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
NoneExample
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 0.5Program 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