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22 "MATH 6050 Spring 2009 \n" }{TEXT 234 17 "Gradient Method \n" }{TEXT 234 4 "and\n" }{TEXT 234 43 "Newton Method for Minimization of Functions" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 0 31 "restart; with(VectorCalculus): " }{TEXT 210 17 "Loading packages " }{TEXT 211 14 "VectorCalculus" }{TEXT 212 5 " and " }{TEXT 213 13 "LinearAlgebra" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "BasisFormat(false);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{SECT 0 {PARA 209 "" 0 "" {TEXT 209 16 "Gradient Methods" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=(x1-1)^2+(2*x1^2-x2)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot3d(f,x1=0..2,x2=1..3,axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gf:=Vector(Gradient(f,[x1,x2]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 30 "Stepsize - using Armijo's rule" }{TEXT 200 23 " for choosing stepsize " }{XPPEDIT 200 0 "Typesetting:-mrow(Ty pesetting:-msup(Typesetting:-mi(\"α\", italic = \"false\", mathv ariant = \"normal\"), Typesetting:-mrow(Typesetting:-mi(\"k\", italic \+ = \"true\", mathvariant = \"italic\")), superscriptshift = \"0\"), Typ esetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo(\".\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Ty pesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6&-I%msupGF$6%-I#miGF$6%Q(αF'/%'italicGQ&fals eF'/%,mathvariantGQ'normalF'-F#6#-F/6%Q\"kF'/F3Q%trueF'/F6Q'italicF'/% 1superscriptshiftGQ\"0F'-I#moGF$6-Q\"~F'F5/%&fenceGF4/%*separatorGF4/% )stretchyGF4/%*symmetricGF4/%(largeopGF4/%.movablelimitsGF4/%'accentGF 4/%'lspaceGQ&0.0emF'/%'rspaceGFX-FE6-Q\".F'F5FHFJFLFNFPFRFTFVFYFD" } {TEXT 200 7 "We fix " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:- mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:- mi(\"s\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-m o(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", 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\"normal\"), Typesetting:-mo(\"=\", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em \"), Typesetting:-mn(\"0.1\", mathvariant = \"normal\"), Typesetting:- mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6&-I#miGF$6%Q(σF'/%'italicGQ&falseF'/%,mathvariantGQ'nor malF'-I#moGF$6-Q\"=F'F2/%&fenceGF1/%*separatorGF1/%)stretchyGF1/%*symm etricGF1/%(largeopGF1/%.movablelimitsGF1/%'accentGF1/%'lspaceGQ,0.2777 778emF'/%'rspaceGFI-I#mnGF$6$Q$0.1F'F2-F66-Q\"~F'F2F9F;F=F?FAFCFE/FHQ& 0.0emF'/FKFT" }{TEXT 200 4 "and " }{XPPEDIT 200 0 "Typesetting:-mrow(T ypesetting:-mi(\"β\", italic = \"false\", mathvariant = \"normal \"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mn( \".5\", mathvariant = \"normal\"), Typesetting:-mo(\" \", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\")); " "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6&-I#miGF$6%Q' βF'/%'italicGQ&falseF'/%,mathvariantGQ'normalF'-I#moGF$6-Q\"=F'F2 /%&fenceGF1/%*separatorGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/% .movablelimitsGF1/%'accentGF1/%'lspaceGQ,0.2777778emF'/%'rspaceGFI-I#m nGF$6$Q#.5F'F2-F66-Q\"~F'F2F9F;F=F?FAFCFE/FHQ&0.0emF'/FKFT" }{TEXT 200 57 "and we need to find the first v1 such that the relation\n" } {TEXT 200 3 " " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-mo( \" \", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo( \" \", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo( \" \", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi( \"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mfen ced(Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic \+ = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:- mi(\"k\", italic = \"true\", mathvariant = \"italic\")), superscriptsh ift = \"0\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo(\"+ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mo(\" \", mathvariant = \"nor mal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typeset ting:-mi(\"α\", italic = \"false\", mathvariant = \"normal\"), T ypesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-mi(\" d\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(T ypesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"k\", italic = \"tr ue\", mathvariant = \"italic\")), superscriptshift = \"0\"), Typesetti ng:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0e m\", rspace = \"0.0em\")), mathvariant = \"normal\"), Typesetting:-mo( \"≦\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"f\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mfenced(Types etting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"k\" , italic = \"true\", mathvariant = \"italic\")), superscriptshift = \" 0\")), mathvariant = \"normal\"), Typesetting:-mo(\"−\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \+ \"0.2222222em\"), Typesetting:-mi(\"ασ\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mo(\" \", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Type setting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-mi(\"d \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Ty pesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"k\", italic = \"tru e\", mathvariant = \"italic\")), superscriptshift = \"0\"), Typesettin g:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mo(\"∇\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.1111111em\") , Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-msup(Typesetting: -mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:- mrow(Typesetting:-mi(\"k\", italic = \"true\", mathvariant = \"italic \")), superscriptshift = \"0\")), mathvariant = \"normal\"));" "-I%mro wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'66-I#moGF$6-Q\"~F'/%,ma thvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF4/%* symmetricGF4/%(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceGQ&0. 0emF'/%'rspaceGFCF+F+F+F+F+-I#miGF$6%Q\"fF'/%'italicGQ%trueF'/F0Q'ital icF'-I(mfencedGF$6$-F#6*-I%msupGF$6%-FG6%Q\"xF'FJFM-F#6#-FG6%Q\"kF'FJF M/%1superscriptshiftGQ\"0F'F+-F,6-Q\"+F'F/F2F5F7F9F;F=F?/FBQ,0.2222222 emF'/FEF`oF+-FG6%Q(αF'/FKF4F/F+-FU6%-FG6%Q\"dF'FJFM-F#6$F+FfnFin F+F/-F,6-Q0≦F'F/F2F5F7F9F;F=F?/FBQ,0.2777778emF'/FEFapFF -FP6$-F#6#FTF/-F,6-Q(−F'F/F2F5F7F9F;F=F?F_oFao-FG6%Q/α&sig ma;F'FJFMF+F+FfoF+-F,6-Q(∇F'F/F2F5F7F9F;F=F?FA/FEQ,0.1111111emF' FFFcp" }{TEXT 200 22 " holds. we start with " }{XPPEDIT 200 0 "Typeset ting:-mrow(Typesetting:-mi(\"α\", italic = \"false\", mathvarian t = \"normal\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mi(\"s\", italic = \"true\", mathvariant = \"italic\"));" "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q(α F'/%'italicGQ&falseF'/%,mathvariantGQ'normalF'-I#moGF$6-Q\"=F'F2/%&fen ceGF1/%*separatorGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movab lelimitsGF1/%'accentGF1/%'lspaceGQ,0.2777778emF'/%'rspaceGFI-F,6%Q\"sF '/F0Q%trueF'/F3Q'italicF'" }{TEXT 200 47 " and if it does not work we \+ chose the value of " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-m i(\"α\", italic = \"false\", mathvariant = \"normal\"), Typesett ing:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0 em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6$-I#miGF$6%Q(αF'/%'italicGQ&falseF'/%,mathvariantG Q'normalF'-I#moGF$6-Q\"~F'F2/%&fenceGF1/%*separatorGF1/%)stretchyGF1/% *symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%'accentGF1/%'lspaceGQ&0 .0emF'/%'rspaceGFI" }{TEXT 200 5 " as " }{XPPEDIT 200 0 "Typesetting: -mrow(Typesetting:-mi(\"αβ\", italic = \"true\", mathvarian t = \"italic\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+mo dulenameG6\"I,TypesettingGI(_syslibGF'6$-I#miGF$6%Q.