1DLAGV2(1)           LAPACK auxiliary routine (version 3.1)           DLAGV2(1)
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NAME

6       DLAGV2  -  the  Generalized Schur factorization of a real 2-by-2 matrix
7       pencil (A,B) where B is upper triangular
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SYNOPSIS

10       SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR,
11                          SNR )
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13           INTEGER        LDA, LDB
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15           DOUBLE         PRECISION CSL, CSR, SNL, SNR
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17           DOUBLE         PRECISION  A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ), B(
18                          LDB, * ), BETA( 2 )
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PURPOSE

21       DLAGV2 computes the Generalized Schur factorization of  a  real  2-by-2
22       matrix  pencil (A,B) where B is upper triangular. This routine computes
23       orthogonal (rotation) matrices given by CSL, SNL and CSR, SNR such that
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25       1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
26          types), then
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28          [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
29          [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
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31          [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
32          [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],
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34       2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
35          then
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37          [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
38          [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
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40          [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
41          [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]
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43          where b11 >= b22 > 0.
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ARGUMENTS

48       A       (input/output) DOUBLE PRECISION array, dimension (LDA, 2)
49               On entry, the 2 x 2 matrix A.  On exit, A is overwritten by the
50               ``A-part'' of the generalized Schur form.
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52       LDA     (input) INTEGER
53               THe leading dimension of the array A.  LDA >= 2.
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55       B       (input/output) DOUBLE PRECISION array, dimension (LDB, 2)
56               On  entry,  the upper triangular 2 x 2 matrix B.  On exit, B is
57               overwritten by the ``B-part'' of the generalized Schur form.
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59       LDB     (input) INTEGER
60               THe leading dimension of the array B.  LDB >= 2.
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62       ALPHAR  (output) DOUBLE PRECISION array, dimension (2)
63               ALPHAI  (output) DOUBLE PRECISION  array,  dimension  (2)  BETA
64               (output)     DOUBLE     PRECISION    array,    dimension    (2)
65               (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the pen‐
66               cil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may be zero.
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68       CSL     (output) DOUBLE PRECISION
69               The cosine of the left rotation matrix.
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71       SNL     (output) DOUBLE PRECISION
72               The sine of the left rotation matrix.
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74       CSR     (output) DOUBLE PRECISION
75               The cosine of the right rotation matrix.
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77       SNR     (output) DOUBLE PRECISION
78               The sine of the right rotation matrix.
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FURTHER DETAILS

81       Based on contributions by
82          Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
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87 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       DLAGV2(1)
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