Exam
Preliminary Exam, Numerical Analysis, August 2011
Fall 2011 · 8 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
**Hermitian Matrix** Let \(A\in\mathbb{C}^{m\times m}\) be hermitian: a) Prove that all eigenvalues of \(A\) are real. b) Prove that if \(x\) and \(y\) are eigenvectors corresponding to distinct eigenvalues, then \(x\) and \(y\) are orthogonal.
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Question 2
**Induced Matrix Norm** a) Give the definition of an induced matrix norm. b) Show that the Frobenius norm \(\|\cdot\|_F\) is not induced by any vector norm.
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Question 3
**Trapezoidal Rule** a) State the formula for the composite trapezoidal rule \(I_n(f)\) to approximate the integral \(I(f)=\int_a^b f(x)dx\) b) Bound the error in \(I_n(f)\) applied to the following integral: \[ \int_0^{\pi/2}\cos(x)dx \] (You do not need to derive the error formula.)
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Question 4
Numerical Quadrature
Show that there is no set of nodes \(x_1,x_2,\ldots ,x_n\) and coefficients \(\alpha _1,\alpha _2,\ldots ,\alpha _n\) such that the quadrature rule \[ \sum _{j=1}^n \alpha _j f(x_j) \] exactly equals to the integral \(\int _a^b f(x)w(x)dx\) for all polynomials \(f(x)\) of degree less than or equal to \(2n\). \(w(x)\) is the weight function.
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Question 5
Existence and Uniqueness of the Interpolating Polynomial
State the theorem about the existence and uniqueness of the interpolating polynomial.
Give the proof. (You can consider any proof of your choice.)
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Question 6
**Linear Multistep Methods** a) Define the linear multistep method (give formula). Explain what we mean by its region of absolute stability. b) Show that the region of absolute stability for the trapezoidal method is the set of all complex \(h\lambda\) with \(\operatorname{Real}(\lambda)<0\).
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Question 7
**CFL Condition** a) Explain what we mean by the CFL (Courant-Friedrichs-Lewy) Condition b) Obtain CFL condition for the upwind scheme (explicit) below: \[ u_{i,j+1}=(1-\lambda)u_{i,j}+\lambda u_{i-1,j}, \] where \(i=1,2,\ldots,N+1\) and \(j=0,1,2,\ldots,M-1\) \[ \lambda=\frac{a\Delta t}{h},\qquad \Delta t=T/M, \] and \(a\) is a positive constant.The initial condition is \(u_{i,0}=g_i\)
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Question 8
Stability of the Scheme
Using the von Neumann method investigate the stability of the implicit upwind scheme: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} - \frac {u_{m+1}^{n+1}-u_m^{n+1}}{h} = f_m^n, \] \[ u_m^0=g_m,\qquad m=0,\pm 1,\pm 2,\ldots ,\qquad n=0,1,\ldots ,[T/\Delta t]-1. \] Comment on the CFL condition for this scheme.