{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 31 "Newton's Method in 2 Dime nsions" }}{PARA 19 "" 0 "" {TEXT -1 22 "Douglas Wilhelm Harder" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "It's rather difficult to imagine \+ Newton's method in N dimensions, so the following will be a 2-dimensio nal version of the technique. The functions are relatively simple to \+ keep the ideas clear." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Consider the system of two functions:\n\n " }{TEXT 256 1 "f" }{TEXT -1 1 "(" }{TEXT 257 1 "v" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "MATRIX([[f[1]( v)], [f[2](v)]]);" "6#-%'MATRIXG6#7$7#-&%\"fG6#\"\"\"6#%\"vG7#-&F*6#\" \"#6#F." }{TEXT -1 4 " = " }{XPPEDIT 18 0 "MATRIX([[v[1]^2+2*v[2]-1], [3*v[1]+v[2]^2-3]]);" "6#-%'MATRIXG6#7$7#,(*$&%\"vG6#\"\"\"\"\"#F-*&F .F-&F+6#F.F-F-F-!\"\"7#,(*&\"\"$F-&F+6#F-F-F-*$&F+6#F.F.F-F6F2" } {TEXT -1 14 " \nwhere\n " }{TEXT 258 1 "v" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "MATRIX( [[v[1]], [v[2]]] )" "6#-%'MATRIXG6#7$7#&%\"vG6# \"\"\"7#&F)6#\"\"#" }{TEXT -1 34 ".\nWe would like to find a vector, \+ " }{TEXT 259 1 "v" }{XPPEDIT 18 0 "``[r]" "6#&%!G6#%\"rG" }{TEXT -1 11 " such that " }{TEXT 260 1 "f" }{TEXT -1 2 "( " }{TEXT 261 1 "v" } {XPPEDIT 18 0 "``[r];" "6#&%!G6#%\"rG" }{TEXT -1 5 " ) = " }{TEXT 262 1 "0" }{TEXT -1 68 ". In Maple, we can find exact solutions by using \+ the two functions " }{MPLTEXT 0 21 5 "solve" }{TEXT -1 5 " and " } {MPLTEXT 0 21 9 "allvalues" }{TEXT -1 113 ", but the results are rathe r large and they involving roots of cubic polynomials, the roots of wh ich are complex." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "solutio ns := [solve( \{v[1]^2 + 2*v[2] - 1, 3*v[1] + v[2]^2 - 3\}, \{v[1], v[ 2]\} )];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solutionsG7$<$/&%\"vG6# \"\"\"F+/&F)6#\"\"#\"\"!<$/F(-%'RootOfG6$,**$)%#_ZG\"\"$F+F+*$)F9F/F+F +F9!\"\"\"#6F+/%&labelG%$_L4G/F-,&*&#F+F/F+*$)F3F/F+F+F=#F+F/F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solutions[1];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<$/&%\"vG6#\"\"\"F(/&F&6#\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalf( allvalues( solutions[2] ) ); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6%<$/&%\"vG6#\"\"#$!+M%>\"oL!\"*/&F& 6#\"\"\"$!+6&49y#F+<$/F-^$$\"+dv/2*)!#5$!+UN0y " 0 "" {MPLTEXT 1 0 448 "plots[display](\n plot3d( v[1]^2 + 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0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000-F16&F3$\")=THvF7FBFB-%'POI NTSG6%7%F4F4F4-%'COLOURG6&F3\"\"\"FKFK-%'SYMBOLG6#%'CIRCLEG-FE6%7%F4F4 $!\"$F)-FI6&F3F5F4F5FL-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "If we plot just \+ these two planes, we see that we have a linear system of two equations and two unknowns.