## TG01ED

### Orthogonal reduction of a descriptor system to a SVD coordinate form

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To compute for the descriptor system (A-lambda E,B,C)
      the orthogonal transformation matrices Q and Z such that the
      transformed system (Q'*A*Z-lambda Q'*E*Z, Q'*B, C*Z) is in an
      SVD (singular value decomposition) coordinate form with
      the system matrices  Q'*A*Z and Q'*E*Z in the form

                   ( A11  A12 )             ( Er  0 )
          Q'*A*Z = (          ) ,  Q'*E*Z = (       ) ,
                   ( A21  A22 )             (  0  0 )

      where Er is an invertible diagonal matrix having on the diagonal
      the decreasingly ordered nonzero singular values of E.
      Optionally, the A22 matrix can be further reduced to the
      SVD form

                   ( Ar  0 )
             A22 = (       ) ,
                   (  0  0 )

      where Ar is an invertible diagonal matrix having on the diagonal
      the decreasingly ordered nonzero singular values of A22.
      The left and/or right orthogonal transformations performed
      to reduce E and A22 are accumulated.

Specification

          SUBROUTINE TG01ED( JOBA, L, N, M, P, A, LDA, E, LDE, B, LDB,
         $                   C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22, TOL,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          JOBA
          INTEGER            INFO, L, LDA, LDB, LDC, LDE, LDQ, LDWORK,
         $                   LDZ, M, N, P, RNKA22, RANKE
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
         $                   DWORK( * ),  E( LDE, * ), Q( LDQ, * ),
         $                   Z( LDZ, * )

Arguments

Mode Parameters

      JOBA    CHARACTER*1
              = 'N':  do not reduce A22;
              = 'R':  reduce A22 to an SVD form.

Input/Output Parameters

      L       (input) INTEGER
              The number of rows of matrices A, B, and E.  L >= 0.

      N       (input) INTEGER
              The number of columns of matrices A, E, and C.  N >= 0.

      M       (input) INTEGER
              The number of columns of matrix B.  M >= 0.

      P       (input) INTEGER
              The number of rows of matrix C.  P >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading L-by-N part of this array must
              contain the state dynamics matrix A.
              On exit, the leading L-by-N part of this array contains
              the transformed matrix Q'*A*Z. If JOBA = 'R', this matrix
              is in the form

                            ( A11  *   *  )
                   Q'*A*Z = (  *   Ar  0  ) ,
                            (  *   0   0  )

              where A11 is a RANKE-by-RANKE matrix and Ar is a
              RNKA22-by-RNKA22 invertible diagonal matrix, with
              decresingly ordered positive diagonal elements.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,L).

      E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
              On entry, the leading L-by-N part of this array must
              contain the descriptor matrix E.
              On exit, the leading L-by-N part of this array contains
              the transformed matrix Q'*E*Z.

                       ( Er  0 )
              Q'*E*Z = (       ) ,
                       (  0  0 )

              where Er is a RANKE-by-RANKE invertible diagonal matrix
              having on the diagonal the decreasingly ordered positive
              singular values of E.

      LDE     INTEGER
              The leading dimension of array E.  LDE >= MAX(1,L).

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading L-by-M part of this array must
              contain the input/state matrix B.
              On exit, the leading L-by-M part of this array contains
              the transformed matrix Q'*B.

      LDB     INTEGER
              The leading dimension of array B.
              LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the state/output matrix C.
              On exit, the leading P-by-N part of this array contains
              the transformed matrix C*Z.

      LDC     INTEGER
              The leading dimension of array C.  LDC >= MAX(1,P).

      Q       (output) DOUBLE PRECISION array, dimension (LDQ,L)
              The leading L-by-L part of this array contains the
              orthogonal matrix Q, which is the accumulated product of
              transformations applied to A, E, and B on the left.

      LDQ     INTEGER
              The leading dimension of array Q.  LDQ >= MAX(1,L).

