Mathematics 171–Numerical Analysis I Review Sheet–Midterm 2

General Information

The second midterm will be given in class Thursday, November 19. This will be a 75 minute in-class exam. You may (and certainly should) use a hand calculator.

Topics to be Covered

The exam will cover the material from the first exam through the material on extrapolation from the discussion we did in class on Thursday, November 12. This includes the material from sections 2.5, 3.1, 3.2, 3.4, 4.1, 4.2 in Burden and Faires:

1. Aitken acceleration and Steffensen’s method for root finding.

2. Polynomial interpolation – the Lagrange, divided difference forms, and Newton forms for the interpolating polynomial of degree ≤ n for S = {(x_{0}, f(x_{0})), …, (x_{n}, f(x_{n}))}.

3. The error formula for interpolation.

4. Cubic spline interpolation (know the definition of a cubic spline and how to use them to approximate values of a function, its derivative, etc. I will not ask you to compute the coefficients in a cubic spline, though. As you probably realize, that could take the whole 75 minutes by hand!)

5. Numerical differentiation methods, their error terms, Richardson Extrapolation.

Proofs to Know

The following are somewhat loose statements of the major theoretical points in the material we have studied. You should review and know exact statements and proofs of these results.

1. Given any set S = {(x_{0}, f(x_{0})), …, (x_{n}, f(x_{n}))} with x_{i} distinct, there exists a unique polynomial of degree ≤ n interpolating S.

2. The error formula for interpolation:

f(x) = p_{n}(x) + \frac{f^{(n+1)}(ξ)}{(n + 1)!}(x − x_{0})(x − x_{1}) ⋯ (x − x_{n})

for some ξ.

Review Problems

From Burden and Faires:

Section 2.5: 7, 8, 11, 14, 15 Section 3.1: Problems like 3 and 7, 15 where the interpolation points (x_{i}, f(x_{i})) come from functions where you can compute and bound the derivatives needed for the error formula (e.g. by graphing), 17 Section 3.2: 1,2,6,7,10,13 Section 3.4: 7,8,9,10, problems like 11,12 if you are given the spline function. Section 4.1: 1,2,16,21,22 Section 4.2: 5, 8, 9

Review Session

Either Tuesday or Wednesday evenings next week are possible. We can set this up in class November 12.