Post on 04-Jun-2018
transcript
8/13/2019 Chapter 4 BMCU 2072
1/140
Gaussian Elimination
8/13/2019 Chapter 4 BMCU 2072
2/140
8/13/2019 Chapter 4 BMCU 2072
3/140
!orward Elimination
=
2.279
2.177
8.106
112144
1864
1525
3
2
1
x
x
x
The (oal of forward elimination is to transform the$oeffi$ient matri) into an upper trian(ular matri)
=
735.0
21.96
8.106
7.000
56.18.40
1525
3
2
1
x
x
x
8/13/2019 Chapter 4 BMCU 2072
4/140
!orward Elimination* &tep 1
[ ] [ ]56.28.126456.21525 = [ ]
[ ]
[ ]56.18.40
56.28.1264
1864
112144
1864
1525
112144
56.18.40
1525
+ivide Equation 1 ', "- and
multipl, it ', ./0 56.225
64=
&u'tra$t the result from
Equation "
&u'stitute new equation for
Equation "
8/13/2019 Chapter 4 BMCU 2072
5/140
!orward Elimination* &tep 1 $ont2
[ ] [ ]76.58.2814476.51525 =[ ]
[ ]
[ ]76.48.160
76.58.28144
112144
76.48.160
56.18.40
1525
112144
56.18.40
1525
+ivide Equation 1 ', "- and
multipl, it ', 1//0 76.525
144=
&u'tra$t the result from
Equation 3
&u'stitute new equation for
Equation 3
8/13/2019 Chapter 4 BMCU 2072
6/140
!orward Elimination* &tep "
76.48.160
56.18.401525
[ ]( ) [ ]46.58.1605.356.18.40 =[ ]
[ ]
[ ]7.000
46.58.160
76.48.160
7.000
56.18.40
1525
+ivide Equation " ', 4/5
and multipl, it ', 41.50
5.38.4
8.16=
&u'tra$t the result from
Equation 3
&u'stitute new equation for
Equation 3
8/13/2019 Chapter 4 BMCU 2072
7/140
!orward Elimination
A set of nequations and nun%nowns
11313212111 ... bxaxaxaxa nn =++++
22323222121 ... bxaxaxaxa nn =++++
nnnnnnn bxaxaxaxa =++++ ...332211
n612 steps of forward elimination
8/13/2019 Chapter 4 BMCU 2072
8/140
!orward Elimination
Step 1!or Equation "0 divide Equation 1 ', and
multipl, ',
)...( 1131321211111
21 bxaxaxaxaa
ann =++++
111
21
111
21
21211
21
121
... ba
axa
a
axa
a
axa
nn
=+++
11a
21a
8/13/2019 Chapter 4 BMCU 2072
9/140
!orward Elimination
1
11
211
11
21212
11
21121 ... b
a
axa
a
axa
a
axa nn =+++
1
11
2121
11
212212
11
2122 ... b
a
abxa
a
aaxa
a
aa nnn =
++
'
2
'
22
'
22 ... bxaxa nn =++
22323222121 ... bxaxaxaxa nn =++++&u'tra$t the result from Equation "
47777777777777777777777777777777777777777777777777
or
8/13/2019 Chapter 4 BMCU 2072
10/140
!orward Elimination
8epeat this pro$edure for the remainin(equations to redu$e the set of equations as
11313212111 ... bxaxaxaxa nn =++++'
2
'
23
'
232
'
22 ... bxaxaxa nn =+++'
3
'
33
'
332
'
32 ... bxaxaxa nn =+++
''
3
'
32
'
2 ... nnnnnn bxaxaxa =+++
End of Step 1
8/13/2019 Chapter 4 BMCU 2072
11/140
Step 28epeat the same pro$edure for the 3rdterm of
Equation 3
!orward Elimination
11313212111 ... bxaxaxaxa nn =++++
'
2
'
23
'
232
'
22 ... bxaxaxa nn =+++
"
3
"
33
"
33 ... bxaxa nn =++
""
3
"
3 ... nnnnn bxaxa =++
End of Step 2
8/13/2019 Chapter 4 BMCU 2072
12/140
!orward Elimination
At the end of n612 !orward Elimination steps0 the s,stem
of equations will loo% li%e
'
2
'
23
'
232
'
22 ... bxaxaxa nn =+++"
3
"
33
"
33 ... bxaxa nn =++
( ) ( )11 = nnnn
nn bxa
11313212111 ... bxaxaxaxa nn =++++
End of Step (n-1)
8/13/2019 Chapter 4 BMCU 2072
13/140
9atri) !orm at End of !orward
Elimination
=
)(n-
n
"
'
n
)(n
nn
"n
"
'
n
''
n
b
b
b
b
x
x
x
x
a
aa
aaa
aaaa
1
3
2
1
3
2
1
1
333
22322
1131211
0000
00
0
8/13/2019 Chapter 4 BMCU 2072
14/140
!orward Elimination* &tep 1
[ ] [ ]56.28.126456.21525 = [ ]
[ ]
[ ]56.18.40
56.28.1264
1864
112144
1864
1525
112144
56.18.40
1525
+ivide Equation 1 ', "- and
multipl, it ', ./0 56.225
64=
&u'tra$t the result from
Equation "
&u'stitute new equation for
Equation "
8/13/2019 Chapter 4 BMCU 2072
15/140
!orward Elimination* &tep 1 $ont2
[ ] [ ]76.58.2814476.51525 =[ ]
[ ]
[ ]76.48.160
76.58.28144
112144
76.48.160
56.18.40
1525
112144
56.18.40
1525
+ivide Equation 1 ', "- and
multipl, it ', 1//0 76.525
144=
&u'tra$t the result from
Equation 3
&u'stitute new equation for
Equation 3
8/13/2019 Chapter 4 BMCU 2072
16/140
!orward Elimination* &tep "
76.48.160
56.18.401525
[ ]( ) [ ]46.58.1605.356.18.40 =[ ]
[ ]
[ ]7.000
46.58.160
76.48.160
7.000
56.18.40
1525
+ivide Equation " ', 4/5
and multipl, it ', 41.50
5.38.4
8.16=
&u'tra$t the result from
Equation 3
&u'stitute new equation for
Equation 3
8/13/2019 Chapter 4 BMCU 2072
17/140
#a$% &u'stitution
&olve ea$h equation startin( from the last equation
E)ample of a s,stem of 3 equations
=
735.0
21.96
8.106
7.00056.18.40
1525
3
2
1
xx
x
8/13/2019 Chapter 4 BMCU 2072
18/140
#a$% &u'stitution &tartin( Eqns
'
2
'
23
'
232
'
22 ... bxaxaxa nn =+++
"
3
"
3
"
33
... bxaxann
=++
( ) ( )11 = n
nn
n
nn bxa
11313212111 ... bxaxaxaxa nn =++++
8/13/2019 Chapter 4 BMCU 2072
19/140
#a$% &u'stitution
&tart with the last equation 'e$ause it has onl, one un%nown
)1(
)1(
=n
nn
n
n
na
bx
8/13/2019 Chapter 4 BMCU 2072
20/140
#a$% &u'stitution
( ) ( )
( ) 1,...,1for
1
1
11
=
=+=
nia
xab
xi
ii
n
ij j
i
ij
i
i
i
)1(
)1(
=
n
nn
nnn
a
bx
( ) ( ) ( ) ( )
( ) 1,...,1for...
