{VERSION 5 0 "Mac OS X" "5.0" } {USTYLETAB {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ord ered List 5" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Lef t Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Headi ng 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Ordered List 4" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Headin g 2256" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3257" -1 205 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "L ist Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 } {PSTYLE "Ordered List 2" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{CSTYLE "Help Va riable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Tex t" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold " -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Page Number " -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ita lic Small" -1 201 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "He lp Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 202 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small201" -1 203 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Na me" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Ou tput" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dicti onary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 } {CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small201203" -1 204 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 205 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 206 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 207 28 "Polynomial Approximation - I" }}{PARA 204 "" 0 "" {TEXT 208 30 "Legandre vs Taylor Polynomials " }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 42 "Wor ksheet by Mike May, S.J., maymk@slu.edu" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 210 8 "Over view" }}{EXCHG {PARA 0 "" 0 "" {TEXT 209 211 "One of the uses of the G ram Schmidt process is to produce orthogonal projections of a vector o nto a subspace. This can also be thought of as finding the best appro ximation in the subspace under the given norm. " }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 415 "Finding Fourier series of functions is an example of this process. When we find a Fourier p olynomial for a function, we are actually approximating a periodic fun ction by the best fitting trig polynomials of specified degree. Best \+ fitting is defined by a least sum of squares rule. To measure the dis tance between two functions we integrate the square of the difference \+ of the two functions. Thus = " }{XPPEDIT 18 0 "Typesetti ng:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\" ), Typesetting:-msubsup(Typesetting:-mo(\"∫\", form = \"\", fence \+ = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\" , stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreg round = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mro w(Typesetting:-mo(\"&uminus0;\", form = \"prefix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[ 0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"1\")), Ty pesetting:-mn(\"1\"), superscriptshift = \"2\", subscriptshift = \"0\" ), Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typ esetting:-mrow(Typesetting:-mi(\"f\"), Typesetting:-mo(\"&ApplyFunctio n;\", form = \"infix\", fence = \"false\", separator = \"false\", lspa ce = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"fa lse\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", m ovablelimits = \"false\", accent = \"false\", font_style_name = \"2D C omment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255 ,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix \", fence = \"true\", separator = \"false\", lspace = \"thinmathspace \", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"fal se\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", font_style_name = \"2D Co mment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255, 255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"x\")), Typesetting:-m o(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", \+ lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0 ,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typ esetting:-mo(\"⁢\", form = \"infix\", fence = \"false\" , separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy \+ = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[ 0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetti ng:-mi(\"g\"), Typesetting:-mo(\"⁡\", form = \"infix\", \+ fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \+ \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmat hspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"po stfix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths pace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), \+ Typesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width \+ = \"0.3 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting:- mo(\"ⅆ\", form = \"prefix\", fence = \"false\", separato r = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x\")), Typesetting :-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6% -I#miGF$6#Q!F'-F#6*F+-I(msubsupGF$6'-I#moGF$63Q&∫F'/%%formGF./%&fe nceGQ&falseF'/%*separatorGF " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 210 17 "Technical \+ details" }}{EXCHG {PARA 0 "" 0 "" {TEXT 209 41 "We start by loading th e packages we need." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with (orthopoly); with(plots):" }}{PARA 11 "" 1 "" {XPPMATH 20 "7(I\"GG6\"I \"HGF$I\"LGF$I\"PGF$I\"TGF$I\"UGF$" }}{PARA 7 "" 1 "" {TEXT 211 49 "Wa rning, the name changecoords has been redefined" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 205 "For this worksheet we will work with Legandre poly nomials. They are orthogonal in the space of continuous functions on \+ [-1, 1] with the usual integration inner product. Look at the help pa ge for details." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?P" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 209 178 "Time for some technical commands . We want to assume that i and j are integers to simplify the integra tions we need to do. We also don't want them printed with a trailing \+ tilde." