GENERALIZED EIGENSOLUTIONS BY JACOBI METHODS

We have developed an algorithm for solving the positive definite gener alized eigenvalue problem Hx = lambda Mx with optimal accuracy in the following sense: For a maching pair of an eigenvalue lambda of H - la mbda M and its computed approximation lambda + delta lambda*, the rel ative error \delta lambda\lambda* is (up to factor of dimensionality) of the same order as eigenvalue perturbation caused by delta H, delta M with \delta H-jj\less than or equal to O(epsilon)root HiiHjj,\delta M(ij)\less than or equal to O(epsilon)root M(ii)M(jj). The relative e rror bound reads \delta lambda\lambda*less than or equal to f(n)epsil on(parallel to H(s)(-1)parallel to 2+parallel to M(s)(-1)parallel to 2 ), where H = diag (root H-ii)H(s)diag (root H-ii, M = diag (root M(ii) )M(s)diag (root M(ii)), f(.) is mildly growing function of matrix dime nsion and epsilon = max{delta > 0 : fl(1 + delta) = 1} is the roundoff unit of floating point computation fl(.). The same error estimate hol ds for floating point solution of the positive definite product eigenv alue problem HMx = lambda x.