## 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.

```
```  None
```
Example

Program Text

```*     MB05ND EXAMPLE PROGRAM TEXT.
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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
```