laed4(3)

NAME
laed4 - 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

  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

Author

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