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "assume('i',intege r): assume('j',integer):\n" }{MPLTEXT 1 0 25 "interface(showassumed=0) :" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 210 66 "An extended example, appr oximating the function f(x) = sin(3*Pi*x)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 209 54 "We will start with the function func(x) = sin(3\271x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "func := x -> sin(5*Pi*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%funcG6\"f*6#I\"xGF$F$6$I)operator GF$I&arrowGF$F$-I$sinGF$6#,$*(\"\"&\"\"\"I#PiG%*protectedGF1F'F1F1F$F$ F$" }}}{SECT 0 {PARA 205 "" 0 "" {TEXT 212 41 "Computing Taylor and Le gandre polynomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT 209 242 "We would l ike to start by computing coefficients. We will find the coefficient s for both the Legangre polynomials and for Taylor polynomials. If P( n,x) is the nth Legandre polynomial, the projection coeffient of f(x) \+ onto P(n,x) should be " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting: -mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msubs up(Typesetting:-mo(\"∫\", form = \"\", fence = \"false\", separato r = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\" &uminus0;\", form = \"prefix\", fence = \"false\", separator = \"false \", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\"), Typesetting:-mn(\"1\")), Typesetting:-mn(\"1\"), superscriptshift = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typese tting:-mi(\"f\"), Typesetting:-mo(\"⁡\", form = \"infix \", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"i nfinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fal se\", accent = \"false\", font_style_name = \"2D Comment\", size = \"1 2\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typese tting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\" , separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinma thspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"po stfix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths pace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mo(\"&Invis ibleTimes;\", form = \"infix\", fence = \"false\", separator = \"false \", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"P\"), Typese tting:-mo(\"⁡\", form = \"infix\", fence = \"false\", se parator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \" false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1 \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0 ,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting: -mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \" true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0 ]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-m n(\"0\"), Typesetting:-mo(\",\", form = \"infix\", fence = \"false\", \+ separator = \"true\", lspace = \"0em\", rspace = \"verythickmathspace \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true \", separator = \"false\", lspace = \"thinmathspace\", rspace = \"very thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Comment\", size \+ = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), \+ Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.3 em\", depth = \+ \"0.0 ex\", linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\" , form = \"prefix\", fence = \"false\", separator = \"false\", lspace \+ = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mi(\"\"));" "-I%mrowG 6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6*F+-I (msubsupGF$6'-I#moGF$63Q&∫F'/%%formGF./%&fenceGQ&falseF'/%*separat orGFF\\qF?