.TH "laic1" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
laic1 \- laic1: condition estimate, step in gelsy
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBclaic1\fP (job, j, x, sest, w, gamma, sestpr, s, c)"
.br
.RI "\fBCLAIC1\fP applies one step of incremental condition estimation\&. "
.ti -1c
.RI "subroutine \fBdlaic1\fP (job, j, x, sest, w, gamma, sestpr, s, c)"
.br
.RI "\fBDLAIC1\fP applies one step of incremental condition estimation\&. "
.ti -1c
.RI "subroutine \fBslaic1\fP (job, j, x, sest, w, gamma, sestpr, s, c)"
.br
.RI "\fBSLAIC1\fP applies one step of incremental condition estimation\&. "
.ti -1c
.RI "subroutine \fBzlaic1\fP (job, j, x, sest, w, gamma, sestpr, s, c)"
.br
.RI "\fBZLAIC1\fP applies one step of incremental condition estimation\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine claic1 (integer job, integer j, complex, dimension( j ) x, real sest, complex, dimension( j ) w, complex gamma, real sestpr, complex s, complex c)"

.PP
\fBCLAIC1\fP applies one step of incremental condition estimation\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CLAIC1 applies one step of incremental condition estimation in
 its simplest version:

 Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
 lower triangular matrix L, such that
          twonorm(L*x) = sest
 Then CLAIC1 computes sestpr, s, c such that
 the vector
                 [ s*x ]
          xhat = [  c  ]
 is an approximate singular vector of
                 [ L      0  ]
          Lhat = [ w**H gamma ]
 in the sense that
          twonorm(Lhat*xhat) = sestpr\&.

 Depending on JOB, an estimate for the largest or smallest singular
 value is computed\&.

 Note that [s c]**H and sestpr**2 is an eigenpair of the system

     diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
                                           [ conjg(gamma) ]

 where  alpha =  x**H*w\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is INTEGER
          = 1: an estimate for the largest singular value is computed\&.
          = 2: an estimate for the smallest singular value is computed\&.
.fi
.PP
.br
\fIJ\fP 
.PP
.nf
          J is INTEGER
          Length of X and W
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is COMPLEX array, dimension (J)
          The j-vector x\&.
.fi
.PP
.br
\fISEST\fP 
.PP
.nf
          SEST is REAL
          Estimated singular value of j by j matrix L
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is COMPLEX array, dimension (J)
          The j-vector w\&.
.fi
.PP
.br
\fIGAMMA\fP 
.PP
.nf
          GAMMA is COMPLEX
          The diagonal element gamma\&.
.fi
.PP
.br
\fISESTPR\fP 
.PP
.nf
          SESTPR is REAL
          Estimated singular value of (j+1) by (j+1) matrix Lhat\&.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          S is COMPLEX
          Sine needed in forming xhat\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is COMPLEX
          Cosine needed in forming xhat\&.
.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

.SS "subroutine dlaic1 (integer job, integer j, double precision, dimension( j ) x, double precision sest, double precision, dimension( j ) w, double precision gamma, double precision sestpr, double precision s, double precision c)"

.PP
\fBDLAIC1\fP applies one step of incremental condition estimation\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DLAIC1 applies one step of incremental condition estimation in
 its simplest version:

 Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
 lower triangular matrix L, such that
          twonorm(L*x) = sest
 Then DLAIC1 computes sestpr, s, c such that
 the vector
                 [ s*x ]
          xhat = [  c  ]
 is an approximate singular vector of
                 [ L       0  ]
          Lhat = [ w**T gamma ]
 in the sense that
          twonorm(Lhat*xhat) = sestpr\&.

 Depending on JOB, an estimate for the largest or smallest singular
 value is computed\&.

 Note that [s c]**T and sestpr**2 is an eigenpair of the system

     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
                                           [ gamma ]

 where  alpha =  x**T*w\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is INTEGER
          = 1: an estimate for the largest singular value is computed\&.
          = 2: an estimate for the smallest singular value is computed\&.
.fi
.PP
.br
\fIJ\fP 
.PP
.nf
          J is INTEGER
          Length of X and W
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is DOUBLE PRECISION array, dimension (J)
          The j-vector x\&.
.fi
.PP
.br
\fISEST\fP 
.PP
.nf
          SEST is DOUBLE PRECISION
          Estimated singular value of j by j matrix L
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is DOUBLE PRECISION array, dimension (J)
          The j-vector w\&.
.fi
.PP
.br
\fIGAMMA\fP 
.PP
.nf
          GAMMA is DOUBLE PRECISION
          The diagonal element gamma\&.
.fi
.PP
.br
\fISESTPR\fP 
.PP
.nf
          SESTPR is DOUBLE PRECISION
          Estimated singular value of (j+1) by (j+1) matrix Lhat\&.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          S is DOUBLE PRECISION
          Sine needed in forming xhat\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is DOUBLE PRECISION
          Cosine needed in forming xhat\&.
.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

