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2.5.2 Discrete Newton     continued...

To symmetrize the resulting matrix , whose columns are the vectors , , from (45) the matrix

is constructed.

With exact arithmetic, discrete Newton methods converge quadratically if each goes to zero as does [16]. However, the roundoff error limits the smallest feasible size of difference interval in practice and, hence, the accuracy (a combination of roundoff and truncation errors) that can be obtained. As the gradient becomes very small, considerable loss of accuracy may also result from cancellation errors in the numerator (a large relative error, , from the subtraction of two quantities, s and , of similar magnitude). Consequently, discrete Newton methods are inappropriate for large-scale problems unless the Hessian has a known sparsity structure and this structure is exploited in the differencing scheme.