/FCQ3verythickmathspaceF'FDFFFHFKFNFPFRFTFWFZFgnFcqFf qF+F+F+-I'mspaceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGF gs/%*linebreakGQ%autoF'-F563Q0ⅆF'F_oF:F=F?FBFDFFFHFKFNFP FRFTFWFZFgnFcqF+" }{TEXT 209 14 " divided by " }{XPPEDIT 18 0 "Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-msubsup(Typesetting:-mo(\"∫\", form = \"\", fe nce = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0 em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinit y\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", f oreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting: -mrow(Typesetting:-mo(\"&uminus0;\", form = \"prefix\", fence = \"fals e\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretc hy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"f alse\", font_style_name = \"2D Comment\", size = \"12\", foreground = \+ \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"1\")), Typesetting:-mn(\"1\"), superscriptshift = \"2\", subscriptshift = \" 0\"), Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-mrow(Typesetting:-mi(\"P\"), Typesetting:-mo(\"&ApplyFunc tion;\", form = \"infix\", fence = \"false\", separator = \"false\", l space = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \" prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmath space\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"n\"), Typesett ing:-mo(\",\", form = \"infix\", fence = \"false\", separator = \"true \", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"fa lse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", f ont_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mi(\"x\")), Typesett ing:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"fal se\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stre tchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsiz e = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")) , Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"fa lse\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stre tchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsi ze = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typ esetting:-mi(\"P\"), Typesetting:-mo(\"⁡\", form = \"inf ix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspa ce = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"2D Comment\", size = \+ \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"tru e\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thi nmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \" infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", font_style_name = \"2D Comment\", size = \" 12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Types etting:-mrow(Typesetting:-mi(\"n\"), Typesetting:-mo(\",\", form = \"i nfix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rsp ace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"fals e\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", font_style_name = \"2D Com ment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,2 55,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \"p ostfix\", fence = \"true\", separator = \"false\", lspace = \"thinmath space\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.3 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting: -mo(\"ⅆ\", form = \"prefix\", fence = \"false\", separat or = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x\")), Typesetting :-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6% -I#miGF$6#Q!F'-F#6*F+-I(msubsupGF$6'-I#moGF$63Q&∫F'/%%formGF./%&fe nceGQ&falseF'/%*separatorGFF\\qF?