lasd4(3)

NAME
lasd4 - lasd4: D&C step: secular equation nonlinear solver

SYNOPSIS

  Functions

subroutine dlasd4 (n, i, d, z, delta, rho, sigma, work, info)
DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.
subroutine slasd4 (n, i, d, z, delta, rho, sigma, work, info)
SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.

Detailed Description

Function Documentation

  subroutine dlasd4 (integer n, integer i, double precision, dimension( * ) d, double precision, dimension( * ) z, double precision, dimension( * ) delta, double precision rho, double precision sigma, double precision, dimension( * ) work, integer info)

DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.

Purpose:

    !>
    !> This subroutine computes the square root of the I-th updated
    !> eigenvalue of a positive symmetric rank-one modification to
    !> a positive diagonal matrix whose entries are given as the squares
    !> of the corresponding entries in the array d, and that
    !>
    !>        0 <= 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 ) * 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, 0 <= 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 .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
    !>         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
    !>         contains the information necessary to construct the
    !>         (singular) eigenvectors.
    !>

    RHO

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

    SIGMA

    !>          SIGMA is DOUBLE PRECISION
    !>         The computed sigma_I, the I-th updated eigenvalue.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension ( N )
    !>         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
    !>         component.  If N = 1, then WORK( 1 ) = 1.
    !>

    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 slasd4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real sigma, real, dimension( * ) work, integer info)

SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.

Purpose:

    !>
    !> This subroutine computes the square root of the I-th updated
    !> eigenvalue of a positive symmetric rank-one modification to
    !> a positive diagonal matrix whose entries are given as the squares
    !> of the corresponding entries in the array d, and that
    !>
    !>        0 <= 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 ) * 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, 0 <= 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 .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
    !>         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
    !>         contains the information necessary to construct the
    !>         (singular) eigenvectors.
    !>

    RHO

    !>          RHO is REAL
    !>         The scalar in the symmetric updating formula.
    !>

    SIGMA

    !>          SIGMA is REAL
    !>         The computed sigma_I, the I-th updated eigenvalue.
    !>

    WORK

    !>          WORK is REAL array, dimension ( N )
    !>         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
    !>         component.  If N = 1, then WORK( 1 ) = 1.
    !>

    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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