      Z       (output) DOUBLE PRECISION array, dimension (LDZ,N)
              The leading N-by-N part of this array contains the
              orthogonal matrix Z, which is the accumulated product of
              transformations applied to A, E, and C on the right.

      LDZ     INTEGER
              The leading dimension of array Z.  LDZ >= MAX(1,N).

      RANKE   (output) INTEGER
              The effective rank of matrix E, and thus also the order
              of the invertible diagonal submatrix Er.
              RANKE is computed as the number of singular values of E
              greater than TOL*SVEMAX, where SVEMAX is the maximum
              singular value of E.

      RNKA22  (output) INTEGER
              If JOBA = 'R', then RNKA22 is the effective rank of
              matrix A22, and thus also the order of the invertible
              diagonal submatrix Ar. RNKA22 is computed as the number
              of singular values of A22 greater than TOL*SVAMAX,
              where SVAMAX is an estimate of the maximum singular value
              of A.
              If JOBA = 'N', then RNKA22 is not referenced.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in determining the rank of E
              and of A22. If TOL > 0, then singular values less than
              TOL*SVMAX are treated as zero, where SVMAX is the maximum
              singular value of E or an estimate of it for A and E.
              If TOL <= 0, the default tolerance TOLDEF = EPS*L*N is
              used instead, where EPS is the machine precision
              (see LAPACK Library routine DLAMCH). TOL < 1.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX(1,MIN(L,N) +
                            MAX(3*MIN(L,N)+MAX(L,N), 5*MIN(L,N), M, P)).

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              > 0:  the QR algorithm has failed to converge when computing
                    singular value decomposition. In this case INFO
                    specifies how many superdiagonals did not converge.
                    This failure is not likely to occur.

Method

      The routine computes the singular value decomposition (SVD) of E,
      in the form

                     ( Er  0 )
            E  = Q * (       ) * Z'
                     (  0  0 )

      and finds the largest RANKE-by-RANKE leading diagonal submatrix
      Er whose condition number is less than 1/TOL. RANKE defines thus
      the effective rank of matrix E.
      If JOBA = 'R' the same reduction is performed on A22 in the
      partitioned matrix

                   ( A11  A12 )
          Q'*A*Z = (          ) ,
                   ( A21  A22 )

      to obtain it in the form

                   ( Ar  0 )
             A22 = (       ) ,
                   (  0  0 )

      with Ar an invertible diagonal matrix.

      The accumulated transformations are also applied to the rest of
      matrices

           B <- Q' * B,  C <- C * Z.

Numerical Aspects

      The algorithm is numerically backward stable and requires
      0( L*L*N )  floating point operations.

