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Exam Preparation Questions1.a. For the following data, create the forwarddifference table and find the Gregory-Newtonfunction. Calculate the value of function at x=-0.5and x=1.5 Use four decimal digits arithmetic.

ix -1 0 1 2 3

if 10 0 -2 -3.75 4

1.b. For the data given in1.a., find the Newtoninterpolation function. Calculate the value of functionat x=-0.5 and x=1.5 Use four decimal digitsarithmetic.

2. Compute the following integral using Simpsonsrule with n=4. Use six decimal digits arithmetic.

4

34

3

16

45dt

t

tt

3. Compute the following integral using Trapezoidalrule with n=4. Use six decimal digits arithmetic.

8

23 24

dxx

x

4.a. Derive the general least square formula for asecond order function in the form of

322

1 axaxay for n data set consisting of

ix and iy .Note: Derivation can be made in two ways:

I. First, write the error term and set the derivative ofit according to 1a , 2a and 3a to zero to find 1a , 2a

and 3a .

II . Using Vandormonde matrix as taught in the class.

4.b. Fit a quadratic polynomial to the following data,(use the formula derived in 4.a).

ix 0 1 2 3

iy -1 0 3 8

4.c. Fit a quadratic polynomial to the following data,(use the formula derived in 4.a). Use two decimaldigits arithmetic.

ix 0 2 3 5 6

iy -3 -1 4 7 10

4.d. Find the Newton interpolation function for thedata given in 4.b.

4.e. Fit a linear polynomial to the following data,

(make use of the formula derived in 4.a by removingfirst row and first column). Use two decimal digitsarithmetic.

ix 3 5 6 9 11

iy 16 23 26.5 37 44

5.a. Find y value for x=0 using Lagrangeinterpolation. Use four decimal digits arithmetic.

ix -3 -2 1 3 6

iy -70 -19 2 26 317

5.b. Find y value for x=7 using Lagrangeinterpolation. Use four decimal digits arithmetic.

ix 2 4 6 8 10

iy 1 1.8 2.1 3.5 4

6.a. Derive the general least square formula for an

exponential function in the form ofxaexay 1

2 forn data set consisting of ix and iy . Note: First write

the error term, let 11 lnat and set the derivative of it

according to 1t and 2a to zero to find 1t and 2a , then

1a .

6.b. Fit a function in the form ofxaexay 1

2 . (By the formula you derived in 6.a.).Use three decimal digits arithmetic.

ix 0.5 1 2 4

iy 0.0625 0.5 4 32

7.a. For the following data, create the forwarddifference table and calculate the first, second andthird numerical derivatives at x=0.

ix 0 0.2 0.4 0.6 0.8 1

if 0 0.5 0.8 0.4 0 0.2

7.b. Find the Gregory-Newton function and computethe value of function at x=0.3 Use six decimal digitsarithmetic.

8.a. Compute dxex 19

0

using Simpsons 3/8th rule

with n=9. If the exact result of the integration is178.957474, calculate your error percentage. Use sixdecimal digits arithmetic.8.b. Compute the integral in 8.a. using Rombergsmethod with n=4.Use six decimal digits arithmetic.

9. For the following data, find the Vandermondeinterpolation function.

ix -5 3 8iy 98 10 150

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Formula to be given in the exam:

n

kik ki

ki

xx

xxx

)(0

)(

n

iii xyxpxf

0

)()()(

nnb

a

b

a

ppppph

xpxf )..(22)()( 1210

3210 )(38

3)()( pppp

hxpxf

b

a

b

a

210 43

)()( ppph

xpxfb

a

b

a

n

i

i fi

sfxf

100)(

h

xxs 0

!

)1)..(2)(1(

i

issss

i

s

r

ir

r

h

fxf )()(

1

1

1

1

21

222

12

121

11

nnnn

n

n

xxx

xxx

xxx

V

nnnn RRR

RR

R

21

2221

11

nn

b

a

Rxf ,)( 12

kk

abh 1)5.0( ki hiap

)()(21

11 bfafhR

2..n

2,

2)(

2

1,1

1

11,

2

kk

ii

kk

Rpf

hR

k

2..n2..n,,14

41-j

1,11,1-j

,

jijijiji

RRR

Answers:1.a. p(-0.5)=2.8515 and p(1.5)=-3.08601.b. p(x)=10-10(x+1)+4(x+1)x-1.2917(x+1)x(x-1)+0.7084(x+1)x(x-1)(x-2)p(-0.5)=2.8515 p(1.5)=-3.0860

2. Simpsons integral value=1.5217493. Trapezoidal rule integral=9.450104

4.a.

i

ii

ii

ii

iii

iii

y

xy

xy

a

a

a

nxx

xxx

xxx 2

3

2

1

2

23

234

4.b. y=x2-14.c. y=0.09x2+1.67x-3.274.d. y=x2-14.e. y=3.5x+5.5

5.a. f(0)=5.01295.b. f(7)=2.70306.a.

iiiii

iii

xyxxaxt

xyxant

lnln

lnln

221

21

6.b. xexy 109.1125.0

7.a. 5.2)0( f 5)0( f 5.62)0( f

7.b. f(0.3)=0.749609

8.a. 088285.17919

0

dxex ,the error is

0.073096%8.b.9. c={3 -5 -2}T p(x)=3x2-5x-2