. Let x[1,2,3,4], y 1,0,1,0]. Determine the natural spline s with s(x)y for j 1,2,3,4 by a calcul...
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
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
2. A natural cubic spline S is defined by S(z) = { $(") = 11 S. (C)=1+ B(-1) + D(x - 1)3 S (2)=1+b(-2) + 3(x - 2)2 + (x - 2) if 1 <3 <2 if 2 <3<3 Find B, D, b, and d. Hint: Recall that a natural cubic spline has S" equal to 0 at the endpoints.
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
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...
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.
Let them be given positive parameters: k and e. Lets consider linear spline S(x) lean on n+1 of equidistant nodes in (-1,1], interpolating f(x) = sin(kx). Determine n, for wchich we will have guarantees that ||f - Slloo<=
Q8 Spline 15 Points Define s(2) = -5 + 8x6x2 + 2x2 on 1 < x <2, and 8(x) = 27 402 + 1822. on 2<< <3 Verify that s() is a cubic spline function on (1,3). Is it a natural spline function on this interval?
QUESTION 4 (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, F (3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system x2 + y = 9, 22 + y2 = 25, x, y > 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as...
Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C CB uniformly from the power set of B and let y be its size. Give the joint p.m.f of (X,Y) and compute E(X – Y). Note: X,Y can take value 0 if you pick the empty set....