.TH "lagv2" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
lagv2 \- lagv2: 2x2 generalized Schur factor
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBdlagv2\fP (a, lda, b, ldb, alphar, alphai, beta, csl, snl, csr, snr)"
.br
.RI "\fBDLAGV2\fP computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular\&. "
.ti -1c
.RI "subroutine \fBslagv2\fP (a, lda, b, ldb, alphar, alphai, beta, csl, snl, csr, snr)"
.br
.RI "\fBSLAGV2\fP computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine dlagv2 (double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( 2 ) alphar, double precision, dimension( 2 ) alphai, double precision, dimension( 2 ) beta, double precision csl, double precision snl, double precision csr, double precision snr)"

.PP
\fBDLAGV2\fP computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
 matrix pencil (A,B) where B is upper triangular\&. This routine
 computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
 SNR such that

 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
    types), then

    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
    [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

    [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],

 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
    then

    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
    [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

    [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]

    where b11 >= b22 > 0\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A\&.
          On exit, A is overwritten by the ``A-part'' of the
          generalized Schur form\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          THe leading dimension of the array A\&.  LDA >= 2\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is DOUBLE PRECISION array, dimension (LDB, 2)
          On entry, the upper triangular 2 x 2 matrix B\&.
          On exit, B is overwritten by the ``B-part'' of the
          generalized Schur form\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          THe leading dimension of the array B\&.  LDB >= 2\&.
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
          ALPHAR is DOUBLE PRECISION array, dimension (2)
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          ALPHAI is DOUBLE PRECISION array, dimension (2)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is DOUBLE PRECISION array, dimension (2)
          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
          pencil (A,B), k=1,2, i = sqrt(-1)\&.  Note that BETA(k) may
          be zero\&.
.fi
.PP
.br
\fICSL\fP 
.PP
.nf
          CSL is DOUBLE PRECISION
          The cosine of the left rotation matrix\&.
.fi
.PP
.br
\fISNL\fP 
.PP
.nf
          SNL is DOUBLE PRECISION
          The sine of the left rotation matrix\&.
.fi
.PP
.br
\fICSR\fP 
.PP
.nf
          CSR is DOUBLE PRECISION
          The cosine of the right rotation matrix\&.
.fi
.PP
.br
\fISNR\fP 
.PP
.nf
          SNR is DOUBLE PRECISION
          The sine of the right rotation matrix\&.
.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
Mark Fahey, Department of Mathematics, Univ\&. of Kentucky, USA 
.RE
.PP

.SS "subroutine slagv2 (real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( 2 ) alphar, real, dimension( 2 ) alphai, real, dimension( 2 ) beta, real csl, real snl, real csr, real snr)"

.PP
\fBSLAGV2\fP computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SLAGV2 computes the Generalized Schur factorization of a real 2-by-2
 matrix pencil (A,B) where B is upper triangular\&. This routine
 computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
 SNR such that

 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
    types), then

    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
    [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

    [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],

 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
    then

    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
    [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

    [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]

    where b11 >= b22 > 0\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A\&.
          On exit, A is overwritten by the ``A-part'' of the
          generalized Schur form\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          THe leading dimension of the array A\&.  LDA >= 2\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is REAL array, dimension (LDB, 2)
          On entry, the upper triangular 2 x 2 matrix B\&.
          On exit, B is overwritten by the ``B-part'' of the
          generalized Schur form\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          THe leading dimension of the array B\&.  LDB >= 2\&.
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
          ALPHAR is REAL array, dimension (2)
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          ALPHAI is REAL array, dimension (2)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is REAL array, dimension (2)
          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
          pencil (A,B), k=1,2, i = sqrt(-1)\&.  Note that BETA(k) may
          be zero\&.
.fi
.PP
.br
\fICSL\fP 
.PP
.nf
          CSL is REAL
          The cosine of the left rotation matrix\&.
.fi
.PP
.br
\fISNL\fP 
.PP
.nf
          SNL is REAL
          The sine of the left rotation matrix\&.
.fi
.PP
.br
\fICSR\fP 
.PP
.nf
          CSR is REAL
          The cosine of the right rotation matrix\&.
.fi
.PP
.br
\fISNR\fP 
.PP
.nf
          SNR is REAL
          The sine of the right rotation matrix\&.
.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
Mark Fahey, Department of Mathematics, Univ\&. of Kentucky, USA 
.RE
.PP

.SH "Author"
.PP 
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