Preliminary Exam, Numerical Analysis, September 1996

Philosophy. Why are matrices important? Why do we multiply a matrix with a vector the way we do?

The QR Factorization. Let \(A\) be an \(m\times n\) matrix, where \(m\ge n\). What is the QR factorization of \(A\)? How do you compute it? What do you use it for?

Gaussian Elimination is LU Factorization. What is the LU factorization of a square matrix? How do you solve a linear system \(Ax=b\) by Gaussian Elimination? Show that for a general \(4\times 4\) system the matrix part of Gaussian Elimination is equivalent to computing the LU factorization of \(A\).

Gaussian Quadrature. What is Gaussian quadrature? How do you pick the weights and knots? Why?

Least Squares. Describe in general the problem of approximating a given function in the Least Squares sense by a linear combination of certain basis functions. Discuss the significance of orthogonal basis functions. If you have a basis that is not orthogonal, how can you obtain an orthogonal basis that spans the same space?

Interpolation. Suppose you are given an arbitrarily smooth function \(f\) of a single variable \(x\), and points \[ x_0<x_1<\cdots <x_n. \] Let \(p\) be the interpolating polynomial of degree \(n\) satisfying \[ p(x_i)=f(x_i),\qquad i=0,1,\ldots ,n. \] Derive an expression for the error \[ f(x)-p(x) \] for a general value of \(x\), in terms of the derivatives of \(f\).

Discretization and Convergence. What is a discretization? What does it mean for a discretization to be convergent? Give a precise definition for a differential equation problem and a discretization of your choice.

Local Truncation Error. What is the local truncation error of a numerical scheme? What does consistency mean? Give a specific and precise example.

Stability. What does stability of a numerical scheme mean? Give a specific example.

Convergence is equivalent to stability and convergence. Describe and prove a specific example for this fundamental result.