?laed1

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?laed1

Used by sstedc/dstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.

Syntax

call slaed1( n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info )

call dlaed1( n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info )

Include Files

mkl.fi

Description

The routine ?laed1 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 eigenvectors of a tridiagonal matrix. ?laed7 handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) 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 ?laed2.

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 ?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

n

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

d, q, work

REAL for slaed1

DOUBLE PRECISION for dlaed1.

Arrays:

d(*) contains the eigenvalues of the rank-1-perturbed matrix. The dimension of d must be at least max(1, n).

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 (4n+n²).

ldq

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

indxq

INTEGER. Array, dimension (n).

On entry, the permutation which separately sorts the two subproblems in d into ascending order.

rho

REAL for slaed1

DOUBLE PRECISION for dlaed1.

The subdiagonal entry used to create the rank-1 modification. This parameter can be modified by ?laed2, where it is input/output.

cutpnt

INTEGER.

The location of the last eigenvalue in the leading sub-matrix. min(1,n) ≤ cutpnt ≤ n/2.

iwork

INTEGER.

Workspace array, dimension (4n).

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

On exit, contains the permutation which will reintegrate the subproblems back into sorted order, that is, d( indxq(i = 1, n )) will be in ascending order.

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

?laed1