On the coding of Jacobi's method of computing eigenvalues and eigenvectors of real, symmetric matrices

Abstract

Jacobi~s method 1'2,3 consists of performing consecutive 2 by 2 rotations on a real, symmetric, matrix, H, with elements H(km), k gm, where the largest magnitudes off-diagonal element, H(iJ), is used as a "pivot". A rotation is defined as the following unitary transformation where the primes indicate the new elements. kJ ii JJ ij = H(km) for k and m ~ i or j : i~(k~) + ~R(kj) for k ~nd ~ ~ i or J = sH[ki) + cH(kJ) for ~ and m i or J : c z {H(il) + 2 tH(i~I + t ~lJJ ~ = e 2 ~(jj)-2 tH(i + t z li = 0 with the last equation leading to:1 s= tc A square matrix U, which is initially set to be a unit matrix, is also successively rotated into a unitary matrix of the eigenvectors by the transformations: U' = cU(kl) + sU(kJ) U,(kJ) =-sU(ki) + cU(kJ) for m / i or J It is important to note that the form of t is always numerically accurate and approaches zero as H'() iJ becomes small. In addition the forms of HI(il) and H'(JJ), which do 33-1