1DLAGV2(1)           LAPACK auxiliary routine (version 3.2)           DLAGV2(1)
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NAME

6       DLAGV2  - computes the Generalized Schur factorization of a real 2-by-2
7       matrix 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
24       1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
25          types), then
26          [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
27          [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
28          [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
29          [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],  2)  if  the
30       pencil (A,B) has a pair of complex conjugate eigenvalues,
31          then
32          [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
33          [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
34          [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
35          [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]
36          where b11 >= b22 > 0.
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ARGUMENTS

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

72       Based on contributions by
73          Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
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77 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008                       DLAGV2(1)
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