\nPlease note, because we already started at the poi nt (0, 0), we did not have to " }{TEXT 279 5 "shift" }{TEXT -1 79 " th ese two planes towards the origin. By evaluating the Jacobian at the \+ point " }{TEXT 287 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 44 " and writing down the system of equations J(" }{TEXT 281 1 "f" }{TEXT -1 2 ")(" }{TEXT 280 1 "v" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "MATRIX([[2*v[1], 2], [3, 2*v[2]]])" "6#-%'MATRIXG6#7$7$*&\"\"#\" \"\"&%\"vG6#F*F*F)7$\"\"$*&F)F*&F,6#F)F*" }{TEXT -1 15 " and solving J (" }{TEXT 282 1 "f" }{TEXT -1 2 ")(" }{TEXT 283 1 "v" }{XPPEDIT 18 0 " ``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "Delta" "6#%&D eltaG" }{TEXT 284 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 5 " = - " }{TEXT 285 1 "f" }{TEXT -1 1 "(" }{TEXT 286 1 "v" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 67 "), this automatic ally shifts the system of equations to the origin." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "plots[display](\n plot3d( 2*v[2] - 1, v[1 ] = -4..4, v[2] = -4..4, color = red ),\n plot3d( 3*v[1] - 3, v[1] = \+ -4..4, v[2] = -4..4, color = blue ),\n plot3d( 0, 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0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000-F1 6&F3F7F7F7-%'POINTSG6%7%F7F7F7-%'COLOURG6&F3\"\"\"FIFI-%'SYMBOLG6#%'CI RCLEG-FC6%7%F7F7$!\"\"F)-FG6&F3F7F4F7FJ-FC6%7%F7F7$!\"$F)-FG6&F3F4F7F4 FJ-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "By rotating the plot, you should be able to convince yourself that there is one unique solution to this system of equations. If we find the solution to this system \+ of equations, namely:\n " }{XPPEDIT 18 0 "MATRIX( [[0, 2], [3, 0]] ) * Delta" "6#*&-%'MATRIXG6#7$7$\"\"!\"\"#7$\"\"$F)\"\"\"%&DeltaGF-" } {TEXT -1 1 " " }{TEXT 271 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\" " }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-MATRIX([[-1], [-3]])" "6#,$-%'MAT RIXG6#7$7#,$\"\"\"!\"\"7#,$\"\"$F+F+" }{TEXT -1 14 " \nwe get that " } {XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT 272 1 "v" }{XPPEDIT 18 0 "`` [1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[1],[0 .5]])" "6#-%'MATRIXG6#7$7#\"\"\"7#-%&FloatG6$\"\"&!\"\"" }{TEXT -1 1 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "And thus, " }{TEXT 273 1 "v " }{XPPEDIT 18 0 "``[2];" "6#&%!G6#\"\"#" }{TEXT -1 3 " = " }{TEXT 274 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT 275 1 "v" }{XPPEDIT 18 0 "`` [1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[1], [ 0.5]])" "6#-%'MATRIXG6#7$7#\"\"\"7#-%&FloatG6$\"\"&!\"\"" }{TEXT -1 58 ". If you insepct the following plot carefully, where the " } {TEXT 276 1 "v" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 14 " is white and " }{TEXT 277 1 "v" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"# " }{TEXT -1 44 " is yellow (you may have to rotate the plot " }{TEXT 278 2 "up" }{TEXT -1 114 " to see the yellow dot), you will see that t he second point is half the distance to the root than the first point. 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0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000-F16&F3$\")=THvF6FBFB-%'POI NTSG6%7%F7F7F7-%'COLOURG6&F3\"\"\"FKFK-%'SYMBOLG6#%'CIRCLEG-FE6%7%F7F7 $!\"\"F)-FI6&F3F7F4F7FL-FE6%7%F7F7$!\"$F)-FI6&F3F4F7F4FL-FE6%7%$FKF)$ \"\"&FTF7-FI6&F3F4F4F7FL-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "If we apply Newton's method in 2 dimensions over and over, we get \+ the following sequence of points:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "v[1] := Vector( [0., 0.] );\nf := v -> Vector([v[1]^ 2+2*v[2]-1, 3*v[1]+v[2]^2-3]);\nJ := v -> Matrix( [[2*v[1], 2], [3, 2* v[2]]] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"\"-%'RTABLEG 6%\")7R!