1. Let T = {(-1,0], 0,1]} denote a subdivision of (-1,1) into two equal subintervals and consider the space of linear sp

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1 Let T 1 0 0 1 Denote A Subdivision Of 1 1 Into Two Equal Subintervals And Consider The Space Of Linear Sp 1 (96.8 KiB) Viewed 27 times

1. Let T = {(-1,0], 0,1]} denote a subdivision of (-1,1) into two equal subintervals and consider the space of linear splines S = P.(T) n°((-1,1]). Consider the following piecewise linear polynomials defined on the subdivision T: S (1 (1-), re[-1,0), 01(x) = I € (0,1), 02(x) = 1 = 1-3 kall, 1€ (-1,1), , 03(2) A re(-1,0), (1+x), re [0, 1]. (a) Use Lemma 4.1 from the notes to find the dimension of S. (b) Show that the set {01(1), 02(2), 03(x)} is linearly independent. Explain why it is a basis set for S. (c) Let x1 = -1, x2 = 0,13 = 1 and let 1 f(0) = 2 + + T- 2 Write down the Gram matrix G associated with the Lagrange interpolation conditions at 11, 12, 13, using the basis {01(1), 02(2), 03(1)}. Use G to find the piecewise linear polynomial interpolant p() of f(2) at the nodes 11, 12, 13. You should write p(x) in the form $9(x), x€ (-1,0), p(x) = 1r(x), 1€ [0,1], ) for some linear polynomials (2),r(2). (d) Find a new basis {V1(2), 12(r), 03(r)} of such that ViXj) = dij,i, j = 1, 2, 3. Write down ;(2) as piecewise linear polynomials (i.e., in a form similar to that of p(2) above). (e) Using the basis {V1(1), 32(2), 03(x)}, derive a Lagrange interpolatory quadra- ture rule for approximating I() = Ls f(x)dr.