## MB05MD

### Matrix exponential for a real non-defective matrix

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

Purpose

      To compute exp(A*delta) where A is a real N-by-N non-defective
      matrix with real or complex eigenvalues and delta is a scalar
      value. The routine also returns the eigenvalues and eigenvectors
      of A as well as (if all eigenvalues are real) the matrix product
      exp(Lambda*delta) times the inverse of the eigenvector matrix
      of A, where Lambda is the diagonal matrix of eigenvalues.
      Optionally, the routine computes a balancing transformation to
      improve the conditioning of the eigenvalues and eigenvectors.

Specification

          SUBROUTINE MB05MD( BALANC, N, DELTA, A, LDA, V, LDV, Y, LDY, VALR,
         $                   VALI, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         BALANC
          INTEGER           INFO, LDA, LDV, LDWORK, LDY, N
          DOUBLE PRECISION  DELTA
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), DWORK(*), V(LDV,*), VALI(*), VALR(*),
         $                  Y(LDY,*)

Arguments

Mode Parameters

      BALANC  CHARACTER*1
              Indicates how the input matrix should be diagonally scaled
              to improve the conditioning of its eigenvalues as follows:
              = 'N':  Do not diagonally scale;
              = 'S':  Diagonally scale the matrix, i.e. replace A by
                      D*A*D**(-1), where D is a diagonal matrix chosen
                      to make the rows and columns of A more equal in
                      norm. Do not permute.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix A.  N >= 0.

      DELTA   (input) DOUBLE PRECISION
              The scalar value delta of the problem.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the matrix A of the problem.
              On exit, the leading N-by-N part of this array contains
              the solution matrix exp(A*delta).

      LDA     INTEGER
              The leading dimension of array A.  LDA >= max(1,N).

      V       (output) DOUBLE PRECISION array, dimension (LDV,N)
              The leading N-by-N part of this array contains the
              eigenvector matrix for A.
              If the k-th eigenvalue is real the k-th column of the
              eigenvector matrix holds the eigenvector corresponding
              to the k-th eigenvalue.
              Otherwise, the k-th and (k+1)-th eigenvalues form a
              complex conjugate pair and the k-th and (k+1)-th columns
              of the eigenvector matrix hold the real and imaginary
              parts of the eigenvectors corresponding to these
              eigenvalues as follows.
              If p and q denote the k-th and (k+1)-th columns of the
              eigenvector matrix, respectively, then the eigenvector
              corresponding to the complex eigenvalue with positive
              (negative) imaginary value is given by
                                        2
              p + q*j (p - q*j), where j  = -1.

      LDV     INTEGER
              The leading dimension of array V.  LDV >= max(1,N).

      Y       (output) DOUBLE PRECISION array, dimension (LDY,N)
              The leading N-by-N part of this array contains an
              intermediate result for computing the matrix exponential.
              Specifically, exp(A*delta) is obtained as the product V*Y,
              where V is the matrix stored in the leading N-by-N part of
              the array V. If all eigenvalues of A are real, then the
              leading N-by-N part of this array contains the matrix
              product exp(Lambda*delta) times the inverse of the (right)
              eigenvector matrix of A, where Lambda is the diagonal
              matrix of eigenvalues.

      LDY     INTEGER
              The leading dimension of array Y.  LDY >= max(1,N).

      VALR    (output) DOUBLE PRECISION array, dimension (N)
      VALI    (output) DOUBLE PRECISION array, dimension (N)
              These arrays contain the real and imaginary parts,
              respectively, of the eigenvalues of the matrix A. The
              eigenvalues are unordered except that complex conjugate
              pairs of values appear consecutively with the eigenvalue
              having positive imaginary part first.

Workspace

      IWORK   INTEGER array, dimension (N)

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK, and if N > 0, DWORK(2) returns the reciprocal
              condition number of the triangular matrix used to obtain
              the inverse of the eigenvector matrix.

      LDWORK  INTEGER
              The length of the array DWORK.  LDWORK >= max(1,4*N).
              For good performance, LDWORK must generally be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = i:  if INFO = i, the QR algorithm failed to compute all
                    the eigenvalues; no eigenvectors have been computed;
                    elements i+1:N of VALR and VALI contain eigenvalues
                    which have converged;
              = N+1:  if the inverse of the eigenvector matrix could not
                    be formed due to an attempt to divide by zero, i.e.,
                    the eigenvector matrix is singular;
              = N+2:  if the matrix A is defective, possibly due to
                    rounding errors.

Method

      This routine is an implementation of "Method 15" of the set of
      methods described in reference [1], which uses an eigenvalue/
      eigenvector decomposition technique. A modification of LAPACK
      Library routine DGEEV is used for obtaining the right eigenvector
      matrix. A condition estimate is then employed to determine if the
      matrix A is near defective and hence the exponential solution is
      inaccurate. In this case the routine returns with the Error
      Indicator (INFO) set to N+2, and SLICOT Library routines MB05ND or
      MB05OD are the preferred alternative routines to be used.

References

      [1] Moler, C.B. and Van Loan, C.F.
          Nineteen dubious ways to compute the exponential of a matrix.
          SIAM Review, 20, pp. 801-836, 1978.

      [2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
          Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
          Ostrouchov, S., and Sorensen, D.
          LAPACK Users' Guide: Second Edition.
          SIAM, Philadelphia, 1995.

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations.

Further Comments

      None

Example

Program Text

    *     MB05MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 20 )
          INTEGER          LDA, LDV, LDY
          PARAMETER        ( LDA = NMAX, LDV = NMAX, LDY = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 4*NMAX )
    *     .. Local Scalars ..
          DOUBLE PRECISION DELTA
          INTEGER          I, INFO, J, N
          CHARACTER*1      BALANC
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), V(LDV,NMAX),
         $                 VALI(NMAX), VALR(NMAX), Y(LDY,NMAX)
          INTEGER          IWORK(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         MB05MD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          BALANC = 'N'
          READ ( NIN, FMT = * ) N, DELTA
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
    *        Find the exponential of the real non-defective matrix A*DELTA.
             CALL MB05MD( BALANC, N, DELTA, A, LDA, V, LDV, Y, LDY, VALR,
         $                VALI, IWORK, DWORK, LDWORK, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 )
                DO 20 I = 1, N
                   WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
       20       CONTINUE
                WRITE ( NOUT, FMT = 99995 ) ( VALR(I), VALI(I), I = 1,N )
                WRITE ( NOUT, FMT = 99994 )
                DO 40 I = 1, N
                   WRITE ( NOUT, FMT = 99996 ) ( V(I,J), J = 1,N )
       40       CONTINUE
                WRITE ( NOUT, FMT = 99993 )
                DO 60 I = 1, N
                   WRITE ( NOUT, FMT = 99996 ) ( Y(I,J), J = 1,N )
       60       CONTINUE
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB05MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB05MD = ',I2)
    99997 FORMAT (' The solution matrix exp(A*DELTA) is ')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' The eigenvalues of A are ',/20(2F5.1,'*j  '))
    99994 FORMAT (/' The eigenvector matrix for A is ')
    99993 FORMAT (/' The inverse eigenvector matrix for A (premultiplied by'
         $        ,' exp(Lambda*DELTA)) is ')
    99992 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     MB05MD EXAMPLE PROGRAM DATA
       4     1.0
       0.5   0.0   2.3  -2.6
       0.0   0.5  -1.4  -0.7
       2.3  -1.4   0.5   0.0
      -2.6  -0.7   0.0   0.5

Program Results

     MB05MD EXAMPLE PROGRAM RESULTS

     The solution matrix exp(A*DELTA) is
      26.8551  -3.2824  18.7409 -19.4430
      -3.2824   4.3474  -5.1848   0.2700
      18.7409  -5.1848  15.6012 -11.7228
     -19.4430   0.2700 -11.7228  15.6012

     The eigenvalues of A are
     -3.0  0.0*j    4.0  0.0*j   -1.0  0.0*j    2.0  0.0*j

     The eigenvector matrix for A is
      -0.7000   0.7000   0.1000  -0.1000
       0.1000  -0.1000   0.7000  -0.7000
       0.5000   0.5000   0.5000   0.5000
      -0.5000  -0.5000   0.5000   0.5000

     The inverse eigenvector matrix for A (premultiplied by exp(Lambda*DELTA)) is
      -0.0349   0.0050   0.0249  -0.0249
      38.2187  -5.4598  27.2991 -27.2991
       0.0368   0.2575   0.1839   0.1839
      -0.7389  -5.1723   3.6945   3.6945

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