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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 1 "\011" }{TEXT 257 4 "\011
\011\011\011" }{TEXT 258 40 "Least Squares - The Two Dimensional Case
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 461 "Solving an over-determined \+
 linear system A x = b is generally impossible, but\nfinding an x0 so \+
that A x0 is as close as possible to b leads to the concept of a\nleas
t squares solution.  One can see that such an x0 must satisfy the\nnor
mal equations\n\011\011\011\011A_tr A x0 =  A_tr b \nwhere A_tr is the
 transpose of the matrix A.  Generically this system will\nhave a uniq
ue solution. We  use this worksheet to study this in the case of \nfit
ting a set of points to a line.\n\n" }{TEXT 260 9 "Problem: " }{TEXT 
261 109 "Find the best fitting straight line  y = cx + d through the f
ollowing points:\n \011\011\011\011(1,2), (1,3),(2,4), (3,5)\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "pointlist := [[1,2],[1,3],[2,4],[3,
5]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*pointlistG7&7$\"\"\"\"\"#7$
F'\"\"$7$F(\"\"%7$F*\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 9 "Solu
tion:" }{TEXT 263 123 " The over-determined system here is \n\011\011
\0111 c + d = 2\n\011\011\0111 c + d = 3\n\011\011\0112 c + d = 4\n
\011\011\0113 c + d = 5\n In matrix form: A x = b:\n" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Wa
rning, the protected names norm and trace have been redefined and unpr
otected\n" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 264 1 "\n" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "A := array([[1,1],[1,1],[2,1],[3,1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7$\"\"\"F*F)7$\"\"
#F*7$\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x :=vector(
[c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7$%\"cG%
\"dG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "b := vector([2,3,4,
5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7&\"\"#\"\"$
\"\"%\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 86 "This step just con
firms we've chosen A and b to have the right over-determined system." 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalm(A &* x = b );" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%'vectorG6#7&,&%\"cG\"\"\"%\"dGF*F(,&F)\"\"#F+
F*,&F)\"\"$F+F*-F%6#7&F-F/\"\"%\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
266 31 " The normal equations N x = m :" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "N := evalm(transpose(A) &* A);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"NG-%'matrixG6#7$7$\"#:\"\"(7$F+\"\"%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "m := evalm(transpose(A) &* b);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"mG-%'vectorG6#7$\"#G\"#9" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 29 "x0 := evalm(inverse(N) &* m);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#x0G-%'vectorG6#7$#\"#9\"#6F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "line := x -> x0[1] * x + x0[2];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%lineGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,
&*&&%#x0G6#\"\"\"F19$F1F1&F/6#\"\"#F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 79 "plot(\{pointlist,line\},\n\011title = `Least Squar
es Solution Vs. Points Connected`);" }}{PARA 13 "" 1 "" {GLPLOT2D 331 
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1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 267 82 "Here's a singular example where the norm
al equations don't have a unique solution." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "pointlist := [[1,1],[1,2],[1,3],[1,4]];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%*pointlistG7&7$\"\"\"F'7$F'\"\"#7$F'\"\"$7$F'
\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:= array([[1,1],[
1,1],[1,1],[1,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrix
G6#7&7$\"\"\"F*F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "b \+
:= vector([1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vect
orG6#7&\"\"\"\"\"#\"\"$\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 92 "
The normal equations N x = m :\nSince A had rank 1, N also has rank 1 \+
and is not invertible.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "N := ev
alm(transpose(A) &* A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'ma
trixG6#7$7$\"\"%F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 77 "Still th
e equations N x = m have solutions - the normal equations always do !
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "m := evalm(transpose(A) &* b);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG-%'vectorG6#7$\"#5F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 270 128 "It is tempting to investigate wh
at happens when the problem is nearly singular.\nWe change the first p
oint to (1.000001,1.00001)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A:= \+
array([[1.000001,1],[1,1],[1,1],[1,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7&7$$\"(,++\"!\"'\"\"\"7$F-F-F.F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "b := vector([1.000001,2,3,4]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7&$\"(,++\"!\"'\"\"#
\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "N := evalm(tr
anspose(A) &* A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6
#7$7$$\"++?++S!\"*$\"(,++%!\"'7$F-\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "m := evalm(transpose(A) &* b);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"mG-%'vectorG6#7$$\"++-++5!\")$\"),++5!\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x0 := evalm(inverse(N) &* m)
;" }}{PARA 8 "" 1 "" {TEXT -1 36 "Error, (in inverse) singular matrix
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Digits;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "Digits := 40;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#S" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x0 := evalm(inverse(N) &* \+
m);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#x0G-%'vectorG6#7$$\"B+++++++
++++++,++'!#E$!B++++++++++++++++'F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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