αβF'/%' italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"~F'/F3Q'normalF'/ %&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeo pGF=/%.movablelimitsGF=/%'accentGF=/%'lspaceGQ&0.0emF'/%'rspaceGFL" } {TEXT 200 12 " and so on.\n" }{TEXT 200 96 "We write an expression for this Armijo condition , later we'll check the sign of this expression " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "arc:=evalf(eval(f,[x1= xkk[1],x2=xkk[2]])- eval(f,[x1=xk[1],x2=xk[2]])-al*sig*DotProduct(Vect or(2,dk),Vector(2,gk)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 20 "We d efine values of " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-mi( \"σ\", italic = \"false\", mathvariant = \"normal\"), Typesettin g:-mo(\",\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.3333333em\"), Typesetting:-mo(\" \", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Type setting:-mi(\"s\", italic = \"true\", mathvariant = \"italic\"), Types etting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", separat or = \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.3333333em\"), Typesetting:-mo(\" \", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), \+ Typesetting:-mi(\"β\", italic = \"false\", mathvariant = \"normal \"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6*-I#miGF$6%Q(σF'/%'italicGQ&falseF' /%,mathvariantGQ'normalF'-I#moGF$6-Q\",F'F2/%&fenceGF1/%*separatorGQ%t rueF'/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%'a ccentGF1/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F66-Q\"~F'F2F9/F< F1F>F@FBFDFFFH/FLFJ-F,6%Q\"sF'/F0F=/F3Q'italicF'F5FN-F,6%Q'βF'F/F 2FN" }{TEXT 200 33 " and the initial point xk=[2,0] " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sig:=.1; s:=1; bet:=0.5;xk:=Vector( [2,0]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 48 "Initial value of the al=s, (variable al denotes " }{XPPEDIT 200 0 "Typesetting:-mrow(Types etting:-mi(\"α\", italic = \"false\", mathvariant = \"normal\")) ;" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q (αF'/%'italicGQ&falseF'/%,mathvariantGQ'normalF'" }{TEXT 200 44 "), variable gk has a value of the gradient " }{XPPEDIT 200 0 "Typese tting:-mrow(Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6#-I#moGF$6-Q\"~F'/%,mathvariantGQ'n ormalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF4/%*symmetricGF4/ %(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceGQ&0.0emF'/%'rspac eGFC" }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-mo(\"∇\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \" 0.1111111em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\" f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mfence d(Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi (\"k\", italic = \"true\", mathvariant = \"italic\")), superscriptshif t = \"0\")), mathvariant = \"normal\"), Typesetting:-mo(\",\", mathvar iant = \"normal\", fence = \"false\", separator = \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333 333em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6(-I#moGF$6-Q&∇F'/%,mathvariantGQ'n ormalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF4/%*symmetricGF4/ %(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceGQ&0.0emF'/%'rspac eGQ,0.1111111emF'-F,6-Q\"~F'F/F2F5F7F9F;F=F?FA/FEFC-I#miGF$6%Q\"fF'/%' italicGQ%trueF'/F0Q'italicF'-I(mfencedGF$6$-F#6#-I%msupGF$6%-FL6%Q\"xF 'FOFR-F#6#-FL6%Q\"kF'FOFR/%1superscriptshiftGQ\"0F'F/-F,6-Q\",F'F/F2/F 6FQF7F9F;F=F?FA/FEQ,0.3333333emF'FG" }{TEXT 200 32 " dk presents a dir ection vector " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-mi(\" \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\" \+ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-msup(Typesetting:-mi(\"d\", italic = \"tr ue\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mo(\" \+ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mi(\"k\", italic = \"true\", mathvariant \+ = \"italic\")), superscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q!F'/%'italicGQ%trueF'/%,math variantGQ'italicF'-I#moGF$6-Q\"~F'/F3Q'normalF'/%&fenceGQ&falseF'/%*se paratorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF =/%'accentGF=/%'lspaceGQ&0.0emF'/%'rspaceGFL-I%msupGF$6%-F,6%Q\"dF'F/F 2-F#6$F5-F,6%Q\"kF'F/F2/%1superscriptshiftGQ\"0F'" }{TEXT 200 14 "(in \+ this case " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-msup(Types etting:-mi(\"d\", italic = \"true\", mathvariant = \"italic\"), Typese tting:-mrow(Typesetting:-mi(\"k\", italic = \"true\", mathvariant = \" italic\")), superscriptshift = \"0\"), Typesetting:-mo(\"=\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \" 0.2777778em\"), Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mo(\"∇\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.1111111em\"), Typesetting:-mi(\"f\", \+ italic = \"true\", mathvariant = \"italic\"), Typesetting:-mfenced(Typ esetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"tru e\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"k \", italic = \"true\", mathvariant = \"italic\")), superscriptshift = \+ \"0\")), mathvariant = \"normal\"), Typesetting:-mi(\"\", italic = \"t rue\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6)-I%msupGF$6%-I#miGF$6%Q\"dF'/%'italicGQ%trueF'/% ,mathvariantGQ'italicF'-F#6#-F/6%Q\"kF'F2F5/%1superscriptshiftGQ\"0F'- I#moGF$6-Q\"=F'/F6Q'normalF'/%&fenceGQ&falseF'/%*separatorGFH/%)stretc hyGFH/%*symmetricGFH/%(largeopGFH/%.movablelimitsGFH/%'accentGFH/%'lsp aceGQ,0.2777778emF'/%'rspaceGFW-FA6-Q*&uminus0;F'FDFFFIFKFMFOFQFS/FVQ, 0.2222222emF'/FYFhn-FA6-Q&∇F'FDFFFIFKFMFOFQFS/FVQ&0.0emF'/FYQ,0.11 11111emF'-F/6%Q\"fF'F2F5-I(mfencedGF$6$-F#6#-F,6%-F/6%Q\"xF'F2F5F8F=FD -F/6%Q!F'F2F5" }{TEXT 200 71 " for the steepest descent method), fk re presents the value of function " }{XPPEDIT 200 0 "Typesetting:-mrow(Ty pesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typ esetting:-mfenced(Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( \"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow (Typesetting:-mi(\"k\", italic = \"true\", mathvariant = \"italic\")), superscriptshift = \"0\")), mathvariant = \"normal\"), Typesetting:-m o(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6%-I#miGF$6%Q\"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'- I(mfencedGF$6$-F#6#-I%msupGF$6%-F,6%Q\"xF'F/F2-F#6#-F,6%Q\"kF'F/F2/%1s uperscriptshiftGQ\"0F'/F3Q'normalF'-I#moGF$6-Q\"~F'FH/%&fenceGQ&falseF '/%*separatorGFP/%)stretchyGFP/%*symmetricGFP/%(largeopGFP/%.movableli mitsGFP/%'accentGFP/%'lspaceGQ&0.0emF'/%'rspaceGFin" }{TEXT 200 14 " a t the point " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-msup(Typ esetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Type setting:-mrow(Typesetting:-mi(\"k\", italic = \"true\", mathvariant = \+ \"italic\")), superscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6#-I%msupGF$6%-I#miGF$6%Q\"xF'/%'italicGQ% trueF'/%,mathvariantGQ'italicF'-F#6#-F/6%Q\"kF'F2F5/%1superscriptshift GQ\"0F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "al:=s; gk:=eval( Vector(gf),[x1=xk[1],x2=xk[2]]);dk:=-gk; fk:=eval(f,[x1=xk[1],x2=xk[2] ]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 20 "xkk presents vector " } {XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\" x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(T ypesetting:-mi(\"k\", italic = \"true\", mathvariant = \"italic\")), s uperscriptshift = \"0\"), Typesetting:-mo(\"+\", mathvariant = \"norma l\", fence = \"false\", separator = \"false\", stretchy = \"false\", s ymmetric = \"false\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\" ), Typesetting:-mi(\"α\", italic = \"false\", mathvariant = \"no rmal\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typeset ting:-mi(\"d\", italic = \"true\", mathvariant = \"italic\"), Typesett ing:-mrow(Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"k\", it alic = \"true\", mathvariant = \"italic\")), superscriptshift = \"0\") );" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%msupGF$ 6%-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-F/6 %Q\"kF'F2F5/%1superscriptshiftGQ\"0F'-I#moGF$6-Q\"+F'/F6Q'normalF'/%&f enceGQ&falseF'/%*separatorGFH/%)stretchyGFH/%*symmetricGFH/%(largeopGF H/%.movablelimitsGFH/%'accentGFH/%'lspaceGQ,0.2222222emF'/%'rspaceGFW- F/6%Q(αF'/F3FHFD-FA6-Q\"~F'FDFFFIFKFMFOFQFS/FVQ&0.0emF'/FYF\\o-F ,6%-F/6%Q\"dF'F2F5-F#6$FhnF:F=" }{TEXT 200 21 " for chosen value of " }{XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-mi(\"α\", italic = \"false\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q(αF'/%'italicGQ&falseF' /%,mathvariantGQ'normalF'" }{TEXT 200 35 " and we check the Armijo con dition " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xkk:=xk+al*dk;ar c;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 132 "The result arc is positiv e, so we decrease value of al, update xkk and check arc again and then repeat it while arc becames negative" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "al:=al*bet;xkk:=xk+al*dk;arc;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "al:=al*bet;xkk:=xk+al*dk; arc;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "al:=al*bet;xkk:=xk+al*dk; arc;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "al:=al*bet;xkk:=xk+al*dk; ar c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "al:=al*bet;xkk:=xk+al *dk; arc;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "al:=al*bet;xkk :=xk+al*dk; arc;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 8 "Now the " } {XPPEDIT 200 0 "Typesetting:-mrow(Typesetting:-mi(\"α\", italic \+ = \"false\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q(αF'/%'italicGQ&falseF'/% ,mathvariantGQ'normalF'" }{TEXT 200 136 " satisfies the Armijo rule an d our point xkk becomes the point xk and we repeat for this point the \+ above procedure of choosing stepsize " }{XPPEDIT 200 0 "Typesetting:-m row(Typesetting:-mi(\"α\", italic = \"false\", mathvariant = \"n ormal\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6$-I#miGF$6%Q(αF'/%'italicGQ&fals eF'/%,mathvariantGQ'normalF'-I#moGF$6-Q\"~F'F2/%&fenceGF1/%*separatorG F1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%'acce ntGF1/%'lspaceGQ&0.0emF'/%'rspaceGFI" }{TEXT 200 54 " but this time ma ke this procedure automatic by using " }{TEXT 200 16 "do while end do " }{TEXT 