Differential Equations and the Matrix Exponential
Problem Description
A linear differential equation y' = Ay with initial condition
y(0) = y_0 has the solution
y(t) = exp(At)y_0,
where the matrix exponential
exp(At) = I + At + (At)^2/2! + (At)^3/3! + ...
The solutions to nonlinear differential equations can be written in integral
forms involving the exponential of the matrix defining
the linear part of the equation and these are the basis of various numerical methods,
in particular the class of exponential integrators.
Two problems arise:

Compute the exponential of the nbyn matrix A.

Compute the action of the exponential of the nbyn matrix A
on a vector b or matrix B: exp(A)*b or exp(A)*B.
The second problem is particularly important when A is large and sparse and it is
impractical to form exp(A) explicitly.
Research
We have made significant improvements to the scaling and
squaring method for computing exp(A),
which is the most widely used method.
The scaling and squaring algorithm used by the MATLAB expm function
was developed in Manchester,
and that algorithm has recently been improved so as to avoid the problem of
overscaling.
We have also made significant progress in computing the Fréchet derivative,
and estimating the condition number,
of the matrix exponential.
In particular,
we have shown how to efficiently intertwine computation
of exp(A) with estimation of its condition number.
Most recently, we have developed an
algorithm for the exp(A)*B problem
that combines the scaling part of the scaling and squaring method
with a truncated Taylor series approximation to the exponential.
In our experiments this algorithm has proved
superior to Krylovbased methods and ODE integrators such as
the MATLAB functions ode45 and ode15s.
Publications (recent)

Awad AlMohy,
and
Nicholas J.
Higham,
Computing the Action of the Matrix Exponential, with an Application
to Exponential Integrators,
SIAM J. Sci. Comp. 33 (2): 488511, 2011.
MATLAB
codes.

Nicholas J.
Higham,
and Awad AlMohy,
Computing Matrix Functions,
Acta Numerica 19: 159208, 2010.

Awad AlMohy,
and
Nicholas J.
Higham,
The Complex Step Approximation to the Fréchet Derivative of a
Matrix Function,
Numer. Alg. 53(1): 133148, 2010.
Published version.

Awad AlMohy,
and
Nicholas J.
Higham,
A New Scaling and Squaring Algorithm for the Matrix Exponential,
SIAM J. Matrix Anal. Appl. 31(3): 970989, 2009.
MATLAB codes.

Nicholas J.
Higham,
Functions of Matrices: Theory and Computation,
SIAM, 2008. xx+425 pages,
ISBN 9780898716467.
Events
People
Our team at Manchester involves:
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