laed1#

Functions

void slaed1(
    const INT  n,
          f32* D,
          f32* Q,
    const INT  ldq,
          INT* indxq,
    const f32  rho,
    const INT  cutpnt,
          f32* work,
          INT* iwork,
          INT* info
);
void slaed1(const INT n, f32 *D, f32 *Q, const INT ldq, INT *indxq, const f32 rho, const INT cutpnt, f32 *work, INT *iwork, INT *info)#

SLAED1 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. SLAED7 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**T*u, u is a vector of length N with ones in the CUTPNT-1 and CUTPNT th elements and zeros elsewhere (0-based).

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 SLAED2.

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED3). 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.

Parameters

in
n

The dimension of the symmetric tridiagonal matrix. N >= 0.

inout
D

Double precision array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

inout
Q

Double precision array, dimension (LDQ, N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

in
ldq

The leading dimension of the array Q. LDQ >= max(1,N).

inout
indxq

Integer array, dimension (N). On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems back into sorted order, i.e. D( INDXQ( I ) ) will be in ascending order (0-based).

in
rho

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

in
cutpnt

The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.

out
work

Double precision array, dimension (4*N + N**2).

out
iwork

Integer array, dimension (4*N).

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

  • > 0: if info = 1, an eigenvalue did not converge.

Functions

void dlaed1(
    const INT  n,
          f64* D,
          f64* Q,
    const INT  ldq,
          INT* indxq,
    const f64  rho,
    const INT  cutpnt,
          f64* work,
          INT* iwork,
          INT* info
);
void dlaed1(const INT n, f64 *D, f64 *Q, const INT ldq, INT *indxq, const f64 rho, const INT cutpnt, f64 *work, INT *iwork, INT *info)#

DLAED1 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. DLAED7 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**T*u, u is a vector of length N with ones in the CUTPNT-1 and CUTPNT th elements and zeros elsewhere (0-based).

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 DLAED2.

The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED3). 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.

Parameters

in
n

The dimension of the symmetric tridiagonal matrix. N >= 0.

inout
D

Double precision array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.

inout
Q

Double precision array, dimension (LDQ, N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.

in
ldq

The leading dimension of the array Q. LDQ >= max(1,N).

inout
indxq

Integer array, dimension (N). On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems back into sorted order, i.e. D( INDXQ( I ) ) will be in ascending order (0-based).

in
rho

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

in
cutpnt

The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.

out
work

Double precision array, dimension (4*N + N**2).

out
iwork

Integer array, dimension (4*N).

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

  • > 0: if info = 1, an eigenvalue did not converge.

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