Using Maple to work out error terms

Question

I have the exponential matrix using a Jordan form matrix and transition matrix: $E(t) = e^{At}= \left[ \begin{array}{ccc} e^{3t}+3te^{2t} & 3e^{3t}-3e^{2t}+6te^{2t} & 6e^{3t}-6e^{2t}+9te^{2t} \\ -2e^{3t}+2e^{2t}-3te^{2t} & -6e^{3t}+7e^{2t}-6te^{2t} & -12e^{3t}+12e^{2t}-9te^{2t} \\ e^{3t}-e^{2t}+te^{2t} & 3e^{3t}-3e^{2t}+2te^{2t} & 6e^{3t}-5e^{2t}+3te^{2t} \\ \end{array} \right]$

where $A = \left[ \begin{array}{ccc} 6 & 9 & 15\\ -5 & -10 & -21\\ 2 & 5 & 11 \\ \end{array} \right]$ and $t$ is an input value.

Question:

Approximate $E(2.5)$ using the Taylor expansion of the exponential up to the order $k$. Give the results $E_k$ for $k = 5,10,20,50,100$ and compute the error terms $e_k = \left|\left|E_k-e^{2.5A}\right|\right|$ in each case. Note here $\left|\left|M\right|\right|$ means the greatest absolute value of the entries of a matrix $M$.

I have easily (using maple) computed the approximation for each order using the sum:

$\sum\limits_{k=0}^{\infty}\frac{1}{k!}(At)^k$

but how do i work out the error terms $e_k$? what Maple command do i use to get $\left|\left|M\right|\right|$

Answer 1

I don't think there is a built-in function for that. There is VectorNorm(v,infinity), but you would first have to flatten the matrix into a vector in order to use that.

Given that the matrices are 3 by 3, it is easier to write a loop than to search for clever tricks:

with(LinearAlgebra):
A:=<<6 | 9 | 15>, <-5 | -10 | -21>, <2 | 5 | 11>>;
max(seq(seq(abs(A[i,j]),i=1..3),j=1..3));

Output: 21.