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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 83 "Same as PivotStrategies
.mws but only one exercise to show differences more clearly." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 259 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;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{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]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'matrixG6
#7%7%$\"&IL$!\"&\"%#f\"$\"&L.\"!\"%7%$\"%AA!\"$\"&5n\"$\"%7'*!\"#7%$!&
6c\"F0$\"&#z^F0$!&bo\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"b2:=vector([795.3,965,2.714]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
b2G-%'vectorG6#7%$\"%`z!\"\"\"$l*$\"%9F!\"$" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 15 "Exact Solution:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "linsolve(A2,b2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7%$\"+?hb-#*!\")$\"+_H#fN&!#5$!+Uhz>&)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 261 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:=3;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%'DigitsG\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
16 "Augmented Matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 98 "AA2:=matrix([[0.33330,1592,1.0333,795.3],
[2.222,16710,96.12,965],[-1.5611,5.1792,-1.6855,2.714]]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$AA2G-%'matrixG6#7%7&$\"&IL$!\"&\"%#f\"$\"&L.
\"!\"%$\"%`z!\"\"7&$\"%AA!\"$\"&5n\"$\"%7'*!\"#\"$l*7&$!&6c\"F0$\"&#z^
F0$!&bo\"F0$\"%9FF7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m21:
=AA2[2,1]/AA2[1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$m21G$\"$n'!
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB2:=addrow(AA2,1,2,
-m21);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$\"&I
L$!\"&\"%#f\"$\"&L.\"!\"%$\"%`z!\"\"7&$\"\"!F6$\"#h\"\"#$\"$#*)F3$!$M%
\"\"\"7&$!&6c\"F0$\"&#z^F0$!&bo\"F0$\"%9F!\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "m31:=BB2[3,1]/BB2[1,1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$m31G$!$o%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "BB2:=addrow(BB2,1,3,-m31);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$BB2G-%'matrixG6#7%7&$\"&IL$!\"&\"%#f\"$\"&L.\"!\"%$\"%`z!\"\"7&$\"
\"!F6$\"#h\"\"#$\"$#*)F3$!$M%\"\"\"7&F5$\"$X(F>$\"$8$!\"#$\"$s$F>" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m32:=BB2[3,2]/BB2[2,2];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$m32G$\"$A\"!\"#" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 26 "BB2:=addrow(BB2,2,3,-m32);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$\"&IL$!\"&\"%#f\"$\"&L.\"!
\"%$\"%`z!\"\"7&$\"\"!F6$\"#h\"\"#$\"$#*)F3$!$M%\"\"\"7&F5$F>F>$!$1\"F
6$\"$,*F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "BB2[3,2]:=0;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$BB2G6$\"\"$\"\"#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x:=backsub(BB2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7%$\"$<\"\"\"!$\"$J&!\"$$!
$])!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "Wrong answer due to almost no influence of small coeffici
ents when performing row operations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 26 "Partial pivotin
g strategy:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(AA2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&$\"$L$!\"$\"%#f\"$\"$.\"!\"#$\"$&z
\"\"!7&$\"$A#F.\"&5n\"$\"$h*!\"\"\"$l*7&$!$c\"F.$\"$=&F.$!$p\"F.$\"$r#
F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"\"$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "BB2:=swapro
w(AA2,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&
$\"%AA!\"$\"&5n\"$\"%7'*!\"#\"$l*7&$\"&IL$!\"&\"%#f\"$\"&L.\"!\"%$\"%`
z!\"\"7&$!&6c\"F9$\"&#z^F9$!&bo\"F9$\"%9FF," }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "m21:=BB2[2,1]/BB2[1,1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$m21G$\"$]\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "BB2:=addrow(BB2,1,2,-m21);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$\"%AA!\"$\"&5n\"$\"%7'*!\"#\"$l
*7&$\"\"!F4$!#\"*\"\"\"$!$M\"!\"\"$\"$]'F47&$!&6c\"!\"%$\"&#z^F@$!&bo
\"F@$\"%9FF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m31:=BB2[3,
1]/BB2[1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$m31G$!$.(!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB2:=addrow(BB2,1,3,-m31);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$\"%AA!\"$\"&
5n\"$\"%7'*!\"#\"$l*7&$\"\"!F4$!#\"*\"\"\"$!$M\"!\"\"$\"$]'F47&F3$\"$<
\"\"\"#$\"$f'F:$\"$\"oF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"m32:=BB2[3,2]/BB2[2,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$m32G$!$
H\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB2:=addrow(BB2
,2,3,-m32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&
$\"%AA!\"$\"&5n\"$\"%7'*!\"#\"$l*7&$\"\"!F4$!#\"*\"\"\"$!$M\"!\"\"$\"$
]'F47&F3F3$!$2\"F4$\"$1*F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "x:=backsub(BB2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vecto
rG6#7%$\"$P\"\"\"!$\"$F&!\"$$!$Z)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "The answer is even worst
 in this case. The entrie 2.222 is certainly larger (absolute value) t
han the other two in its column, however its relative size compared wi
th " }}{PARA 0 "" 0 "" {TEXT -1 65 "the rest of the entries in its row
 is smaller than the other two." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 25 "Scaled Pivoting Strategy:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "e
valm(AA2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&$\"$L$!