>\"-%'MATRIXG6#7$7#$\"\"!F2F0&%'VectorG6#%'columnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"vG6\"6$%)operatorG%&arrowGF(-%' VectorG6#7$,(*$)&9$6#\"\"\"\"\"#F6F6*&F7F6&F46#F7F6F6F6!\"\",(*&\"\"$F 6F3F6F6*$)F9F7F6F6F>F;F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG f*6#%\"vG6\"6$%)operatorG%&arrowGF(-%'MatrixG6#7$7$,$*&\"\"#\"\"\"&9$6 #F4F4F4F37$\"\"$,$*&F3F4&F66#F3F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "UseHardwareFloats := false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2UseHardwareFloatsG%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "v[2] := v[1] + LinearAlgebra:-LinearSolve( J(v[1 ]), -f(v[1]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"#-%'RT ABLEG6%\")Op`7-%'MATRIXG6#7$7#$\"+++++5!\"*7#$\"+++++]!#5&%'VectorG6#% 'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "v[3] := v[2] + \+ LinearAlgebra:-LinearSolve( J(v[2]), -f(v[2]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"$-%'RTABLEG6%\");q`7-%'MATRIXG6#7$7#$\"+++ +D6!\"*7#$!+-++]7!#5&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "v[4] := v[3] + LinearAlgebra:-LinearSolve( J(v[3]), -f(v[3]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"%-%'RTABL EG6%\")%=YH\"-%'MATRIXG6#7$7#$\"+VrN05!\"*7#$\")Tr&y\"!#5&%'VectorG6#% 'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "v[5] := v[4] + \+ LinearAlgebra:-LinearSolve( J(v[4]), -f(v[4]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"&-%'RTABLEG6%\")c:]5-%'MATRIXG6#7$7#$\"+Z5 ++5!\"*7#$\")^pH8!#7&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "v[6] := v[5] + LinearAlgebra:-LinearSolve( J(v[5]), -f(v[5]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"'-%'RTABL EG6%\")cu`7-%'MATRIXG6#7$7#$\"+++++5!\"*7#$!&yl\"!#9&%'VectorG6#%'colu mnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Note how the convergence \+ is approximately quadratic: the second term in the vector has initial ly two zeros, then four, and then nine." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "Just plotting the points, we see that after the first th ree points, the rest are indistinguishable from the root (at the centr e of the plot). The first point is on the edge of the black square." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "plots[display](\n plot3d( v[1]^2 + 2*v[ 2] - 1, v[1] = 0..2, v[2] = -1..1, color = red ),\n plot3d( 3*v[1] + \+ v[2]^2 - 3, v[1] = 0..2, v[2] = -1..1, color = blue ),\n plot3d( 0, v [1] = 0..2, v[2] = -1..1, color = black ),\n plots[pointplot3d]( [[0, 0,0]], color = white, symbol = circle ),\n plots[pointplot3d]( [[1, 0 .5, 0], [1.125, -0.125, 0], [1.005357143, 0.0017857141, 0], [1.0000010 47, 0.000013296951, 0], [1., -0.6578e-9, 0]], color = green, symbol = \+ circle ),\n axes = boxed\n);" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }} {PARA 13 "" 1 "" {GLPLOT3D 781 781 781 {PLOTDATA 3 "6(-%%GRIDG6&;$\"\" !F($\"\"#F(;$!\"\"F($\"\"\"F(X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":C 008000000000000C006AAAAAAAAAAAAC005555555555555C004000000000000C002AAA AAAAAAAAAC001555555555554BFFFFFFFFFFFFFFEBFFD555555555554BFFAAAAAAAAAA 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