Romberg integration
نویسنده
چکیده
1. Mathematical preliminaries Many problems in Numerical Analysis involve the transformation of a continuous problem into a discrete one. Our example here is the calculation of the value of a definite integral which is turned into computing a weighted sum of function values. An important parameter in this discretization is the mesh length or step size, h: b a f (x)dx ≈ h n i=1 f (a + 2i − 1 2 h); h = b − a n. (1) Geometrically speaking we approximate the area under f (x) by the sum of the areas of a succession of rectangles of width h and height equal to the function value at the mid-point. If the integrand is sufficiently smooth the error of this approximation can be expressed as a series of terms involving even powers of the step size h. To see this in a simple case consider the integral of f (x) over a single subinterval 1
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