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Discussion of Jacobi's method for eigenvalues

  • Choose a tolerance ϵ, making it a small number, typically 108 or smaller.
  • Setup a while test where one compares the norm of the newly computed off-diagonal matrix elements off(A)=ni=1nj=1,jia2ij>ϵ.
  • Now choose the matrix elements akl so that we have those with largest value, that is |akl|=maxij|aij|.
  • Compute thereafter τ=(allakk)/2akl, tanθ, cosθ and sinθ.
  • Compute thereafter the similarity transformation for this set of values (k,l), obtaining the new matrix B=S(k,l,θ)TAS(k,l,θ).
  • Compute the new norm of the off-diagonal matrix elements and continue till you have satisfied off(B)ϵ