1
1
,2
1
2,1
1
1,
1
==
+
++
+
nia
xaxaxabxi
ii
n
i
nii
i
iii
i
ii
i
ii
8/13/2019 Chapter 4 BMCU 2072
21/140
E)ample* Thermal Coeffi$ient
A trunnion of diameter 1"3.3:
has to 'e $ooled from a room
temperature of 5;
The equation that (ives the
diametri$ $ontra$tion0 +0 of the
trunnion in dr,6i$e>al$ohol
mi)ture 'oilin( temperature is
41;5
slid throu(h the hu' after$ontra$tin(
8/13/2019 Chapter 4 BMCU 2072
22/140
E)ample* Thermal Coeffi$ient
The e)pression for the thermal e)pansion $oeffi$ient0
is o'tained usin( re(ression anal,sis and hen$e solvin( the followin(
simultaneous linear equations*
!ind the values of usin( Nave Gauss Elimination
=
567992
10041621
100571
102435751086472110267
10864721102672860
102672860242
4
3
2
1
1085
85
5
.
.
.
a
a
a
...
..
.
2
321 TaTaa ++=
321 and,, aaa
8/13/2019 Chapter 4 BMCU 2072
23/140
E)ample* Thermal Coeffi$ient
!orward Elimination* &tep 1
?ields
( )=
2860
24
12
RowRow
=
567992
1017972
100571
1024351086472110267
109957910851830
102672860243
4
3
2
1
1085
75
5
.
.
.
a
a
a
...
..
.
8/13/2019 Chapter 4 BMCU 2072
24/140
8/13/2019 Chapter 4 BMCU 2072
25/140
8/13/2019 Chapter 4 BMCU 2072
26/140
E)ample* Thermal Coeffi$ient
#a$% &u'stitution* &olve for a3usin( the third equation
11
9
2
3
2
3
9
1040661
1053494
1037886
10378861053494
=
=
=
.
.
.a
.a.
8/13/2019 Chapter 4 BMCU 2072
27/140
E)ample* Thermal Coeffi$ient
#a$% &u'stitution* &olve for a"usin( the se$ond equation
3
3
7
2
5 101797.2)109957.9(108518.3 =+ aa
( )
( ) ( )
9
5
1173
5
3
73
2
1000872
1085183
1040661109957.91017972
1085183109957.91017972
=
=
=
.
.
..
.a.a
8/13/2019 Chapter 4 BMCU 2072
28/140
E)ample* Thermal Coeffi$ient
#a$% &u'stitution*&olve for a1usin( the first equation
( ) 435
21 10057110267286024 =++ .a.aa
( )
( ) ( ) ( )
6
11594
3
5
2
4
1
100690524
10406611026710008722860100571
24
102672860100571
=
=
=
.
....
a.a.a
8/13/2019 Chapter 4 BMCU 2072
29/140
=
11
9
6
3
2
1
1040661
1000872
1006905
.
.
.
a
a
a
E)ample* Thermal Coeffi$ient
&olution*The solution ve$tor is
The pol,nomial that passes throu(h the three data
points is then*
( ) 2321 TaTaaT ++=
21196 104066110008721006905 T.T.. +=
8/13/2019 Chapter 4 BMCU 2072
30/140
Nave Gauss Elimination
@itfalls
8/13/2019 Chapter 4 BMCU 2072
31/140
@itfalls of Nave Gauss Elimination
@ossi'le division ', Bero
ar(e round6off errors
8/13/2019 Chapter 4 BMCU 2072
32/140
955
11326
3710
321
321
32
=+
=++=
xxx
xxx
xx
@itfallD1 +ivision ', Bero
=
9
11
3
515
326
7100
3
2
1
x
x
x
8/13/2019 Chapter 4 BMCU 2072
33/140
s division ', Bero an issue hereF
955
14356
1571012
321
321
321
=+
=++
=+
xxx
xxx
xxx
=
9
14
15
515
356
71012
3
2
1
x
x
x
8/13/2019 Chapter 4 BMCU 2072
34/140
s division ', Bero an issue hereF
?E&
28524
14356
1571012
321
321
321
=+
=++
=+
xxx
xxx
xxx
=
28
14
15
5124
356
71012
3
2
1
x
x
x
=
2
5.6
15
192112
5.600
71012
3
2
1
x
x
x
+ivision ', Bero is a possi'ilit, at an, step
of forward elimination
8/13/2019 Chapter 4 BMCU 2072
35/140
@itfallD" ar(e 8ound6off Errors
=
9
751.145
315
7249.23101520
3
2
1
x
x
x
E)a$t &olution
=
1
1
1
3
2
1
x
x
x
8/13/2019 Chapter 4 BMCU 2072
36/140
@itfallD" ar(e 8ound6off Errors
=
9
751.145
315
7249.23101520
3
2
1
x
x
x
&olve it on a $omputer usin( .si(nifi$ant di(its with $hoppin(
=
999995.0
05.1
9625.0
3
2
1
x
x
x
8/13/2019 Chapter 4 BMCU 2072
37/140
@itfallD" ar(e 8ound6off Errors
=
9
751.145
315
7249.23101520
3
2
1
x
x
x
&olve it on a $omputer usin( -si(nifi$ant di(its with $hoppin(
=
99995.0
5.1
625.0
3
2
1
x
x
x
s there a wa, to redu$e the round off errorF
8/13/2019 Chapter 4 BMCU 2072
38/140
Avoidin( @itfalls
n$rease the num'er of si(nifi$ant di(its
+e$reases round6off error
+oes not avoid division ', Bero
8/13/2019 Chapter 4 BMCU 2072
39/140
Avoidin( @itfalls
Gaussian Elimination with @artial @ivotin(
Avoids division ', Bero
8edu$es round off error
8/13/2019 Chapter 4 BMCU 2072
40/140
Gauss Elimination with@artial @ivotin(
8/13/2019 Chapter 4 BMCU 2072
41/140
hat is +ifferent A'out @artial
@ivotin(F
pka
At the 'e(innin( of the kthstep of forward elimination0
find the ma)imum of
nkkkkk aaa .......,,........., ,1+
f the ma)imum of the values is
in thep throw0 ,npk then swit$h rowspand k
8/13/2019 Chapter 4 BMCU 2072
42/140
9atri) !orm at #e(innin( of "nd
&tep of !orward Elimination
=
