Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

152 APPENDIX A. OLD EXAMS
A.6 Final Exam, Fall 2004
[Definitions] Answer all of the following.
P1 (10 pnts) Clearly state Taylor’s Theorem for f(x + h). Include all hypotheses, and state the
conditions on the variable that appears in the error term.
P2 (10 pnts) Write the iteration for Newton’s Method or the Secant Method for finding a root to
the equation f(x) = 0. Your solution should look something like:
x k+1 =?? + ??
??
P3 (10 pnts) Write the general form of the iterated solution to the problem Ax = b. Let Q be
your “splitting matrix,” and use the factor ω. Your solution should look something like:
??x (k) =??x (k−1) +?? or x (k) =??x (k−1) +??
For one of the following variants, describe what Q is, in relation to A : Richardson’s, Jacobi,
Gauss-Seidel.
P4 (10 pnts) Define what it means for the function f(x) ∈ F to be the least squares best approximant
to the data
x 0 x 1 . . . x n
y 0 y 1 . . . y n
[Problems] Answer all of the following.
P1 (10 pnts) Rewrite this system of higher order ODEs as a system of first order ODEs:
⎧
x ′′ (t) = x ′ (t) [x(t) + ty(t)]
y ′′ (t) = y ′ (t) [y(t) − tx(t)]
⎪⎨
x ′ (0) = 1
x(0) = 4
y ′ (0) = −3
⎪⎩
y(0) = −2
P2 (10 pnts) Consider the ODE: {
x ′ (t) = t (x + 1) 2
x(2) = 1/2
Use Euler’s Method to find an approximate value of x(2.1) using a single step.
P3 (10 pnts) Let φ(h) be some computable approximation to the quantity L such that
φ(h) = L + a 4 h 4 + a 8 h 8 + a 12 h 12 + . . .
Combine evaluations of φ(·) to devise a O ( h 8) approximation to L.
P4 (15 pnts) Consider the approximation
f ′ (x) ≈
9f(x + h) − 8f(x) − f(x − 3h)
12h
(a) Derive this approximation using Taylor’s Theorem.
(b) Assuming that f(x) has bounded derivatives, give the accuracy of the above approximation.
Your answer should be something like O ( h ?) .