ExerciseF1Answer.docx
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ExerciseF1Answer.docx
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ExerciseF1Answer
Exercise-1forNumericalMethods
1.P23:
1
Solution:
(a)
(b)
2.P23:
6c.Convertthefollowingtoabinarynumbers:
(c)378
Solution:
378=101111010
3.P23:
9(b).
Solution:
(b)
4.P23:
13(b).
Solution:
1/10≈0.11012×2-3=0.0011012×2-1
1/3≈0.10112×2-1
1/10+1/3≈0.0011012×2-1+0.10112×2-1=0.1110012×2-1
≈0.11102×2-1
1/5≈0.11012×2-2=0.011012×2-1
(1/10+1/3)+1/5≈0.11102×2-1+0.011012×2-1=1.01001×2-1
≈0.10102
E=[(1/10+1/3)+1/5]-0.10102≈0.633333333333333-0.625
≈0.008333333333333
Rx=0.008333333333333/0.633333333333333
=0.013157894736842≈1.3%
5.P37:
1.FindtheerrorExandrelativeerrorRx.Alsodeterminethenumberofsignificantdigitsintheapproximation:
(a)x=2.71828182,x’=2.7182;
(b)y=98350,y’=98000;
(c)z=0.000068,z’=0.00006.
Solution:
(a)x=2.71828182,x’=2.7182;
Ex=0.00008182
Rx=0.0000300998
d=5
(b)y=98350,y’=98000;
Ex=350
Rx=0.003558718
d=3
(c)z=0.000068,z’=0.00006.
Ex=0.000008
Rx=0.117647
d=1
6.P37:
2.
Solution:
Truncationerrorispresentinthissituation.
Theansweris
=0.255307442801339
Ex=1.779866071194292e-008
Rx=Ex/p=6.971461260910335e-008=0.06971461260910335e-006
<0.5*101-7
So,thenumberofsignificantdigitsd=7.
7.P37:
4a.
Solution:
Bothtruncationerrorandlostofsignificancearepresentinthissituation.
F=0.707102999997211
8.P37:
13(a,c).
Solution:
(a)
b<0
(ii)formulax2=1.000000000000000e-003
(i)formulax2=9.999999999763531e-004
(b)
b<0
.
(ii)formulax2=1.000000000000000e-005
(i)formulax2=1.000000338535756e-005
9.P39:
2
Solution:
n
xn
rn
pn
qn
0
1.0000000000
0.9940000000
1.0000000000
1.0000000000
1
0.5000000000
0.4970000000
0.4970000000
0.4970000000
2
0.2500000000
0.2485000000
0.2455000000
0.2425000000
3
0.1250000000
0.1242500000
0.1197500000
0.1092500000
4
0.0625000000
0.0621250000
0.0568750000
0.0306250000
5
0.0312500000
0.0310625000
0.0254375000
-0.0326875000
6
0.0156250000
0.0155312500
0.0097187500
-0.1123437500
7
0.0078125000
0.0077656250
0.0018593750
-0.2481718750
8
0.0039062500
0.0038828125
-0.0020703125
-0.5080859375
9
0.0019531250
0.0019414063
-0.0040351562
-1.0220429688
10
0.0009765625
0.0009707031
-0.0050175781
-2.0470214844
n
xn-rn
xn-pn
xn-qn
0
0.0060000000
0.0000000000
0.0000000000
1
0.0030000000
0.0030000000
0.0030000000
2
0.0015000000
0.0045000000
0.0075000000
3
0.0007500000
0.0052500000
0.0157500000
4
0.0003750000
0.0056250000
0.0318750000
5
0.0001875000
0.0058125000
0.0639375000
6
0.0000937500
0.0059062500
0.1279687500
7
0.0000468750
0.0059531250
0.2559843750
8
0.0000234375
0.0059765625
0.5119921875
9
0.0000117188
0.0059882812
1.0239960938
10
0.0000058594
0.0059941406
2.0479980469
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