Mathematics 172 – Numerical Analysis 2 Review Sheet – Midterm Exam February 25, 1999

General Information

As announced in the course syllabus, the midterm exam for this course will be given in class on Thursday, March 4. You will have the full class period to work on the exam. The format of the exam will be similar to that of the inclass exams last semester – four or five problems, each with several parts. The material to be covered includes the following topics, through the material on Gaussian Elimination from class on February 23 (and Lab 3). Specifically:

Topics To Be Covered

The exam will cover the following material:

1) Adaptive Quadrature (know the error estimate that is the basis of the algorithm and how to apply the algorithm in relatively simple cases like integrals in problems 1,2 from section 4.6).

2) Gaussian Quadrature (know the theory behind the Gaussian Quadrature method – see the Theorems to Know A below – and how to apply it to approximate a definite integral if you are given a table of the roots x_{n,i} and the weights c_{n,i}. I will not ask you to compute roots or weights as we did in Lab 1, though.)

3) Methods for ODE initial value problems: Euler’s Method, the Modified Euler and Midpoint Methods, the Classical 4th order Runge-Kutta method. Be prepared to carry out several steps of each of these methods by hand on a reasonably simple equation. Also know how to estimate their one-step or local truncation errors.

4) Gaussian Elimination for linear systems Ax = b (be able to carry out the algorithm on a 3 × 3 or 4 × 4 system, with or without maximal column pivoting.

Theorems to Know

A) Show that if {p_{0}(x), …, p_{n}(x)} are polynomials orthogonal with respect to the inner product

⟨f, g⟩ = ∫_{a}^{b} f(x)g(x) dx,

with deg(p_{i}) = i all i, and such that p_{n}(x) = 0 has n real roots x_{1}, …, x_{n}, then with weights

c_{i} = ∫_{a}^{b} \frac{∏_{k≠i}(x − x_{k})}{∏_{k≠i}(x_{i} − x_{k})} dx,

the approximate integration rule

∫_{a}^{b} f(x) dx ≐ c_{1}f(x_{1}) + ⋯ + c_{n}f(x_{n})

has degree of precision (at least) 2n − 1. B) The Midpoint Method for the ODE IVP

\{ \begin{matrix}y′ = f(t, y) \\ y(t_{0}) = α \end{matrix}

has one-step error O(h^{3}) (or local truncation error O(h^{2}).

Some Review Problems

Section 4.6/1,2 Section 4.7/1,2,3,4 (given the information in the table on page 227), 6,8 (possible Extra Credit problem) section 5.1/7 (Picard iteration) Section 5.2/3,4, 5 a,b Section 5.4/1,3,10, 13 Section 6.2/Problems like 5, 7 (”partial pivoting” = ”maximal column pivoting”)

Additional Problem

A projectile of mass m = 0.11kg is shot vertically upward with initial velocity v(0) = 8m/sec, from an initial height of 6m. The projectile encounters the usual gravitational force F_{g} = −mg where g = 9.8m/sec^{2}, and air resistance F_{r} = 0.002v^{2}. Thus the differential equation for the velocity function of the projectile is:

0.11 v′(t) = −0.002(v(t))^{2} − 1.078

A) Use Euler’s Method to find the velocity of the projectile at t = 0.1, 0.2, …, 1.0 (step

size h = .1). B) Repeat A), but using the Runge-Kutta method of order 4. Compare your results.

Which should be closer to the true values, and why? C) From the Runge-Kutta values, interpolate to find an approximation to the time when

the projectile reaches its maximum height. Explain how and why you arrived at your answer.

Review Session

I will be happy to hold an afternoon or evening review session for the exam either Monday or Tuesday, March 1, or 2.