Jacobi's method for a symmetric matrix A is described by
A1 = A,
A2 = P1A1Pt1
and, in general,
Ai+1 = PiAiPti.
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

where
c = 2ajksgn(ajj ˆ’ akk) and b = |ajj ˆ’ akk|,
or if ajj = akk,
Pi)jj = (Pi)kk = ˆš2 /2
And
(Pi)kj = ˆ’ (Pi)jk =ˆš2/2.
Develop an algorithm to implement Jacobi's method by setting a21 = 0. Then set a31, a32, a41, a42, a43, . . . , an,1, . . . , an,nˆ’1 in turn to zero. This is repeated until a matrix Ak is computed with

sufficiently small. The eigenvalues of A can then be approximated by the diagonal entries of Ak.

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j+i