2. Consider the cubic spline for a function f on [0, 2] defined by S(x) = { ={ (z. 2x3 + ax2 + rx +1 if 0 < x <1 (x - 1)3 + c(x - 1)2 + d(x - 1) + ß if 1 < x < 2 where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
5. (a) The natural spline S(a) passing through the n+ points is a collection of n cubic functions S,(x) defined in the n intervals x, Sxx Suppose that all the points are equally spaced, with uniform point spacing h=5m-x, for jso,1,..,,n. Ifthe symbols M,, 0, represent the second derivatives of the spline at cach of the mesh points, show that in each intervl-.R-1 For the natural cubic spline (for which M, O and M-=0 ), show that the moments M...
Natural Cubic spline If the and Derivatives at the endpoints are known, ine. Pi" (x)= Y, 4 and Puni (xu) z Yu" for some numbers y," and ya theni 25, + z = 3y[x, 1x23 - į sxy," (al Sunit zou z 3y [xn-1 , xu] + & Sx uni Yu" (6) 10 Derive equations (a) and (b) for the natural cubic Spline
Linear Algebra A natural cubic spline for a function f(x) is defined by ſax° +15x² – 21.75x +14.5 15x521 S(x) = ayx? + 3x² + 2.25x-1.5 25 x 3] Find the value of a1.
Numerical analysis Question 1 A clamped cubic spline Six) on interval [0, 2) is defined by S(x) - 3 + ax + bx2 - 2x06x<1 (3 + c(x - 1) - 4(x - 1)2 + 7(x - 1) 1sx52 where a, b, care constants. Find the values of S'(0) and S(2).
beta is 1 2. ( 20p.) Consider the cubic spline for a function f on (0, 2) defined by 223 + ax2 + rx + 1 if 0 < x < 1 S(X) = (.x - 1)3 + c(x - 1)2 + d(x - 1) + B if i < x < 2 S(x) = { where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
0.19 0.. 1. (Natural Splines) Find the natural cubic spline S(x) satisfying S(0) = 0, S(1/2) = 1, S(1) = 0. Your answer will be 2 cubic polynomials, S.(x), S1(x). Verify that your answer satisfies all the necessary conditions (interpolation, continuity of 1st and 2nd derivatives, boundary conditions). We were unable to transcribe this image
Alpha=9 beta=3 yazarsin 2. ( 20p.) Consider the cubic spline for a function f on (0,2) defined by 2x3 + x² +rx +1 if 0 <x<1 S(x) = (x - 1)3 + c(x - 1)2 + d(x - 1) + B if 1<x<2 = {(2-1) where r,c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
Homework Problem Set 16 x-1)3 for 0 sx$1 a cubic spline? 1. Is the function s (x) = {2(-1)3 for 1 < x 2 -5 +8x-6x2 2x3 for1 sx s 2 Is the function s(x)-[ natural? Let x for j 1,2,3,4 by a calculation on paper 3 a cubic spline? Is it [1,0,1,0]. Determine the natural spline s with s(y)-yi 2. 18x2-2x3 for 2