'
'3
2
1
3
2
1
''
4
'
32
'3
'3332
22322
1131211
0
0
0
n
'
nnnnn
'
n
n'
'
n
''
n
b
b
b
b
x
x
x
x
aaaa
aaa
aaa
aaaa
8/13/2019 Chapter 4 BMCU 2072
43/140
E)ample "ndstep of !E2
=
3
9
8
6
5
431112170
862390
1111240
21670
67.31.5146
5
4
3
2
1
x
x
x
x
x
hi$h two rows would ,ou swit$hF
8/13/2019 Chapter 4 BMCU 2072
44/140
E)ample "ndstep of !E2
=
6
9
8
3
5
21670
862390
1111240
431112170
67.31.5146
5
4
3
2
1
x
x
x
x
x
&wit$hed 8ows
8/13/2019 Chapter 4 BMCU 2072
45/140
Gaussian Elimination
with @artial @ivotin(
A method to solve simultaneous linear
equations of the form [A][X]=[C]
Two steps
1 !orward Elimination
" #a$% &u'stitution
8/13/2019 Chapter 4 BMCU 2072
46/140
!orward Elimination
&ame as nave Gauss elimination method
e)$ept that we swit$h rows 'efore eachof
the n612 steps of forward elimination
8/13/2019 Chapter 4 BMCU 2072
47/140
E)ample* 9atri) !orm at #e(innin(
of "nd
&tep of !orward Elimination
=
'
'
3
2
1
3
2
1
''4
'32
'
3
'
3332
22322
1131211
0
00
n
'
nnnnn'n
n
'
'
n
''
n
b
bb
b
x
xx
x
aaaa
aaaaaa
aaaa
8/13/2019 Chapter 4 BMCU 2072
48/140
9atri) !orm at End of !orward
Elimination
=
)(n-n
"
'
n)(n
nn
"
n
"
'
n
''
n
b
bb
b
x
xx
x
a
aaaaa
aaaa
1
3
2
1
3
2
1
1
333
22322
1131211
0000
000
8/13/2019 Chapter 4 BMCU 2072
49/140
#a$% &u'stitution &tartin( Eqns
'
2
'
23
'
232
'
22 ... bxaxaxa nn =+++
"
3
"
3
"
33 ... bxaxa nn =++
( ) ( )11 = n
nn
n
nn bxa
11313212111 ... bxaxaxaxa nn =++++
8/13/2019 Chapter 4 BMCU 2072
50/140
#a$% &u'stitution
( ) ( )
( ) 1,...,1for11
11
=
=
+=
nia
xab
xi
ii
n
ijj
iij
ii
i
)1(
)1(
=
n
nn
nn
na
bx
8/13/2019 Chapter 4 BMCU 2072
51/140
Gauss Elimination with
@artial @ivotin(E)ample 1
8/13/2019 Chapter 4 BMCU 2072
52/140
E)ample "
=
2279
2177
8106
112144
1864
1525
3
2
1
.
.
.
a
a
a
&olve the followin( set of equations
', Gaussian elimination with partialpivotin(
8/13/2019 Chapter 4 BMCU 2072
53/140
E)ample " Cont
=
2279
2177
8106
112144
1864
1525
3
2
1
.
.
.
a
a
a
1 !orward Elimination
" #a$% &u'stitution
2.279112144
2.1771864
8.1061525
8/13/2019 Chapter 4 BMCU 2072
54/140
!orward Elimination
8/13/2019 Chapter 4 BMCU 2072
55/140
Num'er of &teps of !orward
Elimination
Num'er of steps of forward elimination is
n12=312="
8/13/2019 Chapter 4 BMCU 2072
56/140
!orward Elimination* &tep 1 E)amine a'solute values of first $olumn0 first row
and 'elow
144,64,25
ar(est a'solute value is 1// and e)ists in row 3
&wit$h row 1 and row 3
8.1061525
2.1771864
2.279112144
2.279112144
2.1771864
8.1061525
8/13/2019 Chapter 4 BMCU 2072
57/140
!orward Elimination* &tep 1 $ont2
[ ] [ ]1.1244444.0333.599.634444.02.279112144
=
8.1061525
2.1771864
2.279112144
[ ]
[ ]
[ ]10.53.55560667.20
124.10.44445.33363.99
177.21864
8.1061525
10.535556.0667.20
2.279112144
+ivide Equation 1 ', 1// and
multipl, it ', ./0 4444.0144
64=
&u'tra$t the result from
Equation "
&u'stitute new equation for
Equation "
8/13/2019 Chapter 4 BMCU 2072
58/140
!orward Elimination* &tep 1 $ont2
[ ] [ ]47.481736.0083.200.251736.0279.2112144 =
[ ]
[ ]
[ ]33.588264.0917.20
48.470.17362.08325
106.81525
8.1061525
10.535556.0667.20
2.279112144
33.588264.0917.20
10.535556.0667.20
2.279112144
+ivide Equation 1 ', 1// and
multipl, it ', "-0 1736.0144
25=
&u'tra$t the result from
Equation 3
&u'stitute new equation for
Equation 3
8/13/2019 Chapter 4 BMCU 2072
59/140
!orward Elimination* &tep " E)amine a'solute values of se$ond $olumn0 se$ond row
and 'elow
2.917,667.2
ar(est a'solute value is "H1I and e)ists in row 3
&wit$h row " and row 3
10.535556.0667.20
33.588264.0917.20
2.279112144
33.588264.0917.20
10.535556.0667.20
2.279112144
! d Eli i ti &t " t 2
8/13/2019 Chapter 4 BMCU 2072
60/140
!orward Elimination* &tep " $ont2
[ ] [ ]33.537556.0667.209143.058.330.82642.9170 =
10.535556.0667.20
33.588264.0917.20
2.279112144
[ ]
[ ]
[ ]23.02.000
53.330.75562.6670
53.100.55562.6670
23.02.000
33.588264.0917.20
2.279112144
+ivide Equation " ', "H1I and
multipl, it ', "..I0.9143.0
917.2
667.2=
&u'tra$t the result from
Equation 3
&u'stitute new equation for
Equation 3
8/13/2019 Chapter 4 BMCU 2072
61/140
#a$% &u'stitution
8/13/2019 Chapter 4 BMCU 2072
62/140
#a$% &u'stitution
1.15
2.0
23.0
23.02.0
3
3
=
=
=
a
a
&olvin( for a3
=
2303358
2279
2000
8264091720
112144
23.02.000
33.588264.0917.20
2.279112144
3
2
1
.
