Preliminary Exam, Numerical Analysis, August 2015

Numerical Integration

Find a formula of the form \[ \int _0^{2\pi } f(x)\,dx = A_1 f(0)+A_2 f(\pi ) \] that is exact for any function having the form \[ f(x)=c+d\cos (x). \]

Matrix Properties

Show that if a matrix \(B\in \mathbb {R}^{m\times m}\) is both upper triangular and orthogonal, then it is diagonal.

Singular Value Decomposition (SVD)

a) Consider \(A\in \mathbb {C}^{m\times n}\). Define what we mean by the singular value decomposition of \(A\).

b) Consider \(A\in \mathbb {C}^{m\times m}\) and show that \[ \lvert \det (A)\rvert =\prod _{i=1}^m\sigma _i, \] where \(\{\sigma _i\}\) are the singular values of \(A\).

Overdetermined Linear System

Solve the following overdetermined linear system in the sense of the least squares: \[ z+y=1,\qquad z-y=2,\qquad 4z+2y=4.8. \]

Interpolation

a) State the theorem about the existence and uniqueness of interpolating polynomial. Give a proof.

b) Let \(f(x)=5x^2+x+1\). Find the polynomial of degree 3 that interpolates the values of \(f\) at \(x=-1,0,1,2\).

Linear Multistep Methods

Define linear multistep method (give formula).

a) State necessary and sufficient condition for linear multistep method to be consistent.

b) State root condition for linear multistep method.

c) What can you say about consistency and stability properties of the method: \[ y_{n+1}-3y_n+2y_{n-1}=-h f(y_{n-1})? \]

Example of A-stable Linear Multistep Method

Give an example of A-stable linear multistep method. Justify your answer.

Upwind Scheme

Consider the advection equation \[ u_t-u_x=0,\qquad x_L<x<x_R,\qquad 0<t\le T, \] where \(u(x,0)=g(x)\), and \(u(x_R,t)=u_R(t)\) for \(t>0\).

a) Write the Upwind Scheme for this problem.

b) What is the stencil of the scheme? What is the CFL condition for this method?

c) Investigate the stability of the method using Von Neumann Stability Analysis.