.TH "TESTING/EIG/sdrgsx.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
TESTING/EIG/sdrgsx.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBsdrgsx\fP (nsize, ncmax, thresh, nin, nout, a, lda, b, ai, bi, z, q, alphar, alphai, beta, c, ldc, s, work, lwork, iwork, liwork, bwork, info)"
.br
.RI "\fBSDRGSX\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine sdrgsx (integer nsize, integer ncmax, real thresh, integer nin, integer nout, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) ai, real, dimension( lda, * ) bi, real, dimension( lda, * ) z, real, dimension( lda, * ) q, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) s, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)"

.PP
\fBSDRGSX\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SDRGSX checks the nonsymmetric generalized eigenvalue (Schur form)
 problem expert driver SGGESX\&.

 SGGESX factors A and B as Q S Z' and Q T Z', where ' means
 transpose, T is upper triangular, S is in generalized Schur form
 (block upper triangular, with 1x1 and 2x2 blocks on the diagonal,
 the 2x2 blocks corresponding to complex conjugate pairs of
 generalized eigenvalues), and Q and Z are orthogonal\&.  It also
 computes the generalized eigenvalues (alpha(1),beta(1)), \&.\&.\&.,
 (alpha(n),beta(n))\&. Thus, w(j) = alpha(j)/beta(j) is a root of the
 characteristic equation

     det( A - w(j) B ) = 0

 Optionally it also reorders the eigenvalues so that a selected
 cluster of eigenvalues appears in the leading diagonal block of the
 Schur forms; computes a reciprocal condition number for the average
 of the selected eigenvalues; and computes a reciprocal condition
 number for the right and left deflating subspaces corresponding to
 the selected eigenvalues\&.

 When SDRGSX is called with NSIZE > 0, five (5) types of built-in
 matrix pairs are used to test the routine SGGESX\&.

 When SDRGSX is called with NSIZE = 0, it reads in test matrix data
 to test SGGESX\&.

 For each matrix pair, the following tests will be performed and
 compared with the threshold THRESH except for the tests (7) and (9):

 (1)   | A - Q S Z' | / ( |A| n ulp )

 (2)   | B - Q T Z' | / ( |B| n ulp )

 (3)   | I - QQ' | / ( n ulp )

 (4)   | I - ZZ' | / ( n ulp )

 (5)   if A is in Schur form (i\&.e\&. quasi-triangular form)

 (6)   maximum over j of D(j)  where:

       if alpha(j) is real:
                     |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
           D(j) = ------------------------ + -----------------------
                  max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

       if alpha(j) is complex:
                                 | det( s S - w T ) |
           D(j) = ---------------------------------------------------
                  ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )

           and S and T are here the 2 x 2 diagonal blocks of S and T
           corresponding to the j-th and j+1-th eigenvalues\&.

 (7)   if sorting worked and SDIM is the number of eigenvalues
       which were selected\&.

 (8)   the estimated value DIF does not differ from the true values of
       Difu and Difl more than a factor 10*THRESH\&. If the estimate DIF
       equals zero the corresponding true values of Difu and Difl
       should be less than EPS*norm(A, B)\&. If the true value of Difu
       and Difl equal zero, the estimate DIF should be less than
       EPS*norm(A, B)\&.

 (9)   If INFO = N+3 is returned by SGGESX, the reordering 'failed'
       and we check that DIF = PL = PR = 0 and that the true value of
       Difu and Difl is < EPS*norm(A, B)\&. We count the events when
       INFO=N+3\&.

 For read-in test matrices, the above tests are run except that the
 exact value for DIF (and PL) is input data\&.  Additionally, there is
 one more test run for read-in test matrices:

 (10)  the estimated value PL does not differ from the true value of
       PLTRU more than a factor THRESH\&. If the estimate PL equals
       zero the corresponding true value of PLTRU should be less than
       EPS*norm(A, B)\&. If the true value of PLTRU equal zero, the
       estimate PL should be less than EPS*norm(A, B)\&.

 Note that for the built-in tests, a total of 10*NSIZE*(NSIZE-1)
 matrix pairs are generated and tested\&. NSIZE should be kept small\&.

 SVD (routine SGESVD) is used for computing the true value of DIF_u
 and DIF_l when testing the built-in test problems\&.

 Built-in Test Matrices
 ======================

 All built-in test matrices are the 2 by 2 block of triangular
 matrices

          A = [ A11 A12 ]    and      B = [ B11 B12 ]
              [     A22 ]                 [     B22 ]

 where for different type of A11 and A22 are given as the following\&.
 A12 and B12 are chosen so that the generalized Sylvester equation

          A11*R - L*A22 = -A12
          B11*R - L*B22 = -B12

 have prescribed solution R and L\&.

 Type 1:  A11 = J_m(1,-1) and A_22 = J_k(1-a,1)\&.
          B11 = I_m, B22 = I_k
          where J_k(a,b) is the k-by-k Jordan block with ``a'' on
          diagonal and ``b'' on superdiagonal\&.