/FCQ3 verythickmathspaceF'FDFFFHFKFNFPFRFTFWFZFgn-F,6#Q\"xF'-F563Q\")F'/F9Q( postfixF'F[qF=F]q/FCQ2verythinmathspaceF'F`qFFFHFKFNFPFRFTFWFZFgnF+-F5 63Q1⁢F'FdpF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnF\\pF+F+-I'msp aceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&depthGF^s/%*linebrea kGQ%autoF'-F563Q0ⅆF'F_oF:F=F?FBFDFFFHFKFNFPFRFTFWFZFgnF \\rF+" }{TEXT 209 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " func := x -> sin(5*Pi*x);\n" }{MPLTEXT 1 0 46 "normP[0] := evalf(Int(( P(0,x))^2, x=-1..1)):\n" }{MPLTEXT 1 0 55 "coeffP[0] := evalf(Int(P(0, x)*func(x)*1.0, x=-1..1)):\n" }{MPLTEXT 1 0 40 "taylco[0] := evalf(sub s(x=0,func(x))):\n" }{MPLTEXT 1 0 23 "for i from 1 to 20 do\n" } {MPLTEXT 1 0 46 "normP[i] := evalf(Int((P(i,x))^2, x=-1..1)):\n" } {MPLTEXT 1 0 65 "coeffP[i] := evalf(Int(simplify(P(i,x)*func(x)*1.0), \+ x=-1..1)):\n" }{MPLTEXT 1 0 53 "taylco[i] := evalf(subs(x=0,diff(func( x),x$i))/i!):\n" }{MPLTEXT 1 0 3 "od:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%funcG6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$-I$sinG6$%*protect edGI(_syslibGF$6#,$*(\"\"&\"\"\"I#PiGF.F4F'F4F4F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 89 "Now we define procedures that define the approximating polynomials for specified degrees." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "Lapprox := proc(m,x) \n" }{MPLTEXT 1 0 16 " \+ local count:\n" }{MPLTEXT 1 0 58 " sum(P(count,x)*coeffP[count]/norm P[count],count=0..m):\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 23 "Ta pprox := proc(m,x) \n" }{MPLTEXT 1 0 16 " local count:\n" }{MPLTEXT 1 0 43 " sum(x^count*taylco[count], count=0..m):\n" }{MPLTEXT 1 0 4 " end;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(LapproxG6\"f*6$I\"mGF$I\"xGF$ 6#I&countGF$F$F$-I$sumGF$6$*(-I\"PGF$6$F*F(\"\"\"&I'coeffPGF$F)F2&I&no rmPGF$F)!\"\"/F*;\"\"!F'F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Tap proxG6\"f*6$I\"mGF$I\"xGF$6#I&countGF$F$F$-I$sumGF$6$*&)F(F*\"\"\"&I't aylcoGF$F)F0/F*;\"\"!F'F$F$F$" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 205 0 "" }{TEXT 200 52 "Comparing the fit of Taylor and Legandre polynomia ls" }}{EXCHG {PARA 0 "" 0 "" {TEXT 209 86 "Compare the 3rd degree appr oximations and note that they are substantially different.." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Lapprox(3,x);\n" }{MPLTEXT 1 0 13 "Tapprox(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&$\"+b*y#oV! #5\"\"\"I\"xG6\"F'!\"\"*&$\"+yjNY5!\"*F')F(\"\"$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&$\"+Fjzq:!\")\"\"\"I\"xG6\"F'F'*&$\"+w4kfk!\"(F')F (\"\"$F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 190 "It is worth re calling that Taylor polynomials are \"best fitting\" in the sense of f itting very well at the point of expansion. Compare the graph of the \+ function and the rwo approximations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n:=20:\n" }{MPLTEXT 1 0 51 "pl1 := plot(func(x), x=-1. .1,y=-1..1, color=red):\n" }{MPLTEXT 1 0 58 "pl2 := plot(Lapprox(n,x), x=-1..1,y=-1..1, color=green):\n" }{MPLTEXT 1 0 57 "pl3 := plot(Tappr ox(n,x), x=-1..1,y=-1..1, color=blue):\n" }{MPLTEXT 1 0 29 "display(\{ pl1, pl2, pl3\});\n" }{MPLTEXT 1 0 71 "print(\"func with its 3rd degre Legandre and Taylor approximations.\"):" }}{PARA 13 "" 1 "" {TEXT 213 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6'-%+AXESLABELSG6'Q\"x6\" Q\"yF'-%%FONTG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%'CURVESG6$7iw7$$!\"\" \"\"!$!3Mint&*RBBh!#L7$$!3%pmm\"HU,\"*)*!#=$!3i=kW$e%f.*>$)F=7$$!3^m;H2Jdz#*F=$!3+bI:-O7^!*F=7$$!3vKL$3qwy*F=7$$!3&fmT&Q75y!*F=$!3 i5!Q\"G5%[#**F=7$$!35LeRs#zZ-*F=$!35,=p#)eU#***F=7$$!39***\\iId9(*)F=$ !349Vn04&**)**F=7$$!3YL$eRP8['))F=$!3([Dqk&>Qv(*F=7$$!3mmmmT%p\"e()F=$ !3$[w*Gc79(G*F=7$$!3qlm;/'=3l)F=$!3s&>&)\\>8J`)F=7$$!3%emmmwnMa)F=$!3u \")o6<>-PvF=7$$!3+mm;Hp6O%)F=$!3Zr?&p0CrK'F=7$$!3:mmm\"4m(G$)F=$!3'=m) =59uP\\F=7$$!3iL$eRZD>A)F=$!3nN&eC'3#eT$F=7$$!3)****\\i&[3:\")F=$!3Ol) 3/q=zz\"F=7$$!3Mm;aQUC3!)F=$!3)fB'>md'\\H\"!#>7$$!3\"QLL3i.9!zF=$\"3+K I1*GkDa\"F=7$$!3Mnmm\"p[B!yF=$\"3l./g5:1bIF=7$$!3'3++Dw$H.xF=$\"3!fmhn FXP\\%F=7$$!3RMLLL)QUg(F=$\"3;&4&G&\\bQ#eF=7$$!3\"ommT!R=0vF=$\"3)Hb_M KbK,(F=7$$!3tKL$3i_+I(F=$\"3wHAGe**o4*)F=7$$!3u****\\P8#\\4(F=$\"3zses :'[!*))*F=7$$!3(fmmT&HSoqF=$\"31%zIu5JB%**F=7$$!3ILL$3d%)=/(F=$\"3%)z5 -%yk$y**F=7$$!3i++](=m`,(F=$\"3gO4;Jr3(***F=7$$!3$ommT!y%)))pF=$\"33gZ qncY)***F=7$$!31LL$3UHB'pF=$\"3+R#[?+)\\#)**F=7$$!3Q++]P5\"e$pF=$\"3%R ]xi$=@\\**F=7$$!3rnm;aEH4pF=$\"3qv\"4\"=\\m)*)*F=7$$!3#RLL3FuF)oF=$\"3 X.YqS\\%4$)*F=7$$!3,,+]P2qwnF=$\"3Gf9YGk7\"R*F=7$$!3*pmmT?F1n'F=$\"3UJ &3jB#>\"p)F=7$$!3\")******\\Q*[c'F=$\"3PIN31Y!Rv(F=7$$!3tLL$e\\g\"fkF= $\"3:v-1m7A.mF=7$$!3mnmmTrU`jF=$\"3i)33oC53F&F=7$$!3[++](y$pZiF=$\"3OC 95=4M$z$F=7$$!3YM$ek.M*QhF=$\"3%4y))**\\)3l@F=7$$!3MnmT&Gu,.'F=$\"3Jt& y<>\"*zt%F^s7$$!3?+]PMXT@fF=$!310T$H,&GJ7F=7$$!33LLL$yaE\"eF=$!3^pp9[+ _+HF=7$$!3YmmTN\"for&F=$!3kt_X#*>Q-VF=7$$!3%)****\\([j5i&F=$!3g?[)pP4q g&F=7$$!3ALLeRyEDbF=$!3g4.Z-u\"\\y'F=7$$!3hmmm\">s%HaF=$!3\")[gc[^[4yF =7$$!3]LL$3x&y8_F=$!3c,#>Y5C9W*F=7$$!3Q+++]$*4)*\\F=$!3;Gc'HVb*****F=7 $$!3E++++t_\"y%F=$!3!)e/a*=2pT*F=7$$!38+++]_&\\c%F=$!3mFo)[h8Xv(F=7$$! 3R++++;ggWF=$!39y>-#=/-i'F=7$$!33+++]zCcVF=$!3LcdsnHT3`F=7$$!3z******* H%*=D%F=$!3([>>o#zIaQF=7$$!30+++]1aZTF=$!3qY-9@B(oH#F=7$$!3Xm;aQvx_SF= $!3%**zk6I$y!G)F^s7$$!3RLL3FW,eRF=$\"35y1_A&)H!f'F^s7$$!3M+]i:8DjQF=$ \"3we_J.EcJ@F=7$$!3umm;/#)[oPF=$\"3!\\&e*yl_pb$F=7$$!3=L$e*)p0el$F=$\" 3)>ugEL7q9&F=7$$!3<++v$>BJa$F=$\"36Xeo5&Rid'F=7$$!3(HjF[y)**F=7$$!3zLLek\"fO)HF=$\"3WJJ &y\"fq'***F=7$$!3)RLLLtIf$HF=$\"3cYb`84S\\**F=7$$!3M+]P4<*[#GF=$\"3\"zpF=7$$!3%RLekG'[$R#F=$\"3u T>kn*pZz&F=7$$!3QnmTNz>&H#F=$\"3SR30vscsWF=7$$!3%3+vVe4p>#F=$\"36Fpn) \\lR/$F=7$$!3EMLLL7i)4#F=$\"3/3sOB+&Ha\"F=7$$!3g+]PMLy!*>F=$!3\\h]8x)* pZ9F^s7$$!3#pm;aVXH)=F=$!3cIV?tfMG=F=7$$!3CL$ek`2^x\"F=$!3!R$pa[defMF= 7$$!3c****\\P'psm\"F=$!3SxUmZ_z\"*\\F=7$$!3k***\\iX#ek:F=$!3Pgr4))Gh=j F=7$$!3r*****\\F&*=Y\"F=$!3/4Y*yi*Q\"[(F=7$$!3z***\\P43#f8F=$!3$*Ho+'* z$*\\%)F=7$$!3')****\\74_c7F=$!3%eL\"R))H6*>*F=7$$!3:))F^s$!3X;rx wlJF)*F=7$$!3)3LLL3x%z#)F^s$!3;faF=v,P'*F=7$$!3uJLe9rc&H(F^s$!3m1[r;^; 6\"*F=7$$!3gKL$e9d;J'F^s$!3rR*)*Qv7\"o$)F=7$$!3XLL3xruF`F^s$!30XC2lRdD uF=7$$!3KMLL3s$QM%F^s$!3sP'=#=y,1jF=7$$!3'ym;aQdDG$F^s$!3Mv#)\\g'o2$ \\F=7$$!3T,+]ivF@AF^s$!3$>Yxd,2)=MF=7$$!3'\\L$eRx**f6F^s$!3'pT(RWT07=F =7$$!3\\^omm;zr)*!#@$!3fHv/m_f]:F^s7$$\"3&yK$3_Nl.5F^s$\"3)e4CAs7+d\"F =7$$\"3@CL$3-Dg5#F^s$\"3Y'\\%3el7[KF=7$$\"3e?Le*['R3KF^s$\"3w>Em4.4H[F =7$$\"3%pJL$ezw5VF^s$\"3'eIeL-=cE'F=7$$\"38&)**\\ibQq_F^s$\"3RT-esO# \\O(F=7$$\"3K`mmmJ+IiF^s$\"3aAADVT?(H)F=7$$\"3_@L$3x?'*=(F^s$\"3+hZZVn JT!*F=7$$\"3s*)***\\PQ#\\\")F^s$\"3Md4>ShQ!e*F=7$$\"3Nc;/E]Wn')F^s$\"3 tD*=i8IqH=**F=7$$\"3h*)\\7G$eQq*F^s $\"34G#o1T#=*)**F=7$$\"3ilm\"z\\1A-\"F=$\"3#G%p?()o\"R***F=7$$\"3%*)* \\7Gy%e7\"F=$\"3OQC)>'fC0)*F=7$$\"3GKLLe\"*[H7F=$\"3Q&o[#yOFd$*F=7$$\" 3.***\\PzglL\"F=$\"3aLiESWzM')F=7$$\"3ylm;HCjV9F=$\"3/==n@qjowF=7$$\"3 aKLekSq]:F=$\"30e6)Q'o1'['F=7$$\"3H*******pvxl\"F=$\"3Xep6$oc/7&F=7$$ \"3&*)**\\iSCDw\"F=$\"3C[c$R*zMWOF=7$$\"3g)***\\7JFn=F=$\"3IY'\\R5)zp? F=7$$\"3F)**\\(==-s>F=$\"3YzWPiyQ$R%F^s7$$\"3#z****\\_qn2#F=$!3z))Rafz )H?\"F=7$$\"3/)*\\P%G?\"y@F=$!3qRK$[#[ahFF=7$$\"3=)**\\P/q%zAF=$!3$eO/ $y\"f-D%F=7$$\"3I)*\\7.)>3Q#F=$!3CbzS$zz9j&F=7$$\"3U)***\\i&p@[#F=$!3V %3&pySFqoF=7$$\"3Q)*\\i!>,Zf#F=$!3M\\Bj[o'4/)F=7$$\"3L)**\\(=GB2FF=$!3 d*3*HV#o5'*)F=7$$\"3F)*\\(oWk(>GF=$!3cv[m:R!>g*F=7$$\"3A)****\\2'HKHF= $!3]\\#**)[E]V**F=7$$\"3fJ$e9T`G)HF=$!3[.3\"=2tj***F=7$$\"3)\\m;zu5M.$ F=$!3ziv&=jJi)**F=7$$\"3P)*\\P%3oR3$F=$!3r\"\\F`DUJ\"**F=7$$\"3uJL$3UD X8$F=$!3Uuj4fccx(*F=7$$\"3])**\\P4ScB$F=$!3M$oB/sdFK*F=7$$\"3ElmmmZvOL F=$!3/'pj>AdKj)F=7$$\"3%))***\\iirWMF=$!3(4,)e<#*pdwF=7$$\"3VKLLexn_NF =$!3(*zubK)QCY'F=7$$\"3-mm;a#R1m$F=$!3(y-KSbo<3&F=7$$\"3h******\\2goPF =$!3(\\%HT^.IbNF=7$$\"3\"GLe9\"*Hk'QF=$!3[N`iQWv#3#F=7$$\"3,mm\"H2fU'R F=$!3U1?y^HA6cF^s7$$\"3A**\\PM#)3iSF=$\"3iv/*eWVtt*F^s7$$\"3UKL$eR<*fT F=$\"3duu?s?k&[#F=7$$\"3B**\\P40(oE%F=$\"3P_6`_8HqSF=7$$\"3.mm\"HiBQP% F=$\"3W,*yDbG.a&F=7$$\"3%GLektw2[%F=$\"3xH*G+2gV&oF=7$$\"3l******\\)Hx e%F=$\"3[Y)f=9&QvzF=7$$\"36KLeR*)**)y%F=$\"3&yha`T[dX*F=7$$\"3ckm;H!o- *\\F=$\"3y:XYp:$))***F=7$$\"3nJL$3A_1?&F=$\"3WzkPuUR2&*F=7$$\"3y)*** \\7k.6aF=$\"3X/))*H>hq)zF=7$$\"3al;a84)Q^&F=$\"3lZ4[_5A:pF=7$$\"3IKLe9 as;cF=$\"3^`y9L=IjcF=7$$\"31**\\i:*p&>dF=$\"3Eo*>&GZ!RE%F=7$$\"3#emmmT 9C#eF=$\"3cP4i*RrMv#F=7$$\"37**\\7yI3IfF=$\"3k:fJ%*f/'4\"F=7$$\"3WKLeR $[1pF=$!3-Oqx5>I#*)*F=7$$\"3;L$3xm%*H$ pF=$!3(3Qv9[hY%**F=7$$\"3bKe9Th]fpF=$!3b!\\4G=x(z**F=7$$\"3-LLe9w,')pF =$!3M8kwC\")e(***F=7$$\"3_L3-)3HD,(F=$!3i!z$[BM1)***F=7$$\"3!HLe9cS!Rq F=$!3XZ>![D-7)**F=7$$\"3GKe*[._b1(F=$!30mHpbQ.Z**F=7$$\"3wKLL3N1#4(F=$ !3cwUctuh&*)*F=7$$\"3c***\\(o!*R-tF=$!3s@X*zh$*G*))F=7$$\"3Mmm;HYt7vF= $!3ew&oaT;#GpF=7$$\"3i**\\(o*GP4wF=$!3C*yBP#*3\"edF=7$$\"3\"HL$ek6,1xF =$!3fP'*G@AdbWF=7$$\"3>m;HK%\\E!yF=$!3!p/FA+j00$F=7$$\"3Y*******p(G**y F=$!32SQm)p%Rv:F=7$$\"3ymmTgg/5!)F=$\"3oRD\"y)f'zd\"F^s7$$\"3)HLL3U/37 )F=$\"3&3TqXkBi)=F=7$$\"3=***\\7yi:B)F=$\"3*zxtx9,xb$F=7$$\"3]mmmT6KU$ )F=$\"34q6`eKw@^F=7$$\"3-LLeRZQT%)F=$\"35\\_jwX)4R'F=7$$\"3a****\\P$[/ a)F=$\"3NU\"zS5md](F=7$$\"31mmTN>^R')F=$\"3eIssd)o\"R%)F=7$$\"3fKLLLbd Q()F=$\"3L9RH>pjo\"*F=7$$\"36+]i!*z>W))F=$\"3BsyZ')*>?q*F=7$$\"3`mm\"z W?)\\*)F=$\"3H3%))\\l^*o**F=7$$\"3I+Dcw;j-!*F=$\"3[iqXdX\"*****F=7$$\" 3&HL3_!HWb!*F=$\"3+SkpZ55i**F=7$$\"3rmT&Q8a#3\"*F=$\"3V^@o&>rd&)*F=7$$ \"3Z++]i`1h\"*F=$\"3j,hB1jl\"o*F=7$$\"3Y++DJ&f@E*F=$\"3JS^*Q\\@S;*F=7$ $\"3X++++PDj$*F=$\"36ZQ1GNy:%)F=7$$\"3X++voyMk%*F=$\"3O?kxX5xbuF=7$$\" 3W++]P?Wl&*F=$\"3;X.Xr<93jF=7$$\"3K+]7G:3u'*F=$\"3-8\")e1^z)*[F=7$$\"3 A++v=5s#y*F=$\"3_N*GadLrM$F=7$$\"35+]P40O\"*)*F=$\"3k=I]([M#)p\"F=7$$ \"\"\"F6$\"3Mint&*RBBhF9-%&COLORG6&%$RGBG$F-F5$F6F5Fh\\o-F06$7]x7$F4$ \"3%=vQi*4d^XF^s7$$!3.MLe9r]X**F=$!3!**>uk==93)F^s7$F;$!3cmcp(yh<%=F=7 $$!3')***\\PM@l$)*F=$!32U7PIDhCFF=7$FA$!3:8Mu(=m7^$F=7$FF$!3_1eW(GQ]# \\F=7$FK$!3d1`V%yL1A'F=7$FU$!3GQS0qZY'>)F=7$Fin$!3pdkmmm?z&*F=7$F^o$!3 (ej![-+<@)*F=7$Fco$!3oj.:*eU,***F=7$$!33](oaDS90*F=$!33fOq+]j/5!#<7$Fh o$!3+Cp1)oO$35Ff_o7$$!37;HK*G=\")**)F=$!3/:TO-H555Ff_o7$F]p$!3XE7H[T#* 45Ff_o7$Fbp$!3#>-scT!)z*)*F=7$Fgp$!33lSE3m7%R*F=7$F\\q$!3LR/h%H4Eg)F=7 $Faq$!3Jj\"o=o#QdvF=7$Ffq$!3'GL$yV)\\rH'F=7$F[r$!3?an#HG*4l[F=7$F`r$!3 [=\"*R!z'\\9LF=7$Fer$!3<$Ho<%e![o\"F=7$Fjr$!3R6y5yHku@!#?7$F`s$\"3!f1- $Rm%*H;F=7$Fes$\"3jAW_;Y^8JF=7$Fjs$\"3Q6k%45\\v^%F=7$F_t$\"3@\\6xRqO6e F=7$Fdt$\"3?5%yF#)*ompF=7$Fit$\"3z![:#y-'G\"))F=7$F^u$\"3>iOI>)R>y*F=7 $Fcu$\"3?Dy*)Q?+P)*F=7$Fhu$\"3xL4;yh[v)*F=7$F]v$\"3-HKcLgI(*)*F=7$Fbv$ \"39BQ!\\%=S-**F=7$Fgv$\"3a;!>0hS2*)*F=7$F\\w$\"3%='QxhjJi)*F=7$Faw$\" 3p;.h%R]r\")*F=7$Ffw$\"3:)=zjX\"Hb(*F=7$F[x$\"3S@8%\\CoGM*F=7$F`x$\"3. Jv@)QZWn)F=7$Fex$\"3w'egaPM'pxF=7$Fjx$\"3\"Gez8^_&\\mF=7$F_y$\"3f]tr1T FV`F=7$Fdy$\"3(H'ya8lR&)QF=7$Fiy$\"36z\\nUC)*oAF=7$F^z$\"3UNiAX%3O!eF^ s7$Fcz$!3ErNjz%p68\"F=7$Fhz$!3?$eB\"y(f^\"GF=7$F][l$!3ed1>#ondB%F=7$Fb [l$!3D'>Lk&ox#) Q'=&QyF=7$Fe]l$!3wk<]\")>/&o'F=7$Fj]l$!3rcrbee0]`F=7$F_^l$!3>(*)pV8R-( QF=7$Fd^l$!3e?=4WL?'G#F=7$Fi^l$!3[g2Z\"4d'QzF^s7$F^_l$\"32VBJ(Rx)\\rF^ s7$Fc_l$\"3Q+6Q-_S1AF=7$Fh_l$\"3)*[kIN!=pk$F=7$F]`l$\"3&GKy`13\"\\_F=7 $Fb`l$\"3()*>WtZ4Mo'F=7$Fg`l$\"3TNe$*ey*[!zF=7$F\\al$\"3ig[;^ccv))F=7$ Faal$\"3#)zYoJ=9z%*F=7$Ffal$\"3,\"\\d<0!QWim$HF=7$Fbel$\"3%['*H<'pAN9F=7$Fgel$!3>F=7$Fafl$!3?PZpxp*R`$F=7$Fffl$!3O+jy4*oa/&F=7$F[gl$!3$zg[%3 Qq\\jF=7$F`gl$!3e\"yfkN<$)[(F=7$Fegl$!3-A`mHEOK%)F=7$Fjgl$!3o\"=F_![%z :*F=7$F_hl$!3\\MBxGdXi'*F=7$Fdhl$!3]+i=I*[c*)*F=7$Fihl$!37;u!=;'H3**F= 7$F^il$!3\"3)p?r5V^)*F=7$Fcil$!3AhRgm%)\\D(*F=7$Fhil$!3aAgN,HUJ&*F=7$F ]jl$!3<>U*)\\1X-!*F=7$Fbjl$!3&[l&HxLJh#)F=7$Fgjl$!34;G$zvGcK(F=7$F\\[m $!3bxILlKa$F=7$F`]m$\"3]EKQ@M))eZF=7$Fe]m$\"3GY4)RY.w<'F=7$Fj]m$\"3#\\4DEpC bE(F=7$F_^m$\"3E!ox)>')y!>)F=7$Fd^m$\"3k\\$**oNmD$*)F=7$Fi^m$\"3Rcucc> 4u%*F=7$F^_m$\"34H?p#fF(y'*F=7$Fc_m$\"3<>G[yj\"*>)*F=7$Fh_m$\"3YPgCJcp '*)*F=7$F]`m$\"3U<4`e6_3**F=7$Fb`m$\"3)>c,E6[st*F=7$Fg`m$\"3cchA&[)>5$ *F=7$F\\am$\"3`#eBnS%*=h)F=7$Faam$\"3`CYA?c>rwF=7$Ffam$\"3$zRO][dR^'F= 7$F[bm$\"3>qt0529s^F=7$F`bm$\"3!3t9%F=7$F^dm$!3551$R2c'QbF=7$Fcdm$!3CSJv>\"zBz'F=7$Fhdm$!3'G7' fK?O%)zF=7$F]em$!3ycMOq7UH*)F=7$Fbem$!3)oX&='\\+tf*F=7$Fgem$!3g\">&oE- Hm**F=7$F\\fm$!3iYmL%R5J+\"Ff_o7$Fafm$!3on+j)[XK+\"Ff_o7$Fffm$!3+tXjv@ Dq**F=7$F[gm$!37?Mn^'3[%)*F=7$F`gm$!3/[4D]7Y2%*F=7$Fegm$!3!RJHO*>*3t)F =7$Fjgm$!3qZWQf#4Lw(F=7$F_hm$!3`nu,UAYplF=7$Fdhm$!3b'3T#)3&[$=&F=7$Fih m$!3)fqn$)H]_k$F=7$F^im$!3'3@k)\\+-d@F=7$Fcim$!3/oH)f/Ws:'F^s7$Fhