?laed2

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?laed2

Used by sstedc/dstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Syntax

call slaed2( k, n, n1, d, q, ldq, indxq, rho, z, dlamda, w, q2, indx, indxc, indxp, coltyp, info )

call dlaed2( k, n, n1, d, q, ldq, indxq, rho, z, dlamda, w, q2, indx, indxc, indxp, coltyp, info )

Include Files

mkl.fi

Description

The routine ?laed2 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 entry in the z vector. For each such occurrence the order of the related secular equation problem is reduced by one.

Input Parameters

k

INTEGER. The number of non-deflated eigenvalues, and the order of the related secular equation (0 ≤ k ≤ n).

n

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

n1

INTEGER. The location of the last eigenvalue in the leading sub-matrix; min(1,n) ≤ n1 ≤ n/2.

d, q, z

REAL for slaed2

DOUBLE PRECISION for dlaed2.

Arrays:

d(*) contains the eigenvalues of the two submatrices to be combined. The dimension of d must be at least max(1, n).

q(ldq, *) contains the eigenvectors of the two submatrices in the two square blocks with corners at (1,1), (n1,n1) and (n1+1,n1+1), (n,n). The second dimension of q must be at least max(1, n).

z(*) contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix).

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. Note that elements in the second half of this permutation must first have n1 added to their values.

rho

REAL for slaed2

DOUBLE PRECISION for dlaed2.

On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined.

indx, indxp

INTEGER.

Workspace arrays, dimension (n) each. Array indx contains the permutation used to sort the contents of dlamda into ascending order.

Array indxp contains 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.

coltyp

INTEGER.

Workspace array, dimension (n).

During execution, a label which will indicate which of the following types a column in the q2 matrix is:

1 : non-zero in the upper half only;

2 : dense;

3 : non-zero in the lower half only;

4 : deflated.

Output Parameters

d

On exit, d contains the trailing (n-k) updated eigenvalues (those which were deflated) sorted into increasing order.

q

On exit, q contains the trailing (n-k) updated eigenvectors (those which were deflated) in its last n-k columns.

z

On exit, z content is destroyed by the updating process.

indxq

Destroyed on exit.

rho

On exit, rho has been modified to the value required by ?laed3.

dlamda, w, q2

REAL for slaed2

DOUBLE PRECISION for dlaed2.

Arrays: dlamda(n), w(n), q2(n1²+(n-n1)²).

The array dlamda contains a copy of the first k eigenvalues which is used by ?laed3 to form the secular equation.

The array w contains the first k values of the final deflation-altered z-vector which is passed to ?laed3.

The array q2 contains a copy of the first k eigenvectors which is used by ?laed3 in a matrix multiply (sgemm/dgemm) to solve for the new eigenvectors.

indxc

INTEGER. Array, dimension (n).

The permutation used to arrange the columns of the deflated q matrix into three groups: the first group contains non-zero elements only at and above n1, the second contains non-zero elements only below n1, and the third is dense.

coltyp

On exit, coltyp(i) is the number of columns of type i, for i=1 to 4 only (see the definition of types in the description of coltyp in Input Parameters).

info

INTEGER.

If info = 0, the execution is successful.

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

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?laed2