Exam
Preliminary Exam, Numerical Analysis, August 2017
Fall 2017 · 8 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
-
Question 1
Rank-One Perturbation of the Identity
If \(u\) and \(v\) are \(n\)-vectors, the matrix \(B=I+uv^*\) is known as a rank-one perturbation of the identity. Show that if \(B\) is nonsingular, then its inverse has the form \(B^{-1}=I+\beta uv^*\) for some scalar \(\beta \), and give an expression for \(\beta \). For what \(u\) and \(v\) is \(B\) singular? If it is singular, what is \(\operatorname {null}(B)\)?
-
Question 2
Properties via SVD
Prove that any matrix in \(\mathbb {C}^{m\times n}\) is the limit of a sequence of matrices of full rank. Use the 2-norm for your proof.
-
Question 3
**Numerical Integration** **a)** Establish a numerical integration formula of the form \[ \int_a^b f(x)\,dx \approx Af(a)+Bf'(b) \] that is accurate for polynomials of as high a degree as possible. **b)** Determine the quadrature formula of the form \[ \int_{-2}^{2} f(x)\,dx \approx A_0f(-1)+A_1f(0)+A_2f(1) \] that is accurate for polynomials of degree \(\le2\).
-
Question 4
**Interpolation** **a)** State the theorem about the existence and uniqueness of interpolating polynomial. Give a proof. **b)** Let \(f(x)=7x^2+2x+1\). Find the polynomial of degree 3 that interpolates the values of \(f\) at \(x=-1,0,1,2\).
-
Question 5
Unstable Multistep Method
Consider the numerical method \[ y_{p+1}=3y_p-2y_{p-1} +\frac {h}{2}\left (f(x_p,y_p)-3f(x_{p-1},y_{p-1})\right ), \qquad p\ge 1, \]
for the solution of the initial value problem \[ y'=f(x,y),\qquad y(x_0)=y_0. \] Illustrate with an example of a simple initial value problem that the above numerical scheme is unstable.
-
Question 6
**Linear Multistep Methods** **a)** Define linear multistep method (give formula). Give definition of the 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 \(\operatorname{Real}(\lambda)<0\).
-
Question 7
Heat Equation and Stability of the Scheme
Consider the implicit in time, Backward Euler method for the solution of the heat equation: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} - \frac {u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2} = f_m^{n+1}, \] \[ u_m^0=g_m,\qquad m=0,\pm 1,\pm 2,\ldots ,\qquad n=0,1,\ldots ,[T/\Delta t]-1. \] and investigate the stability of the scheme using the von Neumann analysis.
-
Question 8
**Upwind Scheme** Consider the advection equation \[ u_t-5u_x=0,\qquad x_L