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

.in +1c
.ti -1c
.RI "subroutine \fBzdrgsx\fP (nsize, ncmax, thresh, nin, nout, a, lda, b, ai, bi, z, q, alpha, beta, c, ldc, s, work, lwork, rwork, iwork, liwork, bwork, info)"
.br
.RI "\fBZDRGSX\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine zdrgsx (integer nsize, integer ncmax, double precision thresh, integer nin, integer nout, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( lda, * ) b, complex*16, dimension( lda, * ) ai, complex*16, dimension( lda, * ) bi, complex*16, dimension( lda, * ) z, complex*16, dimension( lda, * ) q, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) s, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)"

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

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

 ZGGES factors A and B as Q*S*Z'  and Q*T*Z' , where ' means conjugate
 transpose, S and T are  upper triangular (i\&.e\&., in generalized Schur
 form), and Q and Z are unitary\&. It also computes the generalized
 eigenvalues (alpha(j),beta(j)), j=1,\&.\&.\&.,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 ZDRGSX is called with NSIZE > 0, five (5) types of built-in
 matrix pairs are used to test the routine ZGGESX\&.

 When ZDRGSX is called with NSIZE = 0, it reads in test matrix data
 to test ZGGESX\&.
 (need more details on what kind of read-in data are needed)\&.

 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\&. triangular form)

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

                     |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
           D(j) = ------------------------ + -----------------------
                  max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

 (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 ZGGESX, 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 same 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 ZGESVD) 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 DLATM5)\&.
.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 DGGESX\&.
.fi
.PP
.br
\fINCMAX\fP 
.PP
.nf
          NCMAX is INTEGER
          Maximum allowable NMAX for generating Kroneker matrix
          in call to ZLAKF2
.fi
.PP
.br
\fITHRESH\fP 
.PP
.nf
          THRESH is DOUBLE PRECISION
          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 INFO not equal to 0\&.)
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDA, NSIZE)
          Copy of A, modified by ZGGESX\&.
.fi
.PP
.br
\fIBI\fP 
.PP
.nf
          BI is COMPLEX*16 array, dimension (LDA, NSIZE)
          Copy of B, modified by ZGGESX\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is COMPLEX*16 array, dimension (LDA, NSIZE)
          Z holds the left Schur vectors computed by ZGGESX\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is COMPLEX*16 array, dimension (LDA, NSIZE)
          Q holds the right Schur vectors computed by ZGGESX\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is COMPLEX*16 array, dimension (NSIZE)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is COMPLEX*16 array, dimension (NSIZE)

          On exit, ALPHA/BETA are the eigenvalues\&.
.fi
.PP
.br
\fIC\fP 
.PP
.nf
          C is COMPLEX*16 array, dimension (LDC, LDC)
          Store the matrix generated by subroutine ZLAKF2, 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 DOUBLE PRECISION array, dimension (LDC)
          Singular values of C
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (LWORK)
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.  LWORK >= 3*NSIZE*NSIZE/2
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array,
                                 dimension (5*NSIZE*NSIZE/2 - 4)
.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 + 2\&.
.fi
.PP
.br
\fIBWORK\fP 
.PP
.nf
          BWORK is LOGICAL array, dimension (NSIZE)
.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 \fB346\fP of file \fBzdrgsx\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.