MATH 211: FINAL PRACTICE
[fix errors!]
Note: Some of these problems you’ll need to reference the book. If these problems were
on the exam, many would include more information on the formulas sheet. Also these
aren’t all the same difficulty or length of time as those on the exam.
Note that it would also be useful to practice problems in the book. This problem list
shouldn’t be considered exhaustive, but is only a collection of sample problems.
Problem 1
Suppose we know f is smooth, and we wish to find a zero of f . We apply Newton’s
method, but observe very poor convergence. What could be the issue? How could the
algorithm be altered to fix this?
Problem 2
Suppose we have the data (1, 2), (2, 3.5), and (4, 6.1). Build the natural cubic spline
polynomial interpolation.
Problem 3
Suppose f(x) = x2. Use divided difference to find the Lagrange Polynomial interpolation
for the query points −2,−1, 0, 1, 2.
Problem 4
Consider f(x) = (|x|+ |1+x|) sin(x). Suppose you want to build a piece-wise polynomial
interpolation of f over [−2, 2], call it P (x), so that if there are N + 1 query points,
|f(x)− P (x)| ≤ C 1
N3
for some C > 0 independent of N . What kind of piecewise polynomial should be used?
What is a problem with finding this interpolation? How can you pick the query points to
maintain this convergence rate.
Date: today.
1
2 MATH 211: FINAL PRACTICE
Problem 5
Let Mf (x0) = f(x0 − h) 2f(x0) ⊕ f(x0 + h). Assume this generates δ numeric error,
i.e.
δ = Mf (x0)− [f(x0 − h)− 2f(x0) + f(x0 + h)].
Then asymptotically what is the error E(h) = |f ′′(x0)− 1h2Mf (x0)|? Find h in terms of δ
that minimizes the error. This is the Second Derivative Midpoint Formula.
Problem 6
(a) Suppose f ∈ C3([a, b]). Which method will give the best order of convergence,
Riemann sums over [a, b], trapezoidal method, simpson’s rule, or some other (n+1)-
point Newton-Cotes formula? If the latter, what n?
(b) Answer the same for f ∈ C5([a, b]).
Problem 7
Suppose f(x) = sin(x), and let xj = j/(3N) for j = 0, · · · 3N + 1. Let h = 1/N . Then if
Λ =
3h
8
N−1∑
k=0
(f(x3j) + 3f(x3j+1) + 3f(x3j+2) + f(x3j+3)),
Λ is the Simpson’s rule approximation to
∫ 1
0 f(x)dx.
(a) Find an error bound in terms of h for | ∫ 10 f(x)dx− Λ|.
(b) Right a similar formula as in Λ, but for approximating
∫ 1
0 f(x)dx using 5-point
closed Newton-Cotes formula (i.e. n = 4). Use 4N + 1 query points.
Problem 8
Suppose y : [0,∞)→ R. Suppose y′(t) = cos(y(t)), y(0) = 0.
(a) Show cos(y(t)) is Lipschitz.
(b) Is there a unique solution on t ∈ [0, 1]?
Problem 8
Show Euler-A’s method is symplectic for p, q ∈ R (so single particle in 1D).
Problem 9
Suppose y : [0,∞) → R. Let y(t) = f(y), y(0) = 0. |f ′(x)| ≤ 1 for all x. Suppose we
wish to use Euler’s to approximate y(T ) for some T > 0. Then let h = T/N be the step
size. Let ωi be the i
th Euler step.
Using Theorem 5.9, find a bound on the approximation |y(T ) − ωN |. If we want the
error to be ε, find N in terms of T to ensure this bound.
MATH 211: FINAL PRACTICE 3
Problem 10
Suppose y : [0,∞)→ R. Let y′ = f(y), y(0) = 0. Consider the algorithm
ω0 = 0
ωi = ωi−1 + hφ(ωi−1).
Find the φ in terms of derivatives of f such that this is the second order Taylor’s Method.
Problem 11
This problem is connected to Problem 10. Suppose n > 0 is an integer. Then consider
the algorithm
ω0 = 0
k1 = α1f(ωi−1)
k2 = α2f(ωi−1 + β1k1)
· · ·
kn = αnf(ωi−1 + βnkn)
ωi = ωi−1 +
n∑
i=1
γiki.
Find n and αi, βi so that this Runge-Kutta scheme has the same order convergence as the
Taylor’s Method for problem 10.
Problem 12
Consider n× n symmetric matrix A with real entries. Suppose for all x ∈ Rn such that
x 6= 0 we have
xTAx > 0.
It is known that such a matrix is diagonalizable, A = PDP−1.
(a) Show the columns of P are eigenvectors.
(b) Show that any pair of eigenvectors of A with different eigenvalues are orthogonal.
(c) Suppose the columns of P are chosen so they all have ‖ · ‖2 norm of 1, and any
columns corresponding to the same eigenvalue are chosen so that they are orthog-
onal. Show P−1 = P T .
Problem 13
Suppose AB = BA, i.e. they commute. Show (AB)2 = A2B2.
4 MATH 211: FINAL PRACTICE
Problem 14
Consider the matrix
A =

0 0 −1 1
1 0 1 2
1 3 1 1
0 1 0 5.

(a) Find the LU decomposition in the form A = P TLU . I.e. find P , L, and U satisfying
this such that P is a permutation, L is lower triangular, and U is upper triangular.
(b) Let e1 = (1, 0, 0, 0)
T . Find A−1e1.
Problem 15
Consider the points x0 < x1 < · · ·xn. Assume xi = ih for h = N−1. Then let f, g ∈
C([0, 1]). The Reimann sum for a general function φ can be written
RN (φ) = h
N−1∑
i=0
f(xi).
Show using Cauchy Schwarz that
|RN (fg)| ≤
√
RN (f2)RN (g2).
Conclude that ∣∣ ∫ 1
0
f(x)g(x)dx
∣∣≤ (∫ 1
0
f(x)2dx
)1/2(∫ 1
0
g(x)2dx
)1/2
.
This is Cauchy Schwarz for the integral inner product 〈f, g〉 = ∫ 10 f(x)g(x)dx and the
continuous L2 norm ‖f‖2 =
(∫ 1
0 f(x)
2dx
)1/2
.
Problem 16
Suppose A is symmetric, and has eigenvalues −5, 2, and 4. What is ‖A‖2?
Problem 17
Suppose L, U , and D are lower triangular, upper triangular, and diagonal respectively,
and A = D − L− U . All entries of these matrices are zero except for
Li−1,i = −1 for 2 ≤ i ≤ n
Ui,i−1 = −1 for 2 ≤ i ≤ n
Dii = 5 for 1 ≤ i ≤ n.
Suppose A = D − L − U . Then the SOR algorithm for solving Ax = b is xn = Txn−1 +
(D − L)−1b, where x0 is arbitrary, and T = (D − L)−1U .
(a) Find a bound for ‖T‖2.
MATH 211: FINAL PRACTICE 5
(b) Using part (a), find a bound for ‖xn − x0‖2. If we plot x vs log(‖xn − x0‖2), we
should see approximately a line. What slope will it have?

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