Accession Number : AD0601064

Title :   REPEATED EXTRAPOLATION TO THE LIMIT IN THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS.

Corporate Author : CALIFORNIA UNIV LOS ANGELES

Personal Author(s) : Gragg,William Bryant

Report Date : JAN 1964

Pagination or Media Count : 111

Abstract : Let the initial-value problem y' = f(x,y), x epsilon a,b , y(a) = s, for a system of differential equations be solved numerically by a one-step method or by a linear multistep method (see Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, 1962). There results a family of sequences y sub n (h) which approximate the exact solution y (x) at the mesh points x = x sub n = a + nh. The parameter h plays an essential role in the construction of the y sub n (h). Under suitable regularity conditions on f it is proved that y sub n (h) has an asymptotic expansion in powers of h, valid as h approaches 0+ and n approaches infinity in such a manner that x = a + nh remains fixed in (a,b . These results provide a rigorous justification for the algorithm of repeated extrapolation to the limit in which successive terms of the expansion for the error y sub n (h) - y(x) are removed by making use of the knowledge of the approximate solutions computed with the values h = 2 to the minus m power h sub O, m = 0,1, . . . . The convergence properties of the algorithm are discussed in some generality and the theory is verified experimentally for a number of classical numerical methods. (Author)

Descriptors :   (*NUMERICAL METHODS AND PROCEDURES, DIFFERENTIAL EQUATIONS), (*DIFFERENTIAL EQUATIONS, NUMERICAL METHODS AND PROCEDURES), SERIES(MATHEMATICS), DIFFERENCE EQUATIONS, MATHEMATICAL ANALYSIS, NONLINEAR DIFFERENTIAL EQUATIONS, MATRICES(MATHEMATICS)

Distribution Statement : APPROVED FOR PUBLIC RELEASE