Sample Midterm 2 Problems, Math 128A
Prof. John A. Strain
1. (a) Write down Newton’s method for finding the root cof a single scalar
equation f(x) = 0, and simplify it for the specific problem of finding the
positive cube root cof a number c3=a > 0. (I.e. write it in an algebraic
form similar to the iteration xn+1 = (1/2)(xn+a/xn) for finding √a.)
(b) Veify that for a= 27 and x0= 4, the error decreases after one Newton
step: |c−x1|<|c−x0|.
(c) Prove that xn→cas n→ ∞ if 1 < c < x0. (Hint: to save algebra, do
the general error estimate first, and specialize to the cube root at the last
minute.)
(d) Replace f0(xn) in Newton’s method by the forward difference approxima-
tion Dhf(x) = (f(x+h)−f(x))/h, for some small h(comparable to the error
en=c−xnat each step). Show that the resulting iteration still converges
quadratically (for a general function fwith f00 bounded) if h=O(|en|) as
n→ ∞.
(e) Would you expect floating-point error in computer evaluation of fto
ruin the conclusion of part (d)?
2. Let P(x) be the quadratic polynomial that interpolates f(x) = cos(x) at
x= 9.0, 9.1 and 9.2 (in radians).
(a) Write down the Lagrange and Newton formulas for P.
(b) Bound the relative error e(x) = |f(x)−P(x)|/|f(x)|on the interval
9.0≤x≤9.2.
(c) Bound the relative error e(x) = |f(x)−P(x)|/|f(x)|on the interval
9.09 ≤x≤9.11. Why is your bound so much smaller or larger than the
bound in (a)?
(d) State the general formula which gives the error on an interval [a, b], in
the degree-npolynomial Pwhich interpolates a smooth function fat n+ 1
points tjin [a, b]. Separate the error into three factors and explain why two
of them are inevitable.
(e) When doing polynomial interpolation of smooth functions, what freedom
do we have to minimize the worst-case error and how should we use it?
3. Let Sjk =p1/n sin(jkπ/n) for j, k = 1,...,n−1.
(a) Show that Sis an orthogonal matrix (and obviously symmetric).
(b) Given values fjfor j= 1,...,n−1, find a trigonometric sine polynomial
σ(t) =
n−1
X
k=1
gksin(kπt)
which interpolates the values fjat the nodes tj=jπ/n.
(c) How would you use the FFT to evaluate all the coefficients gkin O(nlog n)
work?
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