq

INTEGER. Used with real flavors only.

If icompq = 0, compute eigenvalues only.

If icompq = 1, compute eigenvectors of original dense symmetric matrix also. On entry, the array q must contain the orthogonal matrix used to reduce the original matrix to tridiagonal form.

n

INTEGER. The dimension of the symmetric tridiagonal matrix (n≥ 0).

cutpnt

INTEGER. The location of the last eigenvalue in the leading sub-matrix. min(1,n) ≤ cutpnt ≤ n .

qsiz

INTEGER.

The dimension of the orthogonal/unitary matrix used to reduce the full matrix to tridiagonal form; qsiz≥n (for real flavors, qsiz≥n if icompq = 1).

tlvls

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

curlvl

INTEGER. The current level in the overall merge routine, 0 ≤ curlvl ≤ tlvls .

curpbm

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

d

REAL for slaed7/claed7

DOUBLE PRECISION for dlaed7/zlaed7.

Array, dimension at least max(1, n).

Array d(*) contains the eigenvalues of the rank-1-perturbed matrix.

q, work

REAL for slaed7

DOUBLE PRECISION for dlaed7

COMPLEX for claed7

DOUBLE COMPLEX for zlaed7.

Arrays:

q(ldq, *) contains the eigenvectors of the rank-1-perturbed matrix. The second dimension of q must be at least max(1, n).

work(*) is a workspace array, dimension at least (3n+2*qsiz*n) for real flavors and at least (qsiz*n) for complex flavors.

ldq

INTEGER. The leading dimension of the array q; ldq≥ max(1, n).

indxq

INTEGER. Array, dimension (n).

Contains the permutation that separately sorts the two sub-problems in d into ascending order.

rho

REAL for slaed7 /claed7

DOUBLE PRECISION for dlaed7/zlaed7.

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

qstore

REAL for slaed7/claed7

DOUBLE PRECISION for dlaed7/zlaed7.

Array, dimension (n²+1). Serves also as output parameter.

Stores eigenvectors of submatrices encountered during divide and conquer, packed together. qptr points to beginning of the submatrices.

qptr

INTEGER. Array, dimension (n+2). Serves also as output parameter. 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, perm, givptr

INTEGER. Arrays, dimension (n log₂n) each.

The array prmptr(*) 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.

The array perm(*) contains the permutations (from deflation and sorting) to be applied to each eigenblock. This parameter can be modified by ?laed8, where it is output.

The array givptr(*) 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

INTEGER. Array, dimension (2, n log₂n).

Each pair of numbers indicates a pair of columns to take place in a Givens rotation.

givnum

REAL for slaed7/claed7

DOUBLE PRECISION for dlaed7/zlaed7.

Array, dimension (2, n log₂n).

Each number indicates the S value to be used in the corresponding Givens rotation.

iwork

INTEGER.

Workspace array, dimension (4n ).

rwork

REAL for claed7

DOUBLE PRECISION for zlaed7.

Workspace array, dimension (3n+2qsiz*n). Used in complex flavors only.

Output Parameters

d

On exit, contains the eigenvalues of the repaired matrix.

q

On exit, q contains the eigenvectors of the repaired tridiagonal matrix.

indxq

INTEGER. Array, dimension (n).

Contains the permutation that reintegrates the subproblems back into a sorted order, that is,

d(indxq(i = 1, n)) will be in the ascending order.

rho

This parameter can be modified by ?laed8, where it is input/output.

prmptr, perm, givptr

INTEGER. Arrays, dimension (n log₂n) each.

The array prmptr contains an updated list of pointers.

The array perm contains an updated permutation.

The array givptr contains an updated list of pointers.

givcol

This parameter can be modified by ?laed8, where it is output.

givnum

This parameter can be modified by ?laed8, where it is output.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = 1, an eigenvalue did not converge.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?laed7