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

A.4. FIRST MIDTERM, FALL 2004 149
• Construct the Newton form of the polynomial p(x) of lowest degree that interpolates
f(x) at these nodes.
• Suppose that these data were generated by the function f(x) = 40x 3 − 32x 2 − 6x + 15.
Find a numerical upper bound on the error |p(x) − f(x)| over the interval [−0.5, 0.5].
Your answer should be a number.
P14 (15 pnts) • Write out the iteration step of Newton’s Method for finding a zero of the
function f(x). Your iteration should look like
x k+1 = ??
• Write the Newton’s Method iteration for finding 3√ 25/4.
• Letting x 0 = 5 2 , find the x 1 and x 2 for your iteration. Note that 3√ 25/4 ≈ 1.842.
P15 (15 pnts) • Let x be some given number, and let f(·) be a given, “black box” function.
Using Taylor’s Theorem, derive a O (h) approximation to the derivative f ′ (x). Call your
approximation φ(h); the approximation should be defined in terms of f(·) evaluated at
some values, and should probably involve x and h.
• Let f(x) = x 2 + 1. Approximate f ′ (0) using your φ(h), for h = 1 2 .
• For the same f(x) and h, get a O ( h 2) approximation to f ′ (0) using the method of
Richardson’s Extrapolation; your answer should probably include a table like this:
A.4 First Midterm, Fall 2004
D(0, 0)
D(1, 0) D(1, 1)
[Definitions] Answer no more than two of the following.
P1 (12 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 (12 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.
P3 (12 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 =?? + ??
??
[Problems] Answer no more than four of the following.
P1 (14 pnts) Perform bisection on the function graphed below. Let the first interval be [0, 256].
Give the second, third, fourth, and fifth intervals.
P2 (14 pnts) Give a O ( h 4) approximation to sin (h + π/2) . Find a reasonable C such that the
truncation error is no greater than Ch 4 .
P3 (14 pnts) Use Newton’s Method to find a good approximation to 3√ 24. Use x 0 = 3. That is,
iteratively define a sequence with x 0 = 3, such that x k → 3√ 24.
P4 (14 pnts) One of the roots of x 2 − bx + c = 0 as found by the quadratic formula is subject
to subtractive cancellation when |b| ≫ |c| . Which root is it? Rewrite the expression for that
root to eliminate subtractive cancellation.