ExerciseF1Answer.docx

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