SınavHazırlıkSoruları
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7/28/2019 SnavHazrlkSorular
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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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7/28/2019 SnavHazrlkSorular
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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