- A 1 1 Let T F 1 0 0 11 Denote A Subdivision Of 1 1 Into Two Equal Subintervals And Consider The Space Of Li 1 (62.64 KiB) Viewed 18 times
{a = 1 1. Let T = {f-1,0),(0,11} denote a subdivision of (-1,1) into two equal subintervals and consider the space of linear splines S = P(T) (-1,1]). Consider the following piecewise linear polynomials defined on the subdivision 1: (1 - ), Tel-1,0), (1) = 32 2 1 € 0,11, 02(2) LI, 7€-1,1], 1€ |-1,0), 03(2) (1+0), IE [0,1. (a) Use Lemma 4.1 from the notes to find the dimension of S. (b) Show that the set {(), 02(1), 63()} is linearly independent. Explain why it is a basis set for S. (c) Let a = -1,12 = 0,13 - 1 and let 1 - f(2)=|-+|+|-| * + Write down the Gram matrix G associated with the Lagrange interpolation conditions at 11, 12, 13, using the basis {01(), 02(1), 3(1)}. Use G to find the piecewise linear polynomial interpolant p(x) of f(1) at the nodes 11, 12, 13- You should write p(x) in the form p(x) = {9(7), 1 €1-1,0), 1 tr(x), IE [0, 11, for some linear polynomials (2), r(). (d) Find a new basis {41(1), 42(1), %3()} of S such that :(1;) = dj, 2, j = 1,2,3. Write down ;(1) as piecewise linear polynomials (i.e., in a form similar to that of p(x) above). (c) Using the basis {41(2), 32(1), 43()}, derive a Lagrange interpolatory quadra- ture rule for approximating 10) - L163 L adr. f(x)d.