Problem 1. (15) Let xo, x1,... .Xn-1,xn define a set of intervals on the real line, where all intervals have the same length, h = z,-x1-1. It has been shown in class that the computation of the coefficients required for piecewise cubic spline interpolation requires the solution of a linear system of equations C1 C2 T2 rn-3 rn-2 where r (3/h) (f[x,, xt+i] - f[lxi-1,xi]). The matrix, denoted T, is tridiagonal with n - 2 rows and columns; in particular, all the entries except those on the diagonal and in the first subdiagonal and superdiagonal are 0. (This equation corresponds to the so-called "free-boundary" condition in which a-cn-1 =0.) a. Show that T = LLT where L is a triangular matrix with nonzero entries only on the diagonal and first subdiagonal b. Describe how to compute the solution vector c using the factors L and LT. This description should be accompanied by a pseudocode in MATLAB style that could be used to compute the solution. Problem 1. (15) Let xo, x1,... .Xn-1,xn define a set of intervals on the real line, where all intervals have the same length, h = z,-x1-1. It has been shown in class that the computation of the coefficients required for piecewise cubic spline interpolation requires the solution of a linear system of equations C1 C2 T2 rn-3 rn-2 where r (3/h) (f[x,, xt+i] - f[lxi-1,xi]). The matrix, denoted T, is tridiagonal with n - 2 rows and columns; in particular, all the entries except those on the diagonal and in the first subdiagonal and superdiagonal are 0. (This equation corresponds to the so-called "free-boundary" condition in which a-cn-1 =0.) a. Show that T = LLT where L is a triangular matrix with nonzero entries only on the diagonal and first subdiagonal b. Describe how to compute the solution vector c using the factors L and LT. This description should be accompanied by a pseudocode in MATLAB style that could be used to compute the solution