= The linear system y Ay, y(0) = yo, where A is a symmetric matrix, is...

  

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= The linear system y Ay, y(0) = yo, where A is a symmetric matrix, is solved by Euler's method. a Letting en y-y(nh), n = 0,1,..., prove that en 202 max (1+hλ)", Πλεσ(Α) - where (A) is the set of eigenvalues of A and || ||2 is the Euclidean matrix norm (cf. A.1.3.3). . b Demonstrate that for every -1 <<0 and n = 0,1,... it is true that - n²(n-1) < (1+x)"≤e (Hint: Prove first that 1+xe, 1+x+² e for all x ≤0, and then argue that, provided a -1| and |b| are small, it is true that (a-b)"> a"-na"-1b.) c Suppose that the maximal eigenvalue of A is Amax <0. Prove that, as h→0 and nh-t€ [0,t"], Amaxt exh max max = The linear system y Ay, y(0) = yo, where A is a symmetric matrix, is solved by Euler's method. a Letting en y-y(nh), n = 0,1,..., prove that en 202 max (1+hλ)", Πλεσ(Α) - where (A) is the set of eigenvalues of A and || ||2 is the Euclidean matrix norm (cf. A.1.3.3). . b Demonstrate that for every -1 <<0 and n = 0,1,... it is true that - n²(n-1) < (1+x)"≤e (Hint: Prove first that 1+xe, 1+x+² e for all x ≤0, and then argue that, provided a -1| and |b| are small, it is true that (a-b)"> a"-na"-1b.) c Suppose that the maximal eigenvalue of A is Amax <0. Prove that, as h→0 and nh-t€ [0,t"], Amaxt exh max max

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Proving the Inequality for Linear System We have a linear system given by y ... View the full answer