.
.
a
a
a
.
..
8/13/2019 Chapter 4 BMCU 2072
63/140
#a$% &u'stitution $ont2
&olvin( for a2
6719.
917.2
15.18264.033.58917.2
8264.033.58
33.588264.0917.2
32
32
=
=
=
=+a
a
aa
=
23033582279
2000
8264091720112144
3
2
1
.
..
a
aa
.
..
8/13/2019 Chapter 4 BMCU 2072
64/140
#a$% &u'stitution $ont2
&olvin( for a1
2917.0
144
15.167.19122.279144
122.279
2.27912144
321
321
=
=
=
=++
aaa
aaa
=
230
33582279
2000
8264091720112144
3
2
1
.
..
a
aa
.
..
Gaussian Elimination with @artial
8/13/2019 Chapter 4 BMCU 2072
65/140
@ivotin( &olution
=
2279
2177
8106
112144
1864
1525
3
2
1
.
.
.
a
a
a
=
15.1
67.19
2917.0
3
2
1
a
a
a
23.02.000
33.588264.0917.20
2.279112144
Gaussian Elimination without
8/13/2019 Chapter 4 BMCU 2072
66/140
=
2.279
2.1778.106
112144
18641525
3
2
1
x
xx
=
735.0
21.96
8.106
7.000
56.18.40
1525
3
2
1
x
x
x
@artial @ivotin( &olution
=
05.1
7025.19
1705.8
3
2
1
a
a
a
8/13/2019 Chapter 4 BMCU 2072
67/140
Gaussian Elimination withPartial Pivoting Solution
Gaussian Elimination
without Partial Pivoting
Solution
=
05.1
7025.19
1705.8
3
2
1
a
a
a
=
15.1
67.19
2917.0
3
2
1
a
a
a
8/13/2019 Chapter 4 BMCU 2072
68/140
Gauss Elimination with
@artial @ivotin(E)ample "
8/13/2019 Chapter 4 BMCU 2072
69/140
@artial @ivotin(* E)ampleConsider the s,stem of equations
655
901.36099.23
7710
321
321
21
=+
=++
=
xxx
xxx
xx
n matri) form
515
6099.23
0710
3
2
1
x
x
x
6
901.3
7
=
&olve usin( Gaussian Elimination with @artial @ivotin( usin( five
si(nifi$ant di(its with $hoppin(
8/13/2019 Chapter 4 BMCU 2072
70/140
@artial @ivotin(* E)ample!orward Elimination* &tep 1
E)aminin( the values of the first $olumn
J1;J0 J63J0 and J-J or 1;0 30 and -
The lar(est a'solute value is 1;0 whi$h means0 to
follow the rules of @artial @ivotin(0 we swit$hrow1 with row1
=
6
901.3
7
515
6099.23
0710
3
2
1
x
x
x
=
5.2
001.6
7
55.20
6001.00
0710
3
2
1
x
x
x
@erformin( !orward Elimination
8/13/2019 Chapter 4 BMCU 2072
71/140
@artial @ivotin(* E)ample
!orward Elimination* &tep "
E)aminin( the values of the first $olumn
J6;;;1J and J"-J or ;;;;1 and "-
The lar(est a'solute value is "-0 so row " isswit$hed with row 3
=
5.2
001.67
55.20
6001.000710
3
2
1
x
xx
=
001.6
5.27
6001.00
55.200710
3
2
1
x
xx
@erformin( the row swap
8/13/2019 Chapter 4 BMCU 2072
72/140
@artial @ivotin(* E)ample
!orward Elimination* &tep "
@erformin( the !orward Elimination results in*
=
002.6
5.2
7
002.600
55.20
0710
3
2
1
x
x
x
8/13/2019 Chapter 4 BMCU 2072
73/140
@artial @ivotin(* E)ample
#a$% &u'stitution
&olvin( the equations throu(h 'a$% su'stitution
1
002.6
002.63 ==x
15.2
55.2 32 =
=
xx
010
077 321 =
+=
xxx
=
002.6
5.2
7
002.600
55.20
0710
3
2
1
x
x
x
8/13/2019 Chapter 4 BMCU 2072
74/140
@artial @ivotin(* E)ample
[ ]
=
=
1
1
0
3
2
1
x
x
x
X exact[ ]
=
=
1
1
0
3
2
1
x
x
x
X calculated
Compare the $al$ulated and e)a$t solution
The fa$t that the, are equal is $oin$iden$e0 'ut it
does illustrate the advanta(e of @artial @ivotin(
8/13/2019 Chapter 4 BMCU 2072
75/140
K +e$omposition
8/13/2019 Chapter 4 BMCU 2072
76/140
K +e$omposition
K +e$omposition is another method to solve a set of
simultaneous linear equations
hi$h is 'etter0 Gauss Elimination or K +e$ompositionF
To answer this0 a $loser loo% at K de$omposition isneeded
8/13/2019 Chapter 4 BMCU 2072
77/140
9ethod
!or most non6sin(ular matri) [A] that one $ould $ondu$t Nave Gauss
Elimination forward elimination steps0 one $an alwa,s write it as
[A] = [L][U]where
[L] = lower trian(ular matri)
[U] = upper trian(ular matri)
K +e$omposition
8/13/2019 Chapter 4 BMCU 2072
78/140
Low does K +e$omposition wor%F
8/13/2019 Chapter 4 BMCU 2072
79/140
K +e$omposition
Low $an this 'e usedF
Given [A][X] = [C]
1 +e$ompose [A] into [L] and[U]
" &olve [L][] = [C] for []
3 &olve [U][X] = [] for [X]
8/13/2019 Chapter 4 BMCU 2072
80/140
9ethod* [A] +e$ompose to [] and [K]
[ ] [ ][ ]
==
33
2322
131211
3231
21
00
0
1
01
001
u
uu
uuu
ULA
[U] is the same as the $oeffi$ient matri) at the end of the forward
elimination step
[L] is o'tained usin( the multipliersthat were used in the forwardelimination pro$ess
8/13/2019 Chapter 4 BMCU 2072
81/140
!indin( the [U] matri)Ksin( the !orward Elimination @ro$edure of Gauss Elimination
