91Ó°ÊÓ

(a) Consider the problem of cubic polynomial interpolation $$ p\left(x_{i}\right)=y_{i}, \quad i=0,1,2,3 $$ with \(\operatorname{deg}(p) \leq 3\) and \(x_{0}, x_{1}, x_{2}, x_{3}\) distinct. Convert the problem of finding \(p(x)\) to another problem involving the solution of a system of linear equations. Hint: Write $$ p(x)=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3} $$ and determine \(a_{0}, a_{1}, a_{2}\), and \(a_{3}\). Use the interpolation conditions to obtain equations involving \(a_{0}, \ldots, a_{3}\). (b) Expressing the system from (a) in the form (6.11), identify the matrix \(A\) and the vectors \(b\) and \(x\).

Consider the matrix $$ A=\left[\begin{array}{ccccc} 3 & 1 & 0 & \cdots & 0 \\ 1 & 3 & 1 & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & 1 & 3 & 1 \\ 0 & \cdots & 0 & 1 & 3 \end{array}\right] $$ and the vectors $$ b=\left[\begin{array}{c} 4 \\ 5 \\ 5 \\ \vdots \\ 5 \\ 4 \end{array}\right], \quad x=\left[\begin{array}{c} 1 \\ 1 \\ 1 \\ \vdots \\ 1 \\ 1 \end{array}\right] $$ Set up the linear system (6.2) of order \(n\) associated with \(A\) and \(b\). Verify that the given \(x\) solves this linear system.