*Gerald and Wheatley*, Chapter 3- 3.2 Lagrangian Polynomials
- 3.3 Divided Differences
- 3.4 Evenly Spaced Data (Newton-Gregory Polynomial Form)

#### Topics

*CAUTION: These problems are just for practice and are not to be handed in. Some of these old exam problems and answers may not be relevant to your current course. Caveat Usor!*

### Practice Problems:

*In computational problems, use "EXAM PRECISION": Chop to 4 significant (4C) digits only when you write an intermediate or final answer down and continue calculations with those numbers recorded.*

- For problems in the Gerald and Wheatley text, try chapter 3 exercises (pp. 300-307, 5th ed.) #1,2,3,4,11,12,13,14,24,26,28,29,30,31,32.
- Find an approximation to the value f(0.25) using an interpolating
polynomial passing through the 3 points:
- (x(i),f(x(i))= (0.1,0.07972), (0.2,0.1591), (0.3,0.2376),

(Ans.: 0.1984)

- Inverse quadratic interpolation: Suppose the inverse function
x=g(y)=y*exp(y) is given instead of the direct function y=f(x).
- Using y1=1, y2=2, y3=3 find x1, x2, x3 (i.e. xi=x(i)=y(i)*exp(y(i))) to d=3 digits. (note: exp(x) on some calculators is keyed as "x,expx" while it is "x,inv,lnx" on others.)
- Use the (inverse quadratic) interpolating polynominal p2(x) through (xi,yi)=(x(i),y(i)) for i=1 to 3 from (a) to approximate y=f(10) (i.e. the solution of 10=y*exp(y)) to d=3 digits. {Remark: Inverse interpolation, here going from x=g(y) to y=f(x), makes it easier since calculating y=quadratic in x=10 is easier than solving x = quadratic in y when x =10.}

(Ans.: (a) x-vector=(2.71,14.7,60.2); (b) f(10) =1.64+or-0.01 depending on the chops.)

- Compute P_n(n+1), where P_n(x) = "P-sub-n-of-x", for n=0,1,2,3,4 if
P_n(k)=k/(k+1) for k=0 to n.
(Partial Ans.: P_2(3)=0.500 for n=2 only).

- How many multiplications/divisions and additions/subtractions as a
function of n are needed to compute the sum of x^k from k=0 to n by (a)
direct (slow) sum of powers counting exponentiations as equivalent
multiplications and (b) by Newton's (Horner's) rule of fast polynomial
evaluation?. (Hint: sum of k for k=1 to m is m*(m+1)/2).
(Ans.: mults. = n; adds. = n)

- Approximate v(2.738) using an interpolatory polynomial that best
fits the data,

x: 2.600 2.700 2.800 2.900

v: 6.815 7.944 9.299 10.93 .(Final Ans.: v(2.738)=(4c) 8.429 or 8.428 with extra intermediate chops).

- Approximate j3(5.137) by interpolating the data:

x: 5.000 5.200 5.600

j3(x): .3648 .3265 .2298(Final Ans.: j3(5.137)=(4c) .3392 or .3390 with extra chops).

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