112144
1864
1525
( )
112144
56.18.40
1525
56.212;56.225
64
== RowRow
( )
76.48.160
56.18.40
1525
76.513;76.525
144
== RowRow
&tep 1*
8/13/2019 Chapter 4 BMCU 2072
82/140
!indin( the [K] 9atri)
&tep "*
76.48.160
56.18.40
1525
( )
7.000
56.18.40
1525
5.323;5.38.4
8.16
==
RowRow
[ ]
=
7.000
56.18.40
1525
U
9atri) after &tep 1*
8/13/2019 Chapter 4 BMCU 2072
83/140
!indin( the [L] matri)
Ksin( the multipliers used durin( the !orward Elimination @ro$edure
1
01
001
3231
21
56.225
64
11
2121 ===
a
a
76.525
144
11
31
31 ===a
a
!rom the first step
of forward
elimination 1121441864
1525
8/13/2019 Chapter 4 BMCU 2072
84/140
!indin( the [] 9atri)
[ ]
= 15.376.50156.2
001
L
!rom the se$ond
step of forward
elimination
76.48.160
56.18.40
1525
5.38.4
8.16
22
3232 =
==
a
a
8/13/2019 Chapter 4 BMCU 2072
85/140
+oes [][K] = [A]F
[ ] [ ] =
=7.000
56.18.40
1525
15.376.5
0156.2
001
UL F
E l Th l C ffi i t
8/13/2019 Chapter 4 BMCU 2072
86/140
E)ample* Thermal Coeffi$ient
A trunnion of diameter 1"3.3:
has to 'e $ooled from a room
temperature of 5;
The equation that (ives the
diametri$ $ontra$tion + of
the trunnion in dr,6
i$e>al$ohol 'oilin(
temperature is 41;5
throu(h the hu' after
$ontra$tin(
8/13/2019 Chapter 4 BMCU 2072
87/140
E)ample* Thermal Coeffi$ient
The e)pression for the thermal e)pansion $oeffi$ient0 a ! a"# a
$T # a
%T$
is o'tained usin( re(ression anal,sis and hen$e solvin( the followin(
simultaneous linear equations*
!ind the values of a"& a
$&and a
%usin( K +e$omposition
=
56799.2
1004162.1
10057.1
1024357.51086472.11026.7
1086472.11026.72860
1026.72860242
4
3
2
1
1085
85
5
a
a
a
8/13/2019 Chapter 4 BMCU 2072
88/140
E)ample* Thermal Coeffi$ient
Kse !orward Elimination to find the [K] matri)
1085
85
5
1024357.51086472.11026.7
1086472.11026.72860
1026.7286024
( )
102436.5108647.11026.7
109957.9108518.30
1026.7286024
17.11912;17.11924
2860
1085
75
5
==
RowRow
( )
100474.3109957.90
109957.9108518.30
1026.7286024
3025013;3025024
10267
107
75
5
5
==
RowRow.
&tep 1
8/13/2019 Chapter 4 BMCU 2072
89/140
E)ample* Thermal Coeffi$ient
( )
105349.400
109957.9108518.30
1026.7286024
50.25923;50.259108518.3
109957.9
9
75
5
5
7
==
RowRow
[ ]
=9
75
5
105349.400
109957.9108518.30
1026.7286024
U
100474.3109957.90
109957.9108518.30
1026.7286024
107
75
5
This is the matri)
after the 1ststep
&tep "
8/13/2019 Chapter 4 BMCU 2072
90/140
E)ample* Thermal Coeffi$ient
Kse the multipliers from !orward Elimination
1
01
001
3231
21
1711924
2860
11
2121 .
a
a=
==
3025024
10267 5
11
3131 =
== .
a
a
!rom the first step of forward elimination
1085
85
5
1024357.51086472.11026.71086472.11026.72860
1026.7286024
8/13/2019 Chapter 4 BMCU 2072
91/140
E)ample* Thermal Coeffi$ient
[ ]
=
150.25930250
0117.119
001
L
!rom the se$ond step of forward elimination
502591085183109957.9
5
7
22
3232 .
.aa ===
96
65
5
104742.3010957.990
10957.991085183.30
1026.7286024
8/13/2019 Chapter 4 BMCU 2072
92/140
E)ample* Thermal Coeffi$ient
+oes [][K] = [A]F
[ ] [ ] ?
105323.400
109957.9108518.30
1026.7286024
150.25930250
0117.119
001
9
75
5
=
=UL
8/13/2019 Chapter 4 BMCU 2072
93/140
E)ample* Thermal Coeffi$ient
&et [][M] = [C]
=
567992
10041621
100571
15025930250
0117119
0012
4
3
2
1
.
.
.
.
.
( ) 56799250259302501004162117119
10057.1
31
221
4
1
.'.'
.''.
'
=++=+
=
&olve for [M]
8/13/2019 Chapter 4 BMCU 2072
94/140
E)ample* Thermal Coeffi$ient
4
1 10057.1 ='
( )
( )
00217970
1005711711910041621
1711910041621
42
1
2
2
.
...
'..'
==
=
( )
( )
0637880
002179705025910057130250567992
5025930250567992
4
213
.
....
..
=
=
=
[ ]
=
=
0637880
00217970
100571 4
3
2
1
.
.
.
&olve for [M]
8/13/2019 Chapter 4 BMCU 2072
95/140
E)ample* Thermal Coeffi$ient
&et [K][A] = [M]
=
0637880
00217970
100571
105349400
109957.910851830
10267286024 4
3
2
1
9
75
5
.
.
.
a
a
a
.
.
.
( )
( )06378801053484
0021797.0109957.9108518.3
10057.11026.7286024
3
9
3
7
2
5
4
3
5
21
.a.
aa
aaa
=
=+
=++
&olve for A
The 3 equations 'e$ome
8/13/2019 Chapter 4 BMCU 2072
96/140
E)ample* Thermal Coeffi$ient
11
93
1040661
1053494
0637880
=
=
.
.
.a
( )
( ) ( )
9
5
117
5
3
7
2
1000872
1085183
1040661109957.900217970
1085183
109957.900217970
=
=
=
.
.
..
