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.

asigurato7 2022-07-22 Answered
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 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.
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Answers (1)

akademiks1989rz
Answered 2022-07-23 Author has 16 answers
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.
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