200 75 "command (the integer i tells us how many times the v ariable al was change)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 " xk:=xkk; al:=s; i:=0; gk:=eval(Vector(gf),[x1=xk[1],x2=xk[2]]);dk:=-gk ; fk:=eval(f,[x1=xk[1],x2=xk[2]]);ngk:=evalf(Norm(gk,2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 " while arc >=0 do al:=al*bet; xkk: =xk+al*dk; i:=i+1;fk:=eval(f,[x1=xkk[1],x2=xkk[2]]);ngk:=eval(Norm(gk ,2),[x1=xk[1],x2=xk[2]]); end do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 42 "Now we have obtained the correct stepsize " }{XPPEDIT 200 0 "T ypesetting:-mrow(Typesetting:-mi(\"α\", italic = \"false\", math variant = \"normal\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_ syslibGF'6#-I#miGF$6%Q(αF'/%'italicGQ&falseF'/%,mathvariantGQ'no rmalF'" }{TEXT 200 48 " and can continue with making step xk as befor e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "xk:=xkk; al:=s; i:=0; gk:=eval(Vector(gf),[x1=xk[1],x2=xk[2]]);dk:=-gk; fk:=eval(f,[x1=xk[1 ],x2=xk[2]]);ngk:=evalf(Norm(gk,2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 " while arc >=0 do al:=al*bet;xkk:=xk+al*dk; i:=i+1;f k:=eval(f,[x1=xkk[1],x2=xkk[2]]); end do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 214 62 "Again the correct stepsize was found and we make the nex t step" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "xk:=xkk; al:=s; \+ i:=0; gk:=eval(Vector(gf),[x1=xk[1],x2=xk[2]]);dk:=-gk; fk:=eval(f,[x1 =xk[1],x2=xk[2]]);ngk:=eval(Norm(gk,2),[x1=xk[1],x2=xk[2]]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 " while arc >=0 do al:=al*bet ;xkk:=xk+al*dk; i:=i+1;fk:=eval(f,[x1=xkk[1],x2=xkk[2]]); end do;" }} }{EXCHG {PARA 210 "" 0 "" {TEXT 209 19 "Again corrst value " } {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"α\", family = \+ \"Arial\", italic = \"false\", mathvariant = \"normal\"), Typesetting: -mo(\" \", family = \"Arial\", mathvariant = \"normal\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6$-I#miGF$6&Q(αF'/%'familyGQ&ArialF '/%'italicGQ&falseF'/%,mathvariantGQ'normalF'-I#moGF$6.Q\"~F'F/F5/%&fe nceGF4/%*separatorGF4/%)stretchyGF4/%*symmetricGF4/%(largeopGF4/%.mova blelimitsGF4/%'accentGF4/%'lspaceGQ&0.0emF'/%'rspaceGFL" }{TEXT 209 34 " was found and we do the next step" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "xk:=xkk; al:=s; i:=0; gk:=eval(Vector(gf),[x1=xk[1], x2=xk[2]]);dk:=-gk; fk:=eval(f,[x1=xk[1],x2=xk[2]]);ngk:=eval(Norm(gk, 2),[x1=xk[1],x2=xk[2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 " while arc >=0 do al:=al*bet;xkk:=xk+al*dk; i:=i+1;fk:=eval(f,[x1=xk k[1],x2=xkk[2]]); end do;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "xk:=xkk; al:=s; i:=0; gk:=eval(Vector(gf),[x1=xk[1],x2=xk[2]]);dk :=-gk; fk:=eval(f,[x1=xk[1],x2=xk[2]]);ngk:=eval(Norm(gk,2),[x1=xk[1], x2=xk[2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "while arc >= 0 do al:=al*bet;xkk:=xk+al*dk; i:=i+1;fk:=eval(f,[x1=xkk[1],x2=xkk[2] ]); end do;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "xk:=xkk; al :=s; i:=0; gk:=eval(Vector(gf),[x1=xk[1],x2=xk[2]]);dk:=-gk; fk:=eval( f,[x1=xk[1],x2=xk[2]]);ngk:=eval(Norm(gk,2),[x1=xk[1],x2=xk[2]]);\n" } {MPLTEXT 1 0 93 "while arc >=0 do al:=al*bet;xkk:=xk+al*dk; i:=i+1;fk :=eval(f,[x1=xkk[1],x2=xkk[2]]); end do;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "xk:=xkk; al:=s; i:=0; gk:=eval(Vector(gf),[x1=xk[1] ,x2=xk[2]]);dk:=-gk; fk:=eval(f,[x1=xk[1],x2=xk[2]]);ngk:=eval(Norm(gk ,2),[x1=xk[1],x2=xk[2]]);\n" }{MPLTEXT 1 0 93 "while arc >=0 do al:=a l*bet;xkk:=xk+al*dk; i:=i+1;fk:=eval(f,[x1=xkk[1],x2=xkk[2]]); end do; " }}}{SECT 0 {PARA 211 "" 0 "" {TEXT 209 27 "Gradient Method: Automati on" }}{EXCHG {PARA 212 "" 0 "" {TEXT 209 153 "By using the previous ex ample we can write a Maple procedure which automates Gradient Method \+ computations. The input of this procedure will be function " }{TEXT 215 1 "f" }{TEXT 209 16 ", the dimension " }{TEXT 216 1 "n" }{TEXT 209 29 " of the vector x, quantities " }{TEXT 217 1 "s" }{TEXT 209 2 " , " }{TEXT 218 3 "sig" }{TEXT 209 5 " and " }{TEXT 219 3 "bet" }{TEXT 209 42 ", quantity eps (the termination rule is " }{XPPEDIT 2 0 "Typ esetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mo( \"∇\", family = \"Arial\", bold = \"true\", mathvariant = \"bold \", fontweight = \"bold\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.1111111em\"), Typesetting:-mi(\"f\", family = \"Arial\", bold = \+ \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), family = \"Arial\", bold = \"true\", mathvariant = \"bo ld\", fontweight = \"bold\", open = \"∥\", close = \"∥\"), Typ esetting:-mo(\"<\", family = \"Arial\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778e m\", rspace = \"0.2777778em\"), Typesetting:-mi(\"eps\", family = \"Ar ial\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic \", fontweight = \"bold\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesettin gGI(_syslibGF'6%-I(mfencedGF$6)-F#6$-I#moGF$60Q(∇F'/%'familyGQ&A rialF'/%%boldGQ%trueF'/%,mathvariantGQ%boldF'/%+fontweightGF " 0 "" {MPLTEXT 1 0 35 "grad:=proc(f,n,s,sig,bet,eps ,N,x0)\n" }{MPLTEXT 1 0 43 "local al,xk,xkk,dk, fk,gk, ngk, gf, i,j,w; \n" }{MPLTEXT 1 0 46 "gf:=Vector(n,Gradient(f,[seq(x||i,i=1..n)]));\n" }{MPLTEXT 1 0 18 "xk:=Vector(n,x0);\n" }{MPLTEXT 1 0 62 "gk:=eval(gf, [seq(x||j=xk[j],j=1..n)]);ngk:=evalf(Norm(gk,2));\n" }{MPLTEXT 1 0 43 "fk:=eval(f,[seq(x||j=xk[j],j=1..n)]);i:=1;\n" }{MPLTEXT 1 0 37 "while (i<=N and ngk>=eps) do i:=i+1;\n" }{MPLTEXT 1 0 86 "al:=evalf(s); gk: =eval(Vector(gf),[seq(x||j=xk[j],j=1..n)]); ngk:=evalf(Norm(gk,2)); \n " }{MPLTEXT 1 0 25 "dk:=-gk; xkk:=xk+al*dk; \n" }{MPLTEXT 1 0 159 "whi le evalf(eval(f,[seq(x||j=xkk[j],j=1..n)])- eval(f,[seq(x||j=xk[j],j=1 ..n)])-al*sig*DotProduct(Vector(n,dk),Vector(n,gk)))>=0 do al:=al*bet; xkk:=xk+al*dk; \n" }{MPLTEXT 1 0 8 "end do;\n" }{MPLTEXT 1 0 46 "xk:=x kk;fk:=eval(f,[seq(x||j=xk[j],j=1..n)]);\n" }{MPLTEXT 1 0 20 "w[i]:=[x k,fk,ngk]; \n" }{MPLTEXT 1 0 8 "end do;\n" }{MPLTEXT 1 0 10 "print(i); \n" }{MPLTEXT 1 0 3 "w;\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ww:=grad(f,2,1,. 1,0.5,.05,648,[2,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ww [169][1];ww[169][2]; ww[169][3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ww[5];" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 209 24 "You can use \+ the command " }{TEXT 222 6 "print " }{TEXT 209 70 "to print results of your computation. For example, last 10 iterations " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for k from 159 to 169 do print(ww[k]) end \+ do;" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 234 1 " " }{TEXT 223 22 "Newton method: Example" }}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 0 22 "with(V ectorCalculus): " }{TEXT 224 17 "Loading packages " }{TEXT 225 14 "Vec torCalculus" }{TEXT 226 5 " and " }{TEXT 227 13 "LinearAlgebra" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "BasisFormat(false);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=(x1-1)^2+(2*x1^2-x2)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot3d(f,x1=0..4,x2=-1..4 ,axes=boxed);" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 48 "gf:=Grad ient(f,[x1,x2]); hf:=Hessian(f,[x1,x2]);" }{TEXT 228 36 " Computing gr adient and hessian of f" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 0 25 "dk:=LinearSolve(hf,-gf); " }{TEXT 229 17 "Computing vector " } {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"d \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Ty pesetting:-mi(\"k\", italic = \"true\", mathvariant = \"italic\")), su perscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6#-I%msupGF$6%-I#miGF$6%Q\"dF'/%'italicGQ%trueF'/%,mathvari antGQ'italicF'-F#6#-F/6%Q\"kF'F2F5/%1superscriptshiftGQ\"0F'" }}} {EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 0 17 "xkk:=+dk; " }{TEXT 230 24 "Newton method iteration " }{XPPEDIT 2 0 "Typesetting:-mrow(Typ esetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"k\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.222222 2em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), superscript shift = \"0\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6#-I%msupGF$6%-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-F#6%-F/6%Q\"kF'F2F5-I#moGF$6-Q\"+F'/F6Q'normalF'/%&fenceGQ&false F'/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movablel imitsGFE/%'accentGFE/%'lspaceGQ,0.2222222emF'/%'rspaceGFT-I#mnGF$6$Q\" 1F'FA/%1superscriptshiftGQ\"0F'" }}}{EXCHG {PARA 218 "> " 0 "" {MPLTEXT 1 0 65 "xk:=<2,0>; w[0]:=[xk[1],xk[2],evalf(subs(x1=xk[1],x2= xk[2],f))]; " }{TEXT 231 225 "Initial point [2,0], computing the first point after Newton iteration, the array w: two first components of w[ k] are coordinates of the point at iteration k, the last component is \+ the value of objective function at this point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "xk :=evalf(subs(x1=xk[1],x2=xk[2], xkk)); w[1]:=[xk[1],xk[2],subs(x1=xk[1 ],x2=xk[2],f)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xk:=eval (xkk,[x1=xk[1],x2=xk[2]]); w[2]:=[xk[1],xk[2],eval(f,[x1=xk[1],x2=xk[2 ]])];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xk:=eval(xkk,[x1=x k[1],x2=xk[2]]); w[3]:=[xk[1],xk[2],eval(f,[x1=xk[1],x2=xk[2]])];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xk:=eval(xkk,[x1=xk[1],x2=xk [2]]); w[4]:=[xk[1],xk[2],eval(f,[x1=xk[1],x2=xk[2]])];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xk:=eval(xkk,[x1=xk[1],x2=xk[2]]); \+ w[5]:=[xk[1],xk[2],eval(f,[x1=xk[1],x2=xk[2]])];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "w[4][3]; " }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 0 31 "eval(gf,[x1=xk[1],x2=xk[2]]); " }{TEXT 232 39 "Evalu ation of the gradient at the point" }}}{EXCHG {PARA 220 "> " 0 "" {TEXT 209 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plots[contourplot] (f,x1=0..7,x2=-1..6,contours=[.05,0.1.35,1,3,4,6,8,9],numpoints=5500); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "p1:=plots[contourplot]( f,x1=0..7,x2=-1..7,contours=[0.1.35,1,3,4,6,8,9],numpoints=5500):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "p2:=plot([seq([w[k][1],w[k][ 2]],k=0..5)],style=point,symbol=circle,color=blue, thickness=3):" }}} {EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 23 "plots[display](p1,p2); " } {TEXT 233 38 "Visualization of computational results" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "p2:=plot([seq([w[k][1],w[k][2]],k=0..5)], style=line,symbol=circle,color=blue, thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plots[display](p1,p2);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 234 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 125 100 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }