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.

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  ∈  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&#039;t seem to get       |        |        y    0        |              |        2   in my answers.

akademiks1989rz · Accepted Answer

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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