Numerical Analysis Finalterm Exam

Numerical Analysis: Final Exam.

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1. Write the update equation for each following root finding algorithm. (If it is bracketing method,use the form of Xr = function in terms Of(XI.Xu, f(xt), f(xu)). If open method, use the form ofXi+l = function in terms of(Xi , f(Xi),'" ) (50pt)

(a) Secant method (W~ ~)

(b) False position (7}~}:1 ~)

(c) Bisection method (0 1~ ~ )(d) Newton-Raphson method

(e) Modified Secant method (-'T-~ -B W ~ ~ )

(b) '(\r -::-y1- f ('fJ ()("'-- ,,~)___ ~'fC'{:\I')-f(r~J

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Dec. 15, 2014 Chibum Lee1

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Numerical Analysis Final term Exam

2. Consider f(x) = x2 - ax + b for finding approximating solution.

(a) To find the solution of f(x) = 0, derive the recursive formula using Newton Raphson algorithm.(20pt)

(b) Using the formula, perform three iterations for x2 - 13x - 1 = 0 with initial value Xo = 5.What real value is the exact solution of f(x) = 07 Evaluate the true fractional relative errorEt during three iterations. (20pt)

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Dec. 15, 2014 2 Chibum Lee

Numerical Analysis Finalterm Exam

3. An experiment was performed to determine the spring force relation in Hooke's law:

F(l) = k(l - la).

The function F is the force required to stretch the spring to become length l, where the constant la

is the length of the unstretched spring. Suppose measurements are made of the length l, for applied

weights F(l) as given in the following table.

F(l) 2 4 6 8l 7 9 12.5 15.8

(a) Find the least squares approximation for k and la. (Hint: Perform the linear regression assum-

ing l as independent variable.) (40pt)n

(b) Compute the error Br = Let. (5pt)i=l

(c) Compute the coefficient of determination r2 = StStSr (5pt).

\.0\) F:(~)=~ (Q- )0) '= t::Q - \::Qo = Ol'~t- CAc

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Numerical Analysis Fin~l term Exam

1= 11(1 - x - 4x3)dx.-1

4. (a) Find a numerical integratioi rule of the form

.l: f(x)dx = af( -1) + (3f(O) + "If(l).

which is exact for polynomials of the highest possible degree. (30pt)

(b) Approximate the following integrals using the above formula. (10pt)

rc") ~ ~/d: (1'11~) cl~~;

1= Cb-a-) f(~)ttf(~ ++(6')I

6' =d' fC-') --i:4-t(o)-+ f( \)

6I '

~I

~\-') -T 1-',~(6)tf(\) ~I:3. I

6 +4 ~4

Dec. 15, 2014 4 Chibum Lee

Numerical AnalysisI

Finalterm Exam

5. (a) Derive the forward first derivative formula of order O(h2):

I'': . ) = oJ(xo + 2h) + f3f(xo + h) + 'Yf(xo)~ h '

and find cv, (3, and r (Hint: Taylor series expansion.) (35pt)

(b) Using above equation, compute the first derivative approximation of f(x) = x3 t 4x - 10 at

x = 0 with h = ~. CompuJe the analytically true value and determine the relative error Et.

I(15pt)

Dec. 15, 2014 5 Chibum Lee

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et(tJ)= -\ a r f-(-~) = -gci'6 44;f ( ~)=:~1815'0

tiro)=:: ~1. 8t7r +4( ~5.96'4}~3(-/O) s:

:2.)( --L ~r-g" \ 5'"J4-

VCf)~ 30~-t-t "'l. Y(Co)= 4A£b ==- ( :4- ~075 /' X {Ol) =- '3./3- r--- ~~

(b)

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Numerical Analysis Finllterm Exam

y(O) = 1

6. For the given initial-value problem

(a) Find the analytic solution and evaluate at y(I). (20pt)

(b) Using the explicit Euler method with h = 0.5, find the approximated y(I). ~ompute therelative error Et at t = 1 (20pt) I

(c) Using the Runge-Kutta method with h = 1, find the approximated y(l). Compute the relativeerror Et at t = 1. (30pt)

1cPi= 6(k1 + 2k2 + 2k3 + k4) with kl = f(ti, Yi)

1 1k2 = f(ti + 2h, Yi + 2k1hl)

1 1k3 = f(ti + "2h, Yi + "2k2h)

k4 = f(ti + h, Yi + k3h).

Dec. 15, 2014 6 Chibum Lee

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