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MATH 451-1 FINAL EXAMINATION
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- (10 points). Use the Newton’s method to solve the nonlinear equation
y =
3 + ln(y) 2
Write down the formula of the Newton’s method and perform just one iteration start- ing from the initial guess y 0 = 1. (If you wonder why I write the unknown as y instead of the usual x, the expla- nation is that this equation will arise if one applies the implicit scheme in problem 7 to the ODE y′^ = x^2 + ln(y), y(0) = 1, with h = 1.)
- (10 points). Consider the following data
x 1 3 / 2 0 2 f (x) 3 13 / 4 3 5 / 3
- Construct the triangular table of divided differences.
- Write down the Newton’s interpolation polynomial (no need to simplify).
- (12 points). Consider the integral ∫ (^1)
0
1 + x
dx
- Obtain an error bound for the trapezoid rule with 8 subdivisions.
- According to the error estimate formula, at least how many subdivisions do we need in order for the trapezoid rule to be accurate within an error smaller than 0.001? Hint: the error estimate formula for the trapezoid rule is
|error| =
h^2 12
(b − a)|f ′′(ξ)|
- (12 points). Rewrite the following second order ODE in the canonical form of a first order ODE system.
y′′^ = ln(y′) + y/ 2 y(1) = 0 y′(1) = 1
Apply the Euler method to the resulting ODE system. Perform just one iteration with the step size h = 0.5.
- (12 points). Consider the following two schemes for solving the ODE y′^ = f (x, y): an implicit scheme,
yk+1 = yk +
f (xk , yk) + f (xk+1, yk+1) 2
h
and a predictor-corrector scheme,
{ y¯k+1 = yk + f (xk , yk)h yk+1 = yk + f^ (xk,yk^ )+f 2 ( xk+1,y¯k+1)h
Apply the two schemes to the following ODE { y′^ = x^2 + y/ 2 y(0) = 1
Perform just one iteration for each scheme, using the step size h = 1.
- (10 points). Find the LU decomposition of
Show all the intermediate steps.