.
a.a
&olve for A
8/13/2019 Chapter 4 BMCU 2072
97/140
E)ample* Thermal Coeffi$ient
( )
( ) ( )
6
11594
3
5
2
4
1
1006905
24
10406611026710008722860100571
24
102672860100571
=
=
=
.
....
a.a.a
=
11
9
6
3
2
1
1040661
1000872
1006905
.
.
.
a
a
a
8/13/2019 Chapter 4 BMCU 2072
98/140
E)ample* Thermal Coeffi$ient
The solution
ve$tor is
The pol,nomial that passes throu(h the three
data points is then*
=
11
9
6
3
2
1
1040661
1000872
1006905
.
.
.
a
a
a
( )21196
2321
104066110008721006905 T.T..
TaTaaT +=
++=
8/13/2019 Chapter 4 BMCU 2072
99/140
!indin( the inverse of a square matri)
The inverse [#] of a square matri) [A] is defined as
[A][)] = [*] = [)][A]
8/13/2019 Chapter 4 BMCU 2072
100/140
!indin( the inverse of a square matri)
Low $an K +e$omposition 'e used to find the inverseF
Assume the first $olumn of [B] to 'e [b11 b12 bn1]T
Ksin( this and the definition of matri) multipli$ation
!irst $olumn of [B] &e$ond $olumn of [B]
[ ]
=
0
0
1
1
21
11
nb
b
b
A [ ]
=
0
1
0
2
22
12
nb
b
b
A
The remainin( $olumns in [B] $an 'e found in the same manner
8/13/2019 Chapter 4 BMCU 2072
101/140
E)ample* nverse of a 9atri)
!ind the inverse of a square matri) [A]
[ ]
=
112144
1864
1525
A
[ ] [ ] [ ]
==
7000
561840
1525
153765
01562
001
.
..
..
.ULA
Ksin( the de$omposition pro$edure0 the [L] and [U] matri$es are found to 'e
8/13/2019 Chapter 4 BMCU 2072
102/140
E)ample* nverse of a 9atri)
&olvin( for the ea$h $olumn of [B] requires two steps
12&olve [L] [Z] = [C] for [Z]
"2&olve [U] [X] = [Z] for [X]
&tep 1* [ ][ ] [ ]
=
=
0
01
15.376.5
0156.2001
3
2
1
'
''
CL
This (enerates the equations*
05.376.5
056.21
321
21
1
=++
=+ =
'''
'''
8/13/2019 Chapter 4 BMCU 2072
103/140
E)ample* nverse of a 9atri)
&olvin( for [Z]
( )
( ) ( )23
5625317650
537650
562
15620
5620
1
213
12
1
....
'.'.'
.
.
'.'
'
= =
====
=
[ ]
=
=23
562
1
3
2
1
.
.
'
'
'
8/13/2019 Chapter 4 BMCU 2072
104/140
E)ample* nverse of a 9atri)
&olvin( [U][X] = [Z] for [X]
=
3.2
2.56
1
7.000
56.18.40
1525
31
21
11
b
b
b
2.37.0
56.256.18.4
1525
31
3121
312111
=
=
=++
b
bb
bbb
8/13/2019 Chapter 4 BMCU 2072
105/140
E)ample* nverse of a 9atri)
Ksin( #a$%ward &u'stitution
( )
( )04762.0
25
571.49524.05125
51
9524.08.4
571.4560.156.2
8.4
560.156.2
571.47.0
2.3
312111
31
21
31
=
=
=
=
+=
+
=
==
bbb
b
b
b &o the first $olumn ofthe inverse of [A] is*
=
571.4
9524.0
04762.0
31
21
11
b
b
b
8/13/2019 Chapter 4 BMCU 2072
106/140
E)ample* nverse of a 9atri)
8epeatin( for the se$ond and third $olumns of the inverse
&e$ond Column Third Column
=
0
1
0
112144
1864
1525
32
22
12
b
b
b
=
000.5
417.1
08333.0
32
22
12
b
b
b
=
1
0
0
112144
1864
1525
33
23
13
b
b
b
=
429.1
4643.0
03571.0
33
23
13
b
b
b
8/13/2019 Chapter 4 BMCU 2072
107/140
E)ample* nverse of a 9atri)
The inverse of [A] is
[ ]
=
429.1000.5571.44643.0417.19524.0
03571.008333.004762.01
A
To $he$% ,our wor% do the followin( operation
[A][A]61= [*] = [A]61[A]
8/13/2019 Chapter 4 BMCU 2072
108/140
Gauss6&eidel 9ethod
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
109/140
Gauss6&eidel 9ethod
An iterativemethod
#asi$ @ro$edure*
6Al(e'rai$all, solve ea$h linear equation for )i
6Assume an initial (uess solution arra,
6&olve for ea$h )iand repeat
6Kse a'solute relative appro)imate error after ea$h iteration
to $he$% if error is within a pre6spe$ified toleran$e
8/13/2019 Chapter 4 BMCU 2072
110/140
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
111/140
Gauss6&eidel 9ethod
Al(orithmA set of nequations and nun%nowns*
11313212111 ... bxaxaxaxa nn =++++
2323222121 ... bxaxaxaxa n$n =++++
nnnnnnn bxaxaxaxa =++++ ...332211
f*the dia(onal elements arenon6Bero
8ewriteea$h equation solvin(for the $orrespondin( un%nown
e)*
!irst equation0 solve for )1
&e$ond equation0 solve for )"
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
112/140
Gauss6&eidel 9ethod
Al(orithm8ewritin( ea$h equation
11
13132121
1a
xaxaxacx nn
=
nn
nnnnnn
n
nn
nnnnnnnnn
n
nn
a
xaxaxacx
a
xaxaxaxacx
a
xaxaxacx
11,2211
1,1
,122,122,111,11
1
22
232312122
=
=
=
!rom Equation 1
!rom equation "
!rom equation n61
!rom equation n
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
113/140
Gauss &eidel 9ethod
Al(orithmGeneral !orm of ea$h equation
11
1
1
11
1a
xac
x
n
j
j
jj
=
=
22
21
22
2a
xac
x
j
n
jj
j=
=
1,1
1
1
,11
1
=
=nn
n
nj
j
jjnn
na
xac
x
nn
n
njj
jnjn
na
xac
x
=
=1
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
114/140
Gauss &eidel 9ethod
Al(orithmGeneral !orm for an, row iO
.,,2,1,1
nia
xac
xii
n
ijj
jiji
i =
==
Low or where $an this equation 'e usedF
8/13/2019 Chapter 4 BMCU 2072
115/140
8/13/2019 Chapter 4 BMCU 2072
116/140
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
117/140
E)ample* Thermal Coeffi$ient
A trunnion of diameter 1"3.3:
has to 'e $ooled from a room
temperature of 5;
The equation that (ives the
diametri$ $ontra$tion + of
the trunnion in dr,6i$e>al$ohol
'oilin( temperature is 41;5
is (iven ',*
=108
80
)(363.12 dTTD
Figure 1 Trunnion to 'e slid
throu(h the hu' after$ontra$tin(
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
118/140
E)ample* Thermal Coeffi$ient
The e)pression for the thermal e)pansion $oeffi$ient0
is o'tained usin( re(ression anal,sis and hen$e solvin( the followin(
simultaneous linear equations*
!ind the values of a"& a
$&and a
%usin( Gauss6&eidel 9ethod
=
56799.2
1004162.1
10057.1
1024357.51086472.11026.7
1086472.11026.72860
1026.72860242
4
3
2
1
1085
85
5
a
a
a
2321 TaTaa ++=
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
119/140
E)ample* Thermal Coeffi$ient
The s,stem of equations is*
nitial Guess*
Assume an initial (uess of
=
56799.2
1004162.1
10057.1
1024357.51086472.11026.7
1086472.11026.72860
1026.72860242
4
3
2
1
1085
85
5
a
a
a
=
0
0
0
3
2
1
a
a
a
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
120/140
E)ample* Thermal Coeffi$ient
8ewritin( ea$h equationteration 1
=
56799.2
1004162.1
10057.1
1024357.51086472.11026.7
1086472.11026.72860
1026.72860242
4
3
2
1
1085
85
5
a
a
a
( ) 654
1 104042424
01026702860100571
=
= ...
