Jacobis method for a symmetric matrix A is described by A1 = A, A2 = P1A1Pt 1 and, in

Chapter 9, Problem 11

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Jacobis method for a symmetric matrix A is described by A1 = A, A2 = P1A1Pt 1 and, in general, Ai+1 = PiAiPt i . The matrix Ai+1 tends to a diagonal matrix, where Pi is a rotation matrix chosen to eliminate a large off-diagonal element in Ai. Suppose aj,k and ak,j are to be set to 0, where j = k. If ajj = akk , then (Pi)jj = (Pi)kk = + 1 2 1 + b c2 + b2 , (Pi)kj = c 2(Pi)jj c2 + b2 = (Pi)jk , where c = 2ajk sgn(ajj akk ) and b = |ajj akk |, or if ajj = akk , (Pi)jj = (Pi)kk = 2 2 and (Pi)kj = (Pi)jk = 2 2 . Develop an algorithm to implement Jacobis method by setting a21 = 0. Then set a31, a32, a41, a42, a43, ... , an,1, ... , an,n1 in turn to zero.This is repeated until a matrix Ak is computed with n i=1 n j=1j=i |a(k) i j | sufficiently small. The eigenvalues of A can then be approximated by the diagonal entries of Ak .