# laed3 - Man Page

laed3: D&C step: secular equation

## Synopsis

### Functions

subroutine **dlaed3** (k, n, n1, d, q, ldq, rho, dlambda, q2, indx, ctot, w, s, info)**DLAED3** used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

subroutine **slaed3** (k, n, n1, d, q, ldq, rho, dlambda, q2, indx, ctot, w, s, info)**SLAED3** used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

## Detailed Description

## Function Documentation

### subroutine dlaed3 (integer k, integer n, integer n1, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, double precision rho, double precision, dimension( * ) dlambda, double precision, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) ctot, double precision, dimension( * ) w, double precision, dimension( * ) s, integer info)

**DLAED3** used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

**Purpose:**

DLAED3 finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K. It makes the appropriate calls to DLAED4 and then updates the eigenvectors by multiplying the matrix of eigenvectors of the pair of eigensystems being combined by the matrix of eigenvectors of the K-by-K system which is solved here.

**Parameters***K*K is INTEGER The number of terms in the rational function to be solved by DLAED4. K >= 0.

*N*N is INTEGER The number of rows and columns in the Q matrix. N >= K (deflation may result in N>K).

*N1*N1 is INTEGER The location of the last eigenvalue in the leading submatrix. min(1,N) <= N1 <= N/2.

*D*D is DOUBLE PRECISION array, dimension (N) D(I) contains the updated eigenvalues for 1 <= I <= K.

*Q*Q is DOUBLE PRECISION array, dimension (LDQ,N) Initially the first K columns are used as workspace. On output the columns 1 to K contain the updated eigenvectors.

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

*RHO*RHO is DOUBLE PRECISION The value of the parameter in the rank one update equation. RHO >= 0 required.

*DLAMBDA*DLAMBDA is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.

*Q2*Q2 is DOUBLE PRECISION array, dimension (LDQ2*N) The first K columns of this matrix contain the non-deflated eigenvectors for the split problem.

*INDX*INDX is INTEGER array, dimension (N) The permutation used to arrange the columns of the deflated Q matrix into three groups (see DLAED2). The rows of the eigenvectors found by DLAED4 must be likewise permuted before the matrix multiply can take place.

*CTOT*CTOT is INTEGER array, dimension (4) A count of the total number of the various types of columns in Q, as described in INDX. The fourth column type is any column which has been deflated.

*W*W is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.

*S*S is DOUBLE PRECISION array, dimension (N1 + 1)*K Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.

*INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Modified by Francoise Tisseur, University of Tennessee

Definition at line **175** of file **dlaed3.f**.

### subroutine slaed3 (integer k, integer n, integer n1, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, real rho, real, dimension( * ) dlambda, real, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) ctot, real, dimension( * ) w, real, dimension( * ) s, integer info)

**SLAED3** used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

**Purpose:**

SLAED3 finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K. It makes the appropriate calls to SLAED4 and then updates the eigenvectors by multiplying the matrix of eigenvectors of the pair of eigensystems being combined by the matrix of eigenvectors of the K-by-K system which is solved here.

**Parameters***K*K is INTEGER The number of terms in the rational function to be solved by SLAED4. K >= 0.

*N*N is INTEGER The number of rows and columns in the Q matrix. N >= K (deflation may result in N>K).

*N1*N1 is INTEGER The location of the last eigenvalue in the leading submatrix. min(1,N) <= N1 <= N/2.

*D*D is REAL array, dimension (N) D(I) contains the updated eigenvalues for 1 <= I <= K.

*Q*Q is REAL array, dimension (LDQ,N) Initially the first K columns are used as workspace. On output the columns 1 to K contain the updated eigenvectors.

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

*RHO*RHO is REAL The value of the parameter in the rank one update equation. RHO >= 0 required.

*DLAMBDA*DLAMBDA is REAL array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.

*Q2*Q2 is REAL array, dimension (LDQ2*N) The first K columns of this matrix contain the non-deflated eigenvectors for the split problem.

*INDX*INDX is INTEGER array, dimension (N) The permutation used to arrange the columns of the deflated Q matrix into three groups (see SLAED2). The rows of the eigenvectors found by SLAED4 must be likewise permuted before the matrix multiply can take place.

*CTOT*CTOT is INTEGER array, dimension (4) A count of the total number of the various types of columns in Q, as described in INDX. The fourth column type is any column which has been deflated.

*W*W is REAL array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.

*S*S is REAL array, dimension (N1 + 1)*K Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.

*INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Modified by Francoise Tisseur, University of Tennessee

Definition at line **175** of file **slaed3.f**.

## Author

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