Further Comments

      None

Example

Program Text

    *     TG01ED EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          LMAX, NMAX, MMAX, PMAX
          PARAMETER        ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC, LDE, LDQ, LDZ
          PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
         $                   LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( 1, MIN( LMAX, NMAX ) +
         $                            MAX( MMAX, PMAX, 3*MIN( LMAX, NMAX ) +
         $                            MAX( LMAX, NMAX ),
         $                            5*MIN( LMAX, NMAX ) ) ) )
    *     .. Local Scalars ..
          CHARACTER*1      JOBA
          INTEGER          I, INFO, J, L, M, N, P, RANKE, RNKA22
          DOUBLE PRECISION TOL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
         $                 Z(LDZ,NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TG01ED
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX, MIN
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) L, N, M, P, TOL
          JOBA = 'R'
          IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
             WRITE ( NOUT, FMT = 99989 ) L
          ELSE
             IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
                WRITE ( NOUT, FMT = 99988 ) N
             ELSE
                READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
                READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
                IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                   WRITE ( NOUT, FMT = 99987 ) M
                ELSE
                   READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
                   IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                      WRITE ( NOUT, FMT = 99986 ) P
                   ELSE
                      READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *                 Find the transformed descriptor system
    *                 (A-lambda E,B,C).
                      CALL TG01ED( JOBA, L, N, M, P, A, LDA, E, LDE, B, LDB,
         $                         C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22,
         $                         TOL, DWORK, LDWORK, INFO )
    *
                      IF ( INFO.NE.0 ) THEN
                         WRITE ( NOUT, FMT = 99998 ) INFO
                      ELSE
                         WRITE ( NOUT, FMT = 99994 ) RANKE, RNKA22
                         WRITE ( NOUT, FMT = 99997 )
                         DO 10 I = 1, L
                            WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       10                CONTINUE
                         WRITE ( NOUT, FMT = 99996 )
                         DO 20 I = 1, L
                            WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
       20                CONTINUE
                         WRITE ( NOUT, FMT = 99993 )
                         DO 30 I = 1, L
                            WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       30                CONTINUE
                         WRITE ( NOUT, FMT = 99992 )
                         DO 40 I = 1, P
                            WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
       40                CONTINUE
                         WRITE ( NOUT, FMT = 99991 )
                         DO 50 I = 1, L
                            WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
       50                CONTINUE
                         WRITE ( NOUT, FMT = 99990 )
                         DO 60 I = 1, N
                            WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
       60                CONTINUE
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01ED EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01ED = ',I2)
    99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
    99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (' Rank of matrix E   =', I5/
         $        ' Rank of matrix A22 =', I5)
    99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
    99992 FORMAT (/' The transformed state/output matrix C*Z is ')
    99991 FORMAT (/' The left transformation matrix Q is ')
    99990 FORMAT (/' The right transformation matrix Z is ')
    99989 FORMAT (/' L is out of range.',/' L = ',I5)
    99988 FORMAT (/' N is out of range.',/' N = ',I5)
    99987 FORMAT (/' M is out of range.',/' M = ',I5)
    99986 FORMAT (/' P is out of range.',/' P = ',I5)
          END

Program Data

    TG01ED EXAMPLE PROGRAM DATA
      4    4     2     2     0.0
        -1     0     0     3
         0     0     1     2
         1     1     0     4
         0     0     0     0
         1     2     0     0
         0     1     0     1
         3     9     6     3
         0     0     2     0
         1     0
         0     0
         0     1
         1     1
        -1     0     1     0
         0     1    -1     1

Program Results

     TG01ED EXAMPLE PROGRAM RESULTS

     Rank of matrix E   =    3
     Rank of matrix A22 =    1

     The transformed state dynamics matrix Q'*A*Z is
       2.1882  -0.8664  -3.5097  -2.1353
      -0.4569  -0.2146   1.9802   0.3531
      -0.5717  -0.5245  -0.4591   0.4696
      -0.4766  -0.5846   2.1414   0.3086

     The transformed descriptor matrix Q'*E*Z is
      11.8494   0.0000   0.0000   0.0000
       0.0000   2.1302   0.0000   0.0000
       0.0000   0.0000   1.0270   0.0000
       0.0000   0.0000   0.0000   0.0000

     The transformed input/state matrix Q'*B is
      -0.2396  -1.0668
      -0.2656  -0.8393
      -0.7657  -0.1213
       1.1339   0.3780

     The transformed state/output matrix C*Z is
      -0.2499  -1.0573   0.3912  -0.8165
      -0.5225   1.3958   0.8825   0.0000

     The left transformation matrix Q is
      -0.1534   0.5377  -0.6049   0.5669
      -0.0872   0.2536   0.7789   0.5669
      -0.9805  -0.0360   0.0395  -0.1890
      -0.0863  -0.8033  -0.1608   0.5669

     The right transformation matrix Z is
      -0.2612   0.2017  -0.4737   0.8165
      -0.7780   0.4718  -0.0738  -0.4082
      -0.5111  -0.8556  -0.0826   0.0000
      -0.2556   0.0684   0.8737   0.4082

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