Question

[T2] This problem concerns the derivation of the equations used to determine the coefficients of a quadratic spline approximation, S(x), to a function F(x) over an interval [x min, Xmax]. To assist in getting the indices correct, it is suggested you draw a picture that displays the labeling of the values involved. In the computational part of this assignment, you will be setting up and solving these equations. • Assume that n panels are used and the knots {x;} are equispaced and are given by Xi = a + i * h with (Xmax – Xmin) i = 0 ... n where h = . • Assume that the spline S(x) over the ith grid panel is expressed in the form S;(x) = a + bị = (x – x;-1)+ C; * (x - x;-1)* x 6 [x;-1,xi] i = 1 ...n + 1. Note that the index of the • Let the elements of the vector f be given by f; = F(xi-1) elements of the vector f differ from the index of the knots. • Let the elements of the vector q be given by q; = S' (Xi-1) i = 1 ... n + 1. Note that the index of the elements of the vector q differ from the index of the knots.

Answer 1

g'lxo)= ko & g'(xn) =kn clamped conditions co (x) Noenen, g(x) = a (a) n,SULU I cut (a) ant snedn o Step 1 choose ko fkn & determine k,,K2,kz --- Knit using the linear system of ean Step-2 write down the cubic polynomials qj(x) • j = 0,1,2,3, n. making up the cubic spline glx) in the form 7(0) = 2; +a; (X-xj) + 01, (*- *;? + aj, (x-x;3 Where a = aj CN ;) = fj - 9; - q; (03) = kj ºj, = { }(x)) = f (tju f;) - 5 (504 +243) % any) pop (65-furdt Chris

9; (x) = ajo +a; (a- j) + a, (2-x; ²+ Qg (x-x;}}