Differential Equations and the Matrix Exponential
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:
The second problem is particularly important when A is large and sparse and it is
impractical to form exp(A) explicitly.
Compute the exponential of the n-by-n matrix A.
Compute the action of the exponential of the n-by-n matrix A
on a vector b or matrix B: exp(A)*b or exp(A)*B.
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
We have also made significant progress in computing the Fréchet derivative,
and estimating the condition number,
of the matrix exponential.
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 Krylov-based methods and ODE integrators such as
the MATLAB functions ode45 and ode15s.
Computing the Action of the Matrix Exponential, with an Application
to Exponential Integrators,
SIAM J. Sci. Comp. 33 (2): 488-511, 2011.
and Awad Al-Mohy,
Computing Matrix Functions,
Acta Numerica 19: 159-208, 2010.
The Complex Step Approximation to the Fréchet Derivative of a
Numer. Alg. 53(1): 133-148, 2010.
A New Scaling and Squaring Algorithm for the Matrix Exponential,
SIAM J. Matrix Anal. Appl. 31(3): 970-989, 2009.
Functions of Matrices: Theory and Computation,
SIAM, 2008. xx+425 pages,
Our team at Manchester involves:
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