a
( ) ( ) 95862
2 100024310267
0108647211040424286010041621
=
= ..
...a
( ) 1210
9865
3 103269110243575
100024310864721104042410267567992
=
= .
.
.....a
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
121/140
E)ample* Thermal Coeffi$ient
!indin( the a'solute relative appro)imate error
%100100103269.1
0103269.1
%100100100024.3
0100024.3
%100100104042.4
0104042.4
100
12
12
3
9
9
2
6
6
1
=
=
=
=
= =
=
a
a
a
new
i
old
i
new
i
ia x
xx At the end of the first iteration
The ma)imum a'solute
relative appro)imate error is
1;;P
=
12
9
6
3
2
1
1032691
1000243
1040424
.
.
.
a
a
a
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
122/140
E)ample* Thermal Coeffi$ient
teration "
Ksin( from teration 1 the values of aiare found*
=
12
9
6
3
2
1
1032691
1000243
1040424
.
.
.
a
a
a
( ) ( )
6
12594
1
1080214
24
10326911026710002432860100571
=
=
.
....a
( ) ( ) ( )
9
5
12862
2
102291410267
1032691108647211080214286010041621
=
=
..
....a
( )
12
10
9865
3
1047382
10243575
102291410864721108021410267567992
=
=
.
.
.....a
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
123/140
E)ample* Thermal Coeffi$ient
!indin( the a'solute relative appro)imate error
At the end of the se$ond iteration The ma)imum a'soluterelative appro)imate error is
/.3.;P
%360.46100104738.2
)103269.1(104738.2
%007.291001022911.4
100024.310221.4
%2864.8100108021.4
104042.4108021.4
12
1212
3
9
99
2
6
66
1
=
=
=
=
=
=
a
a
a
=
12
9
6
3
2
1
1047382
1022914
1080214
.
.
.
a
a
a
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
124/140
Itrat!on a" a$ a%
1
"
3
/
-
.
//;/"Q1;4.
/5;"1Q1;4.
/H53;Q1;4.
-;.3.Q1;4.
-;H5;Q1;4.
-111"Q1;4.
100
8.2864
3.6300
1.5918
0.6749
0.2593
3;;"/Q1;4H
/""H1Q1;4H
/./I1Q1;4H
/I;"3Q1;4H
/.;33Q1;4H
///1HQ1;4H
100
29.0073
8.9946
1.1922
2.1696
3.6330
413".HQ1;41"
4"/I35Q1;41"
43/H1IQ1;41"
4//;53Q1;41"
4-"3HHQ1;41"
4-HHI"Q1;41"
100
46.3605
29.1527
20.7922
15.8702
12.6290
E)ample* Thermal Coeffi$ient
8epeatin( more iterations0 the followin( values are o'tained
%1a
%2a
%3a
R Noti$e S After si) iterations0 the a'solute relative
appro)imate errors are de$reasin(0 'ut are still hi(h
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
125/140
Itrat!on a" a$ a%
I-I.
-;.H"Q1;4.
-;.H1Q1;4.""--HQ1;4/
";.3;Q1;4/";13HQ1;4H
";13-Q1;4H;;"/"5;;"""1
41/;/HQ1;411
41/;-1Q1;411;;11"-;;1;"H
E)ample* Thermal Coeffi$ient
8epeatin( more iterations0 the followin( values are o'tained
%1a
%2a
%3a
The value of $losel, approa$hes the true value of
=
11
9
6
3
2
1
1040511
1001352
1006915
.
.
.
a
a
a
=
11
9
6
3
2
1
1040661
1000872
1006905
.
.
.
