Math 5070
N. Mackey
Review for Midterm
Disclaimer: This list is meant to be an approximate guide; please note that you are responsible
for all the material discussed in class and assigned for reading.
Bring a calculator (check batteries!), pencils, eraser and blank paper.
1. Precise defintions of continuity, convergence of a sequence.
2. Precise definitions of simple, double, and higher order roots of a function.
3. Be able to clearly state and explain the meaning of the Intermediate Value Theorem; Mean
Value Theorem; Taylor’s theorem (for one variable) with remainder;
4. Understand Taylor’s Theorem, and how it can be used to construct polynomial approximations, and how to use it bound the error in polynomial approximations to non-polynomial
functions.
5. Defintion of rates of convergence, asymptotic error constant. Meaning of linear convergence, quadratic convergence.
6. Describe the floating point, finite precision number system. Understand how numbers
representable in such a system are distributed on the real number line.
7. Derive relative and absolute accuracy of the four basic arithmetic operations, and understand the what the results mean.
8. The IEEE standard for floating point arithmetic contains some special ‘numbers’. What
are they? Give examples of situations which give rise to them. Understand the ‘rules of
arithmetic’ for them.
9. Discuss problems that arise when computing in finite precision arithmetic. Demonstrate
how expressions that are algebraically equivalent in exact arithmetic can behave quite
differently in finite arithmetic Be able to explain the difference.
10. We discussed numerical instability with an example involving a difference equation. Be
prepared to clearly explain the reasons for the instability.
11. Consider the following matlab-code. Assume the function f has been defined.
x = 1;
h = 10. ˆ (−(1:20));
y = (f(x+h)-f(x))./h;
Describe what the variable h contains. Describe what the variable y contains. What does
this code demonstrate? Explain why.
12. Be able to give a clear statement of the main problems addressed in the course so far:
root finding, solving systems of nonlinear equations, polynomial interpolation. In each
case, what is given and what is being found? Be able to give a coherent description of the
various solutions developed for each of these problems.
13. Clearly state and prove the fixed point theorem for a function of a real variable defined on
a closed interval.
14. Give a clear description of the bisection method for root-finding. Be able to write pseudocode for this method, explaining the issues that need to be addressed when writing code.
15. What is the rate of convergence of the bisection method, justify your answer.
16. Given a desired accuracy for the root, how can we predetermine the number of iterations
needed by the bisection method to attain the desired accuracy?
17. Know how to derive Newton’s method for approximating a real root of a real valued
function and be able to give a geometric explanation for the Newton iteration step. Be
prepared to run Newton on your calculator during the exam, and report the sequence of
iterates: display digits in groups of 4 or 5, separated by spaces, mark digits that are in
common between successive iterates, etc.
18. Understand and be able to state clearly under what situations Newton’s method converges.
How fast does it converge in various situations? Be prepared to clearly justify (i.e., prove)
convergence and rates of convergence of Newton’s Method for root finding.
19. What are some of the difficulties with Newton’s method? Describe how variations of
Newton’s Method try to address these difficulties.
20. Understand how the single variable Newton iteration is a special case of the multivariable
iteration. What is the correspondence between the expressions in the multivariable Newton
iteration and the expressions in the single variable Newton iteration? Know how to perform
multivariable Newton on a system of two non-linear equations in two unknowns.
21. Describe the following methods for root finding: inverse linear interpolation, secant method,
safeguarded Newton. Use figures and words. Which of these, if any, require a bracket at
each step?
22. Clearly describe how inverse quadratic interpolation is used for root-finding.
23. What is Newton’s form for the interpolating polynomial? What is Lagrange’s form? What
is the natural form? Do these forms yield different polynomials? What are the advantages
and disadvantages of each?
Know how to compute the interpolation polynomial (Newton and Lagrange) by hand,
given the interpolation points.
24. Understand and be able to clearly state the theorem on the error in polynomial interpolation. Graduate students should know how to prove this theorem.
25. What is Runge’s function, and what does it demonstrate?
26. What is the idea and motivation behind piece-wise polynomial interpolation?
27. Explain the hat or chapeau functions that are used in piece-wise linear interpolation.
28. What is the difference between Hermite interpolation and spline interpolation?
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