## 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 , 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
```   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.

 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.

```
Numerical Aspects
```                            3
The algorithm requires 0(N ) operations.

```
```  None
```
Example

Program Text

```*     MB05MD 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, 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
```