.TH "laed4" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
laed4 \- laed4: D&C step: secular equation nonlinear solver
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBdlaed4\fP (n, i, d, z, delta, rho, dlam, info)"
.br
.RI "\fBDLAED4\fP used by DSTEDC\&. Finds a single root of the secular equation\&. "
.ti -1c
.RI "subroutine \fBslaed4\fP (n, i, d, z, delta, rho, dlam, info)"
.br
.RI "\fBSLAED4\fP used by SSTEDC\&. Finds a single root of the secular equation\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "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)"

.PP
\fBDLAED4\fP used by DSTEDC\&. Finds a single root of the secular equation\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>         The length of all arrays\&.
!> 
.fi
.PP
.br
\fII\fP 
.PP
.nf
!>          I is INTEGER
!>         The index of the eigenvalue to be computed\&.  1 <= I <= N\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is DOUBLE PRECISION array, dimension (N)
!>         The components of the updating vector\&.
!> 
.fi
.PP
.br
\fIDELTA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
!>          RHO is DOUBLE PRECISION
!>         The scalar in the symmetric updating formula\&.
!> 
.fi
.PP
.br
\fIDLAM\fP 
.PP
.nf
!>          DLAM is DOUBLE PRECISION
!>         The computed lambda_I, the I-th updated eigenvalue\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>         = 0:  successful exit
!>         > 0:  if INFO = 1, the updating process failed\&.
!> 
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
!>  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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP

.PP
Definition at line \fB144\fP of file \fBdlaed4\&.f\fP\&.
.SS "subroutine slaed4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real dlam, integer info)"

.PP
\fBSLAED4\fP used by SSTEDC\&. Finds a single root of the secular equation\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>         The length of all arrays\&.
!> 
.fi
.PP
.br
\fII\fP 
.PP
.nf
!>          I is INTEGER
!>         The index of the eigenvalue to be computed\&.  1 <= I <= N\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          D is REAL array, dimension (N)
!>         The original eigenvalues\&.  It is assumed that they are in
!>         order, D(I) < D(J)  for I < J\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          Z is REAL array, dimension (N)
!>         The components of the updating vector\&.
!> 
.fi
.PP
.br
\fIDELTA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
!>          RHO is REAL
!>         The scalar in the symmetric updating formula\&.
!> 
.fi
.PP
.br
\fIDLAM\fP 
.PP
.nf
!>          DLAM is REAL
!>         The computed lambda_I, the I-th updated eigenvalue\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>         = 0:  successful exit
!>         > 0:  if INFO = 1, the updating process failed\&.
!> 
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
!>  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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP

.PP
Definition at line \fB144\fP of file \fBslaed4\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.