The linear system y′=Ay,y(0)=y_0, where A is a symmetric matrix, is solved by the Euler method. Let e_n=y_n−y(nh) Where y_n denotes the Euler approximation and y(nh) the exact solution (h is Euler step size). Prove that ||e_n||_2=||y_0||_2max|(1+h lambda)^n−e^(nh lambda)| Where lambda in sigma(A) where sigma(A) is the set of eigenvalues of A.

Question

The linear system

y^{'} = A y, y (0) = y_{0}

, where

A

is a symmetric matrix, is solved by the Euler method.
Let

e_{n} = y_{n} - y (n h)

Where

y_{n}

denotes the Euler approximation and

y (n h)

the exact solution (

h

is Euler step size).
Prove that

| | e_{n} | |_{2} = | | y_{0} | |_{2} m a x | (1 + h λ)^{n} - e^{n h λ} |

Where

λ \in σ (A)

where

σ (A)

is the set of eigenvalues of

A

.
I have tried various approaches such as writing

e_{n}

as the error bound of the Euler method and taking the norm, but I can't seem to get

| | y_{0} | |_{2}

in my answers.

Answer 1

Given that A is symmetric it can be diagonalised. Therefore we can write down the solution of the system of ODEs directly, and simply subtract the discreteized solution front the exact.

Answers (1)

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