.TH "laed9" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
laed9 \- laed9: D&C step: secular equation
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBdlaed9\fP (k, kstart, kstop, n, d, q, ldq, rho, dlambda, w, s, lds, info)"
.br
.RI "\fBDLAED9\fP used by DSTEDC\&. Finds the roots of the secular equation and updates the eigenvectors\&. Used when the original matrix is dense\&. "
.ti -1c
.RI "subroutine \fBslaed9\fP (k, kstart, kstop, n, d, q, ldq, rho, dlambda, w, s, lds, info)"
.br
.RI "\fBSLAED9\fP used by SSTEDC\&. Finds the roots of the secular equation and updates the eigenvectors\&. Used when the original matrix is dense\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine dlaed9 (integer k, integer kstart, integer kstop, integer n, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, double precision rho, double precision, dimension( * ) dlambda, double precision, dimension( * ) w, double precision, dimension( lds, * ) s, integer lds, integer info)"

.PP
\fBDLAED9\fP used by DSTEDC\&. Finds the roots of the secular equation and updates the eigenvectors\&. Used when the original matrix is dense\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DLAED9 finds the roots of the secular equation, as defined by the
 values in D, Z, and RHO, between KSTART and KSTOP\&.  It makes the
 appropriate calls to DLAED4 and then stores the new matrix of
 eigenvectors for use in calculating the next level of Z vectors\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIK\fP 
.PP
.nf
          K is INTEGER
          The number of terms in the rational function to be solved by
          DLAED4\&.  K >= 0\&.
.fi
.PP
.br
\fIKSTART\fP 
.PP
.nf
          KSTART is INTEGER
.fi
.PP
.br
\fIKSTOP\fP 
.PP
.nf
          KSTOP is INTEGER
          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
          are to be computed\&.  1 <= KSTART <= KSTOP <= K\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of rows and columns in the Q matrix\&.
          N >= K (delation may result in N > K)\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is DOUBLE PRECISION array, dimension (N)
          D(I) contains the updated eigenvalues
          for KSTART <= I <= KSTOP\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is DOUBLE PRECISION array, dimension (LDQ,N)
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.  LDQ >= max( 1, N )\&.
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
          RHO is DOUBLE PRECISION
          The value of the parameter in the rank one update equation\&.
          RHO >= 0 required\&.
.fi
.PP
.br
\fIDLAMBDA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector\&.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          S is DOUBLE PRECISION array, dimension (LDS, K)
          Will contain the eigenvectors of the repaired matrix which
          will be stored for subsequent Z vector calculation and
          multiplied by the previously accumulated eigenvectors
          to update the system\&.
.fi
.PP
.br
\fILDS\fP 
.PP
.nf
          LDS is INTEGER
          The leading dimension of S\&.  LDS >= max( 1, K )\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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
.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
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP

.SS "subroutine slaed9 (integer k, integer kstart, integer kstop, integer n, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, real rho, real, dimension( * ) dlambda, real, dimension( * ) w, real, dimension( lds, * ) s, integer lds, integer info)"

.PP
\fBSLAED9\fP used by SSTEDC\&. Finds the roots of the secular equation and updates the eigenvectors\&. Used when the original matrix is dense\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SLAED9 finds the roots of the secular equation, as defined by the
 values in D, Z, and RHO, between KSTART and KSTOP\&.  It makes the
 appropriate calls to SLAED4 and then stores the new matrix of
 eigenvectors for use in calculating the next level of Z vectors\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIK\fP 
.PP
.nf
          K is INTEGER
          The number of terms in the rational function to be solved by
          SLAED4\&.  K >= 0\&.
.fi
.PP
.br
\fIKSTART\fP 
.PP
.nf
          KSTART is INTEGER
.fi
.PP
.br
\fIKSTOP\fP 
.PP
.nf
          KSTOP is INTEGER
          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
          are to be computed\&.  1 <= KSTART <= KSTOP <= K\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of rows and columns in the Q matrix\&.
          N >= K (delation may result in N > K)\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL array, dimension (N)
          D(I) contains the updated eigenvalues
          for KSTART <= I <= KSTOP\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is REAL array, dimension (LDQ,N)
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.  LDQ >= max( 1, N )\&.
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
          RHO is REAL
          The value of the parameter in the rank one update equation\&.
          RHO >= 0 required\&.
.fi
.PP
.br
\fIDLAMBDA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is REAL array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector\&.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          S is REAL array, dimension (LDS, K)
          Will contain the eigenvectors of the repaired matrix which
          will be stored for subsequent Z vector calculation and
          multiplied by the previously accumulated eigenvectors
          to update the system\&.
.fi
.PP
.br
\fILDS\fP 
.PP
.nf
          LDS is INTEGER
          The leading dimension of S\&.  LDS >= max( 1, K )\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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
.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
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP

.SH "Author"
.PP 
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