| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
IB |
62 |
IB |
2012 |
Numerical Analysis |
|
Paper 2, Section II, D |
2012 |
Let $\left{P_{n}\right}_{n=0}^{\infty}$ be the sequence of monic polynomials of degree $n$ orthogonal on the interval $[-1,1]$ with respect to the weight function $w$.
Prove that each $P_{n}$ has $n$ distinct zeros in the interval $(-1,1)$.
Let $P_{0}(x)=1, P_{1}(x)=x-a_{1}$, and let $P_{n}$ satisfy the following three-term recurrence relation:
$$P_{n}(x)=\left(x-a_{n}\right) P_{n-1}(x)-b_{n}^{2} P_{n-2}(x), \quad n \geqslant 2$$
Set
$$A_{n}=\left[\begin{array}{ccccc}
a_{1} & b_{2} & 0 & \cdots & 0 \\
b_{2} & a_{2} & b_{3} & \ddots & \vdots \\
0 & \ddots & \ddots & \ddots & 0 \\
\vdots & \ddots & b_{n-1} & a_{n-1} & b_{n} \\
0 & \cdots & 0 & b_{n} & a_{n}
\end{array}\right]$$
Prove that $P_{n}(x)=\operatorname{det}\left(x I-A_{n}\right), n \geqslant 1$, and deduce that all the eigenvalues of $A_{n}$ are distinct and reside in $(-1,1)$.