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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 44 "GAUSSIAN ELIMINATION AND PIVOTING STARTEGIES" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Example 1 (Burden Secti
on 6.2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A1:=matri
x([[0.003000,59.14],[5.291,-6.130]]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "b1:=vector([59.17,46.78]);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 257 15 "Exact Solution:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "linsolve(A1
,b1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 258 49 "Using finite arithmetic and Gaussian Elim
ination." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
29 "Definition of the arithmetic:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=4;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A
A1:=matrix([[0.003000,59.14,59.17],[5.291,-6.130,46.78]]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m21:=AA1[2,1]/AA1[1,1];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB1:=addrow(AA1,1,2,-m21);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "BB1[2,1]:=0;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x:=backsub(BB1);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Wrong answer du
e to a relative small pivot element, which results in a large multipli
er." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 260 37 "Exercise 6 (d) Section 6.2 (modified)" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 
"System definition:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=10;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "A2:=matrix(
[[0.33330,1592,1.0333],[2.222,16710,96.12],[-1.5611,5.1792,-1.6855]]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "b2:=vector([795.3,965,2
.714]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 263 15 "Exact Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "linsolve(A2,b2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 49 "Using finite arithmetic a
nd Gaussian Elimination." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 29 "Definition of the arithmetic:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Augmented Matrix" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "AA2:
=matrix([[0.33330,1592,1.0333,795.3],[2.222,16710,96.12,965],[-1.5611,
5.1792,-1.6855,2.714]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"m21:=AA2[2,1]/AA2[1,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"BB2:=addrow(AA2,1,2,-m21);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "m31:=BB2[3,1]/BB2[1,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "BB2:=addrow(BB2,1,3,-m31);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "m32:=BB2[3,2]/BB2[2,2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "BB2:=addrow(BB2,2,3,-m32);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "BB2[3,2]:=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "x:=backsub(BB2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 13 "Wrong answer." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 26 "Partial
 pivoting strategy:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=4;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "evalm(AA1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "BB1:=swaprow(AA1,1,2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "m21:=BB1[2,1]/BB1[1,1];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "BB1:=addrow(BB1,1,2,-m21);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "x:=backsub(BB1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 23 "Same as exact solution." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 25 "Exercise 6 (d) (modified)" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eva
lm(AA2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Using Partial Pivoting:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B
B2:=swaprow(AA2,1,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m2
1:=BB2[2,1]/BB2[1,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB
2:=addrow(BB2,1,2,-m21);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"m31:=BB2[3,1]/BB2[1,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"BB2:=addrow(BB2,1,3,-m31);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "m32:=BB2[3,2]/BB2[2,2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "BB3:=addrow(BB2,2,3,-m32);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "x:=backsub(BB3);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Wrong answer. " }}{PARA 0 "
" 0 "" {TEXT -1 43 "Add partial pivoting in second column also." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "BB3:=swaprow(BB2,2,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "m32:=BB3[3,2]/BB3[2,2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "BB4:=addrow(BB3,2,3,-m32);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "x:=backsub(BB4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Scaled Pivoting Strategy:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "evalm(AA2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "s1:= m
ax(abs(AA2[1,1]),abs(AA2[1,2]),abs(AA2[1,3]));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "e1:=abs(AA2[1,1])/s1;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 52 "s2:= max(abs(AA2[2,1]),abs(AA2[2,2]),abs(AA2[2,3
]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e2:=abs(AA2[2,1])/s
2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "s3:= max(abs(AA2[3,1]
),abs(AA2[3,2]),abs(AA2[3,3]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "e3:=abs(AA2[3,1])/s3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "BB2:=swaprow(AA2,1,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "m21:=BB2[2,1]/BB2[1,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "BB2:=addrow(BB2,1,2,-m21);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "m31:=BB2[3,1]/BB2[1,1];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "BB2:=addrow(BB2,1,3,-m31);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "BB2[3,1]:=0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "evalm(BB2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 125 "Using the same scales obtained in the \+
previous step we can decide what row will be in the pivot position in \+
this second step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "e2:=BB2[2,2]/s2;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "e3:=BB2[3,2]/s1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Since the two scaled entries in
 this column are the same, we don't make any in change the matrix." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "m32:=BB2[3,2]/BB2[2,2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "BB2:=addrow(BB2,2,3,-m32);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "x:=backsub(BB2);" }}}{EXCHG {PARA 0 "> " 0 "" 
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