Algorithm 379: Squank (Simpson Quadrature used adaptivity—noise killed) [D1]

comment Functional procedure nielin, of the integer type, solves a system of simultaneous nonlinear algebraic or trans-cendental equations. Let us consider a given system of n equations with n variables: A kth approximation of the solution of the system (1) is supposed to be given: Yo (~) = (y~), y~), ..-, y~)). (2) If for every i, Ifi(Yo(~))[ < e, (3) where e > 0 is a given number, then the approximation (2) is considered as a solution of the system (1), otherwise a further approximation is calculated. Let h (k) > 0 be given and construct the n new points: For every function of the system (1) a new interpolating polynomial of the first order is constructed on the points (2) and (4) such that: w,(y~k)) = f, A solution of the linear system: wi(y~ , y2 , "'', Y,) = O, i = 1, 2, "', n, (6) is used as the (kT1)-th successive approximation. The special choice of the interpolation points (2) and (4) assures existence and uniqueness of the interpolating poly-nomials wi (5). Namely, the kth approximation has for the ith function the form: n w,(~> (Y) = /,(r2)) + ~ g~(yj-y~(~)' j, (7) i-1 where g~) = (f,(y]k)) _ f,(y~))/h(k). The solution of the system (6) where w~ is given by (7) can be written in the form (see [2]): y(~+l) = y~k) _ (1/a(k))z~) X h Ck), i = 1, 2, ..., n, (9) where z (~) = (z~ k), z~ (k), ..., z(= k)) is a solution of the following linear system: n ~f~(y(k))