a
a
a
from Gauss elimination2
E)ample* Thermal Coeffi$ient
8/13/2019 Chapter 4 BMCU 2072
126/140
a p e e a Coe $ e t
The pol,nomial that passes throu(h the three data points is then
( ) 2321 TaTaaT ++=
21196 104051.11000135.21006915.5 TT +=
Gauss6&eidel 9ethod* @itfall
8/13/2019 Chapter 4 BMCU 2072
127/140
Even thou(h done $orre$tl,0 the answer ma, not $onver(in( to the
$orre$t answer
This is a pitfall of the Gauss6&iedel method* not all s,stems of
equations will $onver(e
s there a fi)F
ne $lass of s,stem of equations alwa,s $onver(es* ne with a diagonally
dominant$oeffi$ient matri)
+ia(onall, dominant* [A] in [A] [X] = [C] is dia(onall, dominant if*
=
n
jj
ijaa
!1
!! =
>n
ijj
ijii aa1
for all iO and for at least one iO
G & id l 9 th d @itf ll
8/13/2019 Chapter 4 BMCU 2072
128/140
Gauss6&eidel 9ethod* @itfall
[ ]
=
116123
14345
3481.52
+ia(onall, dominant* The $oeffi$ient on the dia(onal must 'e at leastequal to the sum of the other $oeffi$ients in that row and at least one row
with a dia(onal $oeffi$ient (reater than the sum of the other $oeffi$ients
in that row
=
1293496
55323
5634124
]#[
hi$h $oeffi$ient matri) is dia(onall, dominantF
9ost ph,si$al s,stems do result in simultaneous linear equations that
have dia(onall, dominant $oeffi$ient matri$es
Gauss6&eidel 9ethod* E)ample "
8/13/2019 Chapter 4 BMCU 2072
129/140
p
Given the s,stem of equations
15312 321 x-xx =+
2835 321 xxx =++
761373 321 =++ xxx
=
1
0
1
3
2
1
x
x
x
ith an initial (uess of
The $oeffi$ient matri) is*
[ ]
=
1373
351
5312
A
ill the solution $onver(e usin( the
Gauss6&iedel methodF
Gauss6&eidel 9ethod* E)ample "
8/13/2019 Chapter 4 BMCU 2072
130/140
p
[ ]
=
1373
351
5312
A
Che$%in( if the $oeffi$ient matri) is dia(onall, dominant
43155 232122 =+=+== aaa
10731313 323133 =+=+== aaa
8531212 131211 =+=+== aaa
The inequalities are all true and at least one row is stritly(reater than*
Therefore* The solution should $onver(e usin( the Gauss6&iedel 9ethod
Gauss6&eidel 9ethod* E)ample "
8/13/2019 Chapter 4 BMCU 2072
131/140
p
=
76
28
1
1373
351
5312
%
$
a
a
a
8ewritin( ea$h equation
12
531 321
xxx
+=
5
328 312
xx
x
=
13
7376 213
xxx
=
ith an initial (uess of
=
1
0
1
3
2
1
x
x
x
( ) ( )50000.0
12
150311 =
+=x
( ) ( )9000.4
5
135.028
2
=
=x
( ) ( )0923.3
13
9000.4750000.03763 =
=x
Ga ss &eidel 9ethod E ample "
8/13/2019 Chapter 4 BMCU 2072
132/140
Gauss6&eidel 9ethod* E)ample "
The a'solute relative appro)imate error
%00.10010050000.0
0000.150000.01
=
=a
%00.1001009000.4
09000.42a
==
%662.671000923.3
0000.10923.3
3a
=
=
The ma)imum a'solute relative error after the first iteration is 1;;P
Gauss6&eidel 9ethod* E)ample "
8/13/2019 Chapter 4 BMCU 2072
133/140
p
=
8118.3
7153.3
14679.0
3
2
1
x
x
x
After teration D1
( ) ( )14679.0
12
0923.359000.4311 =
+=x
( ) ( ) 7153.35
0923.3314679.0282 ==x
( ) ( )8118.3
13
900.4714679.03763 =
=x
&u'stitutin( the ) values into the
equationsAfter teration D"
=
0923.3
9000.4
5000.0
3
2
1
x
x
x
Gauss6&eidel 9ethod* E)ample "
8/13/2019 Chapter 4 BMCU 2072
134/140
p
teration D" a'solute relative appro)imate error
%61.24010014679.0
50000.014679.01a
=
=
%889.311007153.3
9000.47153.32a ==
%874.181008118.3
0923.38118.33a
=
=
The ma)imum a'solute relative error after the first iteration is "/;.1P
This is mu$h lar(er than the ma)imum a'solute relative error o'tained in
iteration D1 s this a pro'lemF
Gauss &eidel 9ethod* E)ample "
8/13/2019 Chapter 4 BMCU 2072
135/140
Itrat!on a1 a2 a3
12
3456
0.500000.14679
0.742750.946750.991770.99919
100.00240.61
80.23621.5464.53910.74307
4.90003.7153
3.16443.02813.00343.0001
100.0031.889
17.4084.49960.824990.10856
3.09233.8118
3.97083.99714.00014.0001
67.66218.876
4.00420.657720.0743830.00101
Gauss6&eidel 9ethod* E)ample "
8epeatin( more iterations0 the followin( values are o'tained
%1a
%2a
%3a
=
4
31
3
2
1
x
xx
=
0001.4
0001.399919.0
3
2
1
x
xx
The solution o'tained is $lose to the e)a$t solution of
Gauss &eidel 9ethod* E)ample 3
8/13/2019 Chapter 4 BMCU 2072
136/140
Gauss6&eidel 9ethod* E)ample 3
Given the s,stem of equations
761373 321 =++ xxx
2835 321 =++ xxx
15312 321 =+ xxx
ith an initial (uess of
=
1
0
1
3
2
1
x
x
x
8ewritin( the equations
3
13776 321
xx
x
=
5
328 312
xxx
=
5
3121 213
=
xxx
Gauss6&eidel 9ethod* E)ample 3
8/13/2019 Chapter 4 BMCU 2072
137/140
Itrat!on a1 A2 a3
12
3456
21.000$196.15
$1995.0$20149
2.0364105
$2.0579105
95.238110.71
109.83109.90109.89109.89
0.8000014.421
$116.021204.6$12140
1.2272105
100.0094.453
112.43109.63109.92109.89
50.680$462.30
4718.1$47636
4.8144105
$4.8653106
98.027110.96
109.80109.90109.89109.89
Condu$tin( si) iterations0 the followin( values are o'tained
%1a
%2a
%3a
The values are not $onver(in(
+oes this mean that the Gauss6&eidel method $annot 'e usedF
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
138/140
The Gauss6&eidel 9ethod $an still 'e used
The $oeffi$ient matri) is not
dia(onall, dominant [ ]
=
5312
351
1373
A
#ut this is the same set of
equations used in e)ample D"0
whi$h did $onver(e [ ]
=
1373
351
5312
A
f a s,stem of linear equations is not dia(onall, dominant0 $he$% to see if
rearran(in( the equations $an form a dia(onall, dominant matri)
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
139/140
Not ever, s,stem of equations $an 'e rearran(ed to have a
dia(onall, dominant $oeffi$ient matri)
'serve the set of equations
3321 =++ xxx9432 321 =++ xxx
97 321 =++ xxx
hi$h equations2 prevents this set of equation from havin( a
dia(onall, dominant $oeffi$ient matri)F
Gauss6&eidel 9ethod
8/13/2019 Chapter 4 BMCU 2072
140/140
&ummar,
6Advanta(es of the Gauss6&eidel 9ethod
6Al(orithm for the Gauss6&eidel 9ethod
6@itfalls of the Gauss6&eidel 9ethod