.SS "subroutine slaic1 (integer job, integer j, real, dimension( j ) x, real sest, real, dimension( j ) w, real gamma, real sestpr, real s, real c)"

.PP
\fBSLAIC1\fP applies one step of incremental condition estimation\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SLAIC1 applies one step of incremental condition estimation in
 its simplest version:

 Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
 lower triangular matrix L, such that
          twonorm(L*x) = sest
 Then SLAIC1 computes sestpr, s, c such that
 the vector
                 [ s*x ]
          xhat = [  c  ]
 is an approximate singular vector of
                 [ L      0  ]
          Lhat = [ w**T gamma ]
 in the sense that
          twonorm(Lhat*xhat) = sestpr\&.

 Depending on JOB, an estimate for the largest or smallest singular
 value is computed\&.

 Note that [s c]**T and sestpr**2 is an eigenpair of the system

     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
                                           [ gamma ]

 where  alpha =  x**T*w\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is INTEGER
          = 1: an estimate for the largest singular value is computed\&.
          = 2: an estimate for the smallest singular value is computed\&.
.fi
.PP
.br
\fIJ\fP 
.PP
.nf
          J is INTEGER
          Length of X and W
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is REAL array, dimension (J)
          The j-vector x\&.
.fi
.PP
.br
\fISEST\fP 
.PP
.nf
          SEST is REAL
          Estimated singular value of j by j matrix L
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is REAL array, dimension (J)
          The j-vector w\&.
.fi
.PP
.br
\fIGAMMA\fP 
.PP
.nf
          GAMMA is REAL
          The diagonal element gamma\&.
.fi
.PP
.br
\fISESTPR\fP 
.PP
.nf
          SESTPR is REAL
          Estimated singular value of (j+1) by (j+1) matrix Lhat\&.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          S is REAL
          Sine needed in forming xhat\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is REAL
          Cosine needed in forming xhat\&.
.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

.SS "subroutine zlaic1 (integer job, integer j, complex*16, dimension( j ) x, double precision sest, complex*16, dimension( j ) w, complex*16 gamma, double precision sestpr, complex*16 s, complex*16 c)"

.PP
\fBZLAIC1\fP applies one step of incremental condition estimation\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZLAIC1 applies one step of incremental condition estimation in
 its simplest version:

 Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
 lower triangular matrix L, such that
          twonorm(L*x) = sest
 Then ZLAIC1 computes sestpr, s, c such that
 the vector
                 [ s*x ]
          xhat = [  c  ]
 is an approximate singular vector of
                 [ L       0  ]
          Lhat = [ w**H gamma ]
 in the sense that
          twonorm(Lhat*xhat) = sestpr\&.

 Depending on JOB, an estimate for the largest or smallest singular
 value is computed\&.

 Note that [s c]**H and sestpr**2 is an eigenpair of the system

     diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
                                           [ conjg(gamma) ]

 where  alpha =  x**H * w\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is INTEGER
          = 1: an estimate for the largest singular value is computed\&.
          = 2: an estimate for the smallest singular value is computed\&.
.fi
.PP
.br
\fIJ\fP 
.PP
.nf
          J is INTEGER
          Length of X and W
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is COMPLEX*16 array, dimension (J)
          The j-vector x\&.
.fi
.PP
.br
\fISEST\fP 
.PP
.nf
          SEST is DOUBLE PRECISION
          Estimated singular value of j by j matrix L
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is COMPLEX*16 array, dimension (J)
          The j-vector w\&.
.fi
.PP
.br
\fIGAMMA\fP 
.PP
.nf
          GAMMA is COMPLEX*16
          The diagonal element gamma\&.
.fi
.PP
.br
\fISESTPR\fP 
.PP
.nf
          SESTPR is DOUBLE PRECISION
          Estimated singular value of (j+1) by (j+1) matrix Lhat\&.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          S is COMPLEX*16
          Sine needed in forming xhat\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is COMPLEX*16
          Cosine needed in forming xhat\&.
.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

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.