# laed4 - Man Page

laed4: D&C step: secular equation nonlinear solver

## Synopsis

### Functions

subroutine **dlaed4** (n, i, d, z, delta, rho, dlam, info)**DLAED4** used by DSTEDC. Finds a single root of the secular equation.

subroutine **slaed4** (n, i, d, z, delta, rho, dlam, info)**SLAED4** used by SSTEDC. Finds a single root of the secular equation.

## Detailed Description

## Function Documentation

### subroutine dlaed4 (integer n, integer i, double precision, dimension( * ) d, double precision, dimension( * ) z, double precision, dimension( * ) delta, double precision rho, double precision dlam, integer info)

**DLAED4** used by DSTEDC. Finds a single root of the secular equation.

**Purpose:**

This subroutine computes the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.

**Parameters***N*N is INTEGER The length of all arrays.

*I*I is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N.

*D*D is DOUBLE PRECISION array, dimension (N) The original eigenvalues. It is assumed that they are in order, D(I) < D(J) for I < J.

*Z*Z is DOUBLE PRECISION array, dimension (N) The components of the updating vector.

*DELTA*DELTA is DOUBLE PRECISION array, dimension (N) If N > 2, DELTA contains (D(j) - lambda_I) in its j-th component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5 for detail. The vector DELTA contains the information necessary to construct the eigenvectors by DLAED3 and DLAED9.

*RHO*RHO is DOUBLE PRECISION The scalar in the symmetric updating formula.

*DLAM*DLAM is DOUBLE PRECISION The computed lambda_I, the I-th updated eigenvalue.

*INFO*INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed.

**Internal Parameters:**

Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line **144** of file **dlaed4.f**.

### subroutine slaed4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real dlam, integer info)

**SLAED4** used by SSTEDC. Finds a single root of the secular equation.

**Purpose:**

This subroutine computes the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.

**Parameters***N*N is INTEGER The length of all arrays.

*I*I is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N.

*D*D is REAL array, dimension (N) The original eigenvalues. It is assumed that they are in order, D(I) < D(J) for I < J.

*Z*Z is REAL array, dimension (N) The components of the updating vector.

*DELTA*DELTA is REAL array, dimension (N) If N > 2, DELTA contains (D(j) - lambda_I) in its j-th component. If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5 for detail. The vector DELTA contains the information necessary to construct the eigenvectors by SLAED3 and SLAED9.

*RHO*RHO is REAL The scalar in the symmetric updating formula.

*DLAM*DLAM is REAL The computed lambda_I, the I-th updated eigenvalue.

*INFO*INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed.

**Internal Parameters:**

Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line **144** of file **slaed4.f**.

## Author

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