Exam
Preliminary Exam, Numerical Analysis, January 2018
Spring 2018 · 8 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Projections
Let \(P\in \mathbb {C}^{N\times N}\) be a projection matrix. Prove that \(\ker (P)\perp \operatorname {range}(P)\) if and only if \(P=P^*\). (I.e., you are being asked to justify the definition of an orthogonal projection matrix.)
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Question 2
Hadamard’s inequality
Let \(A\in \mathbb {C}^{N\times N}\) be comprised of columns \(a_i\in \mathbb {C}^N\), \(i=1,\ldots ,N\). Prove that \[ |\det A|\le \prod _{i=1}^{N}\|a_i\|_2. \] You may use, for example, a QR decomposition/Gram-Schmidt argument.
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Question 3
Quadrature
Consider the quadrature rule \[ \int _{-1}^{1} f(x)\,dx \approx w_{-1}f(-1)+w_0f(0)+w_1f(1)+w_0'f'(0). \] Compute weights \(w_{-1}\), \(w_0\), \(w_1\), and \(w_0'\) such that this quadrature rule is exact for polynomials up to degree 3.
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Question 4
Finite difference formulas
Given \(h>0\), compute weights \(w_j\) for the following 5-point finite difference formula for the third derivative \(f^{(3)}(x)\): \[ f^{(3)}(x)\approx \sum _{j=-2}^{2}w_j f(x+jh) \] \[ = w_{-2}f(x-2h)+w_{-1}f(x-h)+w_0f(x)+w_1f(x+h)+w_2f(x+2h). \] What is the order of accuracy of your formula?
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Question 5
**Absolute stability for ODEs** For the ODE \(y'=f(y)\), and given \(p>0\), a linear multistep scheme has the form \[ \sum_{j=0}^{p}a_jy_{n+j} + \Delta t\sum_{j=0}^{p}b_jf(y_{n+j})=0 \] where \(\Delta t\) is the timestep, \(a_p=1\), and \(y_n\) is the numerical approximation to \(y(t=n\Delta t)\), where the starting time is \(t=0\). **a.** Define the region of absolute stability for the above scheme. **b.** Compute and sketch the region of absolute stability for the backward (i.e., implicit) Euler scheme.
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Question 6
Trapezoid rule
For the ODE \(y'=f(t,y)\), the Trapezoid rule is \[ \frac {y_{n+1}-y_n}{\Delta t} = \frac {1}{2}\left (f(t_n,y_n)+f(t_{n+1},y_{n+1})\right ). \] Compute the region of absolute stability for the Trapezoid rule. Is this method \(A\)-stable?
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Question 7
Unconditionally stable scheme
For the one-dimensional heat equation, \[ u_t=u_{xx}, \] consider the Crank-Nicolson, central difference scheme \[ \frac {u_j^{n+1}-u_j^n}{\Delta t} = \frac {1}{2} \left ( \frac {u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{h^2} + \frac {u_{j+1}^{n}-2u_j^{n}+u_{j-1}^{n}}{h^2} \right ) \] where \(\Delta t\) and \(h\) are temporal and spatial stepsizes, respectively, and where \(u_j^n\) is the numerical approximation to \(u(x=jh,t=n\Delta t)\), with \(x_0=0\) and the initial time \(t=0\). Use von Neumann stability analysis to show that this scheme is unconditionally stable.
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Question 8
Central scheme
Consider the one-dimensional advection equation, \[ u_t=cu_x, \] where \(c\in \mathbb {R}\) is the wavespeed. Use von Neumann stability analysis to determine the values of \(c\) for which the following the central difference, forward Euler scheme is stable: \[ \frac {u_j^{n+1}-u_j^n}{\Delta t} = c\frac {u_{j+1}^{n}-u_{j-1}^{n}}{2h}. \]