 Type 2:  A11 = (a_ij) = ( 2(\&.5-sin(i)) ) and
          B11 = (b_ij) = ( 2(\&.5-sin(ij)) ) for i=1,\&.\&.\&.,m, j=i,\&.\&.\&.,m
          A22 = (a_ij) = ( 2(\&.5-sin(i+j)) ) and
          B22 = (b_ij) = ( 2(\&.5-sin(ij)) ) for i=m+1,\&.\&.\&.,k, j=i,\&.\&.\&.,k

 Type 3:  A11, A22 and B11, B22 are chosen as for Type 2, but each
          second diagonal block in A_11 and each third diagonal block
          in A_22 are made as 2 by 2 blocks\&.

 Type 4:  A11 = ( 20(\&.5 - sin(ij)) ) and B22 = ( 2(\&.5 - sin(i+j)) )
             for i=1,\&.\&.\&.,m,  j=1,\&.\&.\&.,m and
          A22 = ( 20(\&.5 - sin(i+j)) ) and B22 = ( 2(\&.5 - sin(ij)) )
             for i=m+1,\&.\&.\&.,k,  j=m+1,\&.\&.\&.,k

 Type 5:  (A,B) and have potentially close or common eigenvalues and
          very large departure from block diagonality A_11 is chosen
          as the m x m leading submatrix of A_1:
                  |  1  b                            |
                  | -b  1                            |
                  |        1+d  b                    |
                  |         -b 1+d                   |
           A_1 =  |                  d  1            |
                  |                 -1  d            |
                  |                        -d  1     |
                  |                        -1 -d     |
                  |                               1  |
          and A_22 is chosen as the k x k leading submatrix of A_2:
                  | -1  b                            |
                  | -b -1                            |
                  |       1-d  b                     |
                  |       -b  1-d                    |
           A_2 =  |                 d 1+b            |
                  |               -1-b d             |
                  |                       -d  1+b    |
                  |                      -1+b  -d    |
                  |                              1-d |
          and matrix B are chosen as identity matrices (see SLATM5)\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fINSIZE\fP 
.PP
.nf
          NSIZE is INTEGER
          The maximum size of the matrices to use\&. NSIZE >= 0\&.
          If NSIZE = 0, no built-in tests matrices are used, but
          read-in test matrices are used to test SGGESX\&.
.fi
.PP
.br
\fINCMAX\fP 
.PP
.nf
          NCMAX is INTEGER
          Maximum allowable NMAX for generating Kroneker matrix
          in call to SLAKF2
.fi
.PP
.br
\fITHRESH\fP 
.PP
.nf
          THRESH is REAL
          A test will count as 'failed' if the 'error', computed as
          described above, exceeds THRESH\&.  Note that the error
          is scaled to be O(1), so THRESH should be a reasonably
          small multiple of 1, e\&.g\&., 10 or 100\&.  In particular,
          it should not depend on the precision (single vs\&. double)
          or the size of the matrix\&.  THRESH >= 0\&.
.fi
.PP
.br
\fININ\fP 
.PP
.nf
          NIN is INTEGER
          The FORTRAN unit number for reading in the data file of
          problems to solve\&.
.fi
.PP
.br
\fINOUT\fP 
.PP
.nf
          NOUT is INTEGER
          The FORTRAN unit number for printing out error messages
          (e\&.g\&., if a routine returns IINFO not equal to 0\&.)
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA, NSIZE)
          Used to store the matrix whose eigenvalues are to be
          computed\&.  On exit, A contains the last matrix actually used\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of A, B, AI, BI, Z and Q,
          LDA >= max( 1, NSIZE )\&. For the read-in test,
          LDA >= max( 1, N ), N is the size of the test matrices\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is REAL array, dimension (LDA, NSIZE)
          Used to store the matrix whose eigenvalues are to be
          computed\&.  On exit, B contains the last matrix actually used\&.
.fi
.PP
.br
\fIAI\fP 
.PP
.nf
          AI is REAL array, dimension (LDA, NSIZE)
          Copy of A, modified by SGGESX\&.
.fi
.PP
.br
\fIBI\fP 
.PP
.nf
          BI is REAL array, dimension (LDA, NSIZE)
          Copy of B, modified by SGGESX\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is REAL array, dimension (LDA, NSIZE)
          Z holds the left Schur vectors computed by SGGESX\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is REAL array, dimension (LDA, NSIZE)
          Q holds the right Schur vectors computed by SGGESX\&.
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
          ALPHAR is REAL array, dimension (NSIZE)
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          ALPHAI is REAL array, dimension (NSIZE)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is REAL array, dimension (NSIZE)
 \\verbatim
          On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is REAL array, dimension (LDC, LDC)
          Store the matrix generated by subroutine SLAKF2, this is the
          matrix formed by Kronecker products used for estimating
          DIF\&.
.fi
.PP
.br
\fILDC\fP 
.PP
.nf
          LDC is INTEGER
          The leading dimension of C\&. LDC >= max(1, LDA*LDA/2 )\&.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          S is REAL array, dimension (LDC)
          Singular values of C
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (LWORK)
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.
          LWORK >= MAX( 5*NSIZE*NSIZE/2 - 2, 10*(NSIZE+1) )
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (LIWORK)
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
          LIWORK is INTEGER
          The dimension of the array IWORK\&. LIWORK >= NSIZE + 6\&.
.fi
.PP
.br
\fIBWORK\fP 
.PP
.nf
          BWORK is LOGICAL array, dimension (LDA)
.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:  A routine returned an error code\&.
.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

.PP
Definition at line \fB356\fP of file \fBsdrgsx\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.