.ipynb .pdf

Matrix exponential

Let $\mathbf{A}(t)$ be an $m \times m$ matrix whose entries depend on $t$. Then

$$\frac{d \mathbf{u}}{d t} = \mathbf{A}(t)\mathbf{u}$$

is a linear system of differential equations. If the matrix $\mathbf{A}$ is independent of time, it is a linear, constant-coefficient system.

The solution to a linear, constant-coefficient IVP, given also $\mathbf{u}(0)=\mathbf{u}_0$, is formally

$$\mathbf{u}(t) = e^{t\mathbf{A}} \mathbf{u}_0,$$

where $e^{t\mathbf{A}}$ is a matrix exponential, which can be defined using Taylor series or by other means. This result is a seamless generalization of the scalar case, $m=1$.

A = [ -2 5; -1 0 ]

2×2 Array{Int64,2}:
 -2  5
 -1  0

u0 = [1,0]
t = LinRange(0,6,600)     # times for plotting
u = zeros(length(t),length(u0))
for j=1:length(t)
    ut = exp(t[j]*A)*u0 
    u[j,:] = ut'
end

using Plots
plot(t,u,label=["\$u_1\$" "\$u_2\$"],xlabel="t",ylabel="u(t)")

However, while the matrix exponential is a vital theoretical tool, computing it is too slow to be a practical numerical method.

Convergence of Euler’s method Predator-prey system