Maths Question:

1. Consider the matrix A E R3\" defined as 2 1 o A: 1 2 .212 1'] v2 2 (a) Verify that A is symmetric and positive denite and identify the eigenvalues of maxi mum and minimum absolute value. say A1 and A3. respectively. [4] {b} To solve a linear system with matrix A. we use the nonpreconditioned stationary Richardson method. Compute the value of the optimal acceleration parameter air for this method and estimate the minimum number of iterations kw\" so that the error HEfA-"MMJIHA is strictly smaller than 10'\". Assume that ||e':"}||,4 = 1 and that om is used in the iterations. [We recall that ||v||,q = W for all v E R3.) [T] {c} Compute the Cholesky factorisation of the matrix A, i.e._ determine the upper trian gular matrix R such that A = RTE. Write all your calculations. [4] {d} To estimate the eigenvalue .313 of minimum absolute value of the matrix A. we use the inverse power method. Please refer to algorithm (IE-9} in the lecture notes and remember that E|v|| = of + v+ u; for all v = {U1,1321L'3}T E R3. Assume that arm =(1.1.~f}1' and pi\") = {1. 1.01\"\". {i} Using the Cholesky factorisation of the matrix A obtained in part (c). perform one iteration of the inverse power method to compute an\"), pm, and pill-l. Explain your steps and write all your calculations. [8] (ii) Explain how to obtain the requested approximation of Jo, and of its eigenvector once the inverse power method converges. [2]