\"$\"%#f\"$\"$.\"!\"#$\"$&z\"\"!7&$\"$A#F.\"&5n\"$\"$h*!\"\"\"$l*7&$!$
c\"F.$\"$=&F.$!$p\"F.$\"$r#F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "s1:= max(abs(AA2[1,1]),abs(AA2[1,2]),abs(AA2[1,3]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#s1G\"%#f\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "e1:=abs(AA2[1,1])/s1;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#e1G$\"$4#!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "s
2:= max(abs(AA2[2,1]),abs(AA2[2,2]),abs(AA2[2,3]));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#s2G\"&5n\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "e2:=abs(AA2[2,1])/s2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2
G$\"$L\"!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "s3:= max(ab
s(AA2[3,1]),abs(AA2[3,2]),abs(AA2[3,3]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#s3G$\"$=&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "e3:=abs(AA2[3,1])/s3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3
G$\"$,$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "BB2:=swaprow
(AA2,1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$
!&6c\"!\"%$\"&#z^F,$!&bo\"F,$\"%9F!\"$7&$\"%AAF3\"&5n\"$\"%7'*!\"#\"$l
*7&$\"&IL$!\"&\"%#f\"$\"&L.\"F,$\"%`z!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "m21:=BB2[2,1]/BB2[1,1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$m21G$!$U\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "BB2:=addrow(BB2,1,2,-m21);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$!&6c\"!\"%$\"&#z^F,$!&bo\"F,$\"
%9F!\"$7&$\"\"!F6$\"$n\"\"\"#$\"$P*!\"\"$\"$p*F67&$\"&IL$!\"&\"%#f\"$
\"&L.\"F,$\"%`zF<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m31:=B
B2[3,1]/BB2[1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$m31G$!$8#!\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB2:=addrow(BB2,1,3,-m3
1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$!&6c\"!
\"%$\"&#z^F,$!&bo\"F,$\"%9F!\"$7&$\"\"!F6$\"$n\"\"\"#$\"$P*!\"\"$\"$p*
F67&$\"\"\"F3$\"$f\"FA$\"$q'F3$\"$'zF6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "BB2[3,1]:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$B
B2G6$\"\"$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ev
alm(BB2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&$!$c\"!\"
#$\"$=&F*$!$p\"F*$\"$r#F*7&$\"\"!F3$\"$n\"\"\"#$\"$P*!\"\"$\"$p*F37&F3
$\"$f\"\"\"\"$\"$q'!\"$$\"$'zF3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 125 "Using the same scales obtained in t
he previous step we can decide what row will be in the pivot position \+
in this second step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "e2:=BB2[2,2]/s2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#e2G$\"$***!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "e3:=BB2[3,2]/s1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#e3G$\"$***!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 94 "Since the two scaled entries in this column are eq
ual, we don't make any in change the matrix." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m32:=BB2[3,2]/BB2
[2,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$m32G$\"$_*!\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BB2:=addrow(BB2,2,3,-m32);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BB2G-%'matrixG6#7%7&$!&6c\"!\"%$
\"&#z^F,$!&bo\"F,$\"%9F!\"$7&$\"\"!F6$\"$n\"\"\"#$\"$P*!\"\"$\"$p*F67&
$!\"!F6F5$!$D)!\"#$\"$/(F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "x:=backsub(BB2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vecto
rG6#7%$\"$B*!\"\"$\"$P&!\"$$!$`)F+" }}}}{MARK "60" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }