Chapter 37, From Lanczos to Gauss Quadrature Video Solutions, Numerical Linear Algebra

Problem 1

The standard recurrence relation for Legendre polynomials is
\[
P_{n}(x)=\frac{2 n-1}{n} x P_{n-1}(x)-\frac{n-1}{n} P_{n-2}(x)
\]
with initial values $P_{0}(x)=1, P_{1}(x)=x$
(a) Confirm that (37.12) gives the polynomials $P_{2}(x)$ and $P_{3}(x)$ of (7.11)
(b) $\operatorname{since}\left\{P_{n}(x)\right\}$ and $\left\{q_{n+1}(x)\right\}$ are normalized differently, (37.12) is not the same as the recurrence (37.5) with coefficients $(37.6) .$ Write down the two tridiagonal matrices corresponding to these formulas, and derive the relationship between them.
(c) Use the result of (b) to determine a formula for $q_{n+1}(1),$ or equivalently, for $\left\|P_{n}\right\|$

Chris Trentman

Numerade Educator

02:03

Problem 2

Show based on the definition of orthogonality that $q_{n+1}(x)$ has $n$ distinct zeros, all contained in the open interval $(-1,1) .$ (The fact that they are distinct also follows from Exercise $25.1,$ but here, use a direct argument.)

Raymond Matshanda

Numerade Educator

03:09

Problem 3

The problem of interpolating $n$ data values $\left\{y_{j}\right\}$ in $n$ distinct data points $\left\{x_{j}\right\}$ by a polynomial of degree $\leq n-1$ was expressed in (11.4) as a square Vandermonde linear system of equations.
(a) Prove that this Vandermonde matrix is nonsingular by arguing that if the interpolation problem has a solution, it must be unique.
(b) Write down the analogous system of equations implicit in Theorem 37.2 Using the result of (a), prove this theorem.

Lucas Finney

Numerade Educator

03:28

Problem 4

(a) Write a six-line MATLAB program that computes the nodes and weights for the $n$ -point Gauss-Legendre quadrature formula and applies these numbers to compute the approximate integral of the function $f$
(b) Taking $f(x)=e^{x}$ and $n=4,$ confirm the example in the text. Then plot $\left|I\left(e^{x}\right)-I_{n}\left(e^{x}\right)\right|$ on a log scale for $n=1,2, \ldots, 40$ and comment on the results.
(c) Produce a similar plot for $f(x)=e^{|x|},$ and comment.

Stanley Enemuo

Numerade Educator

01:19

Problem 5

The program of Exercise 37.4 computes the zeros of Legendre polynomials, also known as Legendre points in $[-1,1] .$ The zeros of Chebyshev polynomials, Chebyshev points, are given by the explicit formula $(36.9) .$ Perform a sequence of calculations to generate numbers and plots illustrating as elegantly as you can that in the limit $n \rightarrow \infty,$ both Legendre and Chebyshev points approach the limiting density distribution $\mu(x)=\pi^{-1}\left(1-x^{2}\right)^{-1 / 2}$ (in the notation of Exercise 36.2 ). Produce further plots and numbers to explore the question: how close are Legendre points to Chebyshev points for various values of $n ?$

Carson Merrill

Numerade Educator