Let / be defined on [a, b] and let the nodes a xq < x\ < xi = b he given. A quadratic

Chapter 3, Problem 34

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Let / be defined on [a, b] and let the nodes a xq < x\ < xi = b he given. A quadratic spline interpolating function S consists of the quadratic polynomial 5o(a) = no + boix - xq) + co(x - xq)2 on [xq, Xi] and the quadratic polynomial S\ (x) = , + Mx - X|) + C\(x - x,)- on [xi,x2],such that i. 5(xo) = f(xo), S(xi) = f(xi), and Sfe) = /fe), ii. 5eCl [xo,x2l. Show that conditions (i) and (ii) lead to five equations in the six unknowns Aq, cq, a\, h\, and C|. The problem is to decide what additional condition to impose to make the solution unique. Does the condition S e C 2 [xo, X2J lead to a meaningful solution?