## MB05ND

### Matrix exponential and integral for a real matrix

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

Purpose

      To compute

      (a)    F(delta) =  exp(A*delta) and

      (b)    H(delta) =  Int[F(s) ds] from s = 0 to s = delta,

      where A is a real N-by-N matrix and delta is a scalar value.

Specification

          SUBROUTINE MB05ND( N, DELTA, A, LDA, EX, LDEX, EXINT, LDEXIN,
         $                   TOL, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LDA, LDEX, LDEXIN, LDWORK, N
          DOUBLE PRECISION  DELTA, TOL
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), DWORK(*), EX(LDEX,*), EXINT(LDEXIN,*)

Arguments

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) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array must contain the
              matrix A of the problem. (Array A need not be set if
              DELTA = 0.)

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

      EX      (output) DOUBLE PRECISION array, dimension (LDEX,N)
              The leading N-by-N part of this array contains an
              approximation to F(delta).

      LDEX    INTEGER
              The leading dimension of array EX.  LDEX >= MAX(1,N).

      EXINT   (output) DOUBLE PRECISION array, dimension (LDEXIN,N)
              The leading N-by-N part of this array contains an
              approximation to H(delta).

      LDEXIN  INTEGER
              The leading dimension of array EXINT.  LDEXIN >= MAX(1,N).

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in determining the order of the
              Pade approximation to H(t), where t is a scale factor
              determined by the routine. A reasonable value for TOL may
              be SQRT(EPS), where EPS is the machine precision (see
              LAPACK Library routine DLAMCH).

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.

      LDWORK  INTEGER
              The length of the array DWORK. LDWORK >= MAX(1,N*(N+1)).
              For optimum performance LDWORK should be larger (2*N*N).

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              > 0:  if INFO = i, the (i,i) element of the denominator of
                    the Pade approximation is zero, so the denominator
                    is exactly singular;
              = N+1:  if DELTA = (delta * frobenius norm of matrix A) is
                    probably too large to permit meaningful computation.
                    That is, DELTA > SQRT(BIG), where BIG is a
                    representable number near the overflow threshold of
                    the machine (see LAPACK Library Routine DLAMCH).

Method

      This routine uses a Pade approximation to H(t) for some small
      value of t (where 0 < t <= delta) and then calculates F(t) from
      H(t). Finally, the results are re-scaled to give F(delta) and
      H(delta). For a detailed description of the implementation of this
      algorithm see [1].

References

      [1] Benson, C.J.
          The numerical evaluation of the matrix exponential and its
          integral.
          Report 82/03, Control Systems Research Group,
          School of Electronic Engineering and Computer
          Science, Kingston Polytechnic, January 1982.

      [2] Ward, R.C.
          Numerical computation of the matrix exponential with accuracy
          estimate.
          SIAM J. Numer. Anal., 14, pp. 600-610, 1977.

      [3] Moler, C.B. and Van Loan, C.F.
          Nineteen Dubious Ways to Compute the Exponential of a Matrix.
          SIAM Rev., 20, pp. 801-836, 1978.

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations.

Further Comments

      None

Example

Program Text

    *     MB05ND EXAMPLE PROGRAM TEXT.
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 20 )
          INTEGER          LDA, LDEX, LDEXIN, LDWORK
          PARAMETER        ( LDA = NMAX, LDEX = NMAX, LDEXIN = NMAX,
         $                   LDWORK = NMAX*( NMAX+1 ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION DELTA, TOL
          INTEGER          I, INFO, J, N
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), EX(LDEX,NMAX),
         $                 EXINT(LDEXIN,NMAX)
          INTEGER          IWORK(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         MB05ND
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, DELTA, TOL
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
    *        Find the matrix exponential of A*DELTA and its integral.
             CALL MB05ND( N, DELTA, A, LDA, EX, LDEX, EXINT, LDEXIN, TOL,
         $                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 ) ( EX(I,J), J = 1,N )
       20       CONTINUE
                WRITE ( NOUT, FMT = 99995 )
                DO 40 I = 1, N
                   WRITE ( NOUT, FMT = 99996 ) ( EXINT(I,J), J = 1,N )
       40       CONTINUE
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB05ND EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB05ND = ',I2)
    99997 FORMAT (' The solution matrix exp(A*DELTA) is ')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' and its integral is ')
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     MB05ND EXAMPLE PROGRAM DATA
       5     0.1     0.0001
       5.0   4.0   3.0   2.0   1.0
       1.0   6.0   0.0   4.0   3.0
       2.0   0.0   7.0   6.0   5.0
       1.0   3.0   1.0   8.0   7.0
       2.0   5.0   7.0   1.0   9.0

Program Results

     MB05ND EXAMPLE PROGRAM RESULTS

     The solution matrix exp(A*DELTA) is
       1.8391   0.9476   0.7920   0.8216   0.7811
       0.3359   2.2262   0.4013   1.0078   1.0957
       0.6335   0.6776   2.6933   1.6155   1.8502
       0.4804   1.1561   0.9110   2.7461   2.0854
       0.7105   1.4244   1.8835   1.0966   3.4134

     and its integral is
       0.1347   0.0352   0.0284   0.0272   0.0231
       0.0114   0.1477   0.0104   0.0369   0.0368
       0.0218   0.0178   0.1624   0.0580   0.0619
       0.0152   0.0385   0.0267   0.1660   0.0732
       0.0240   0.0503   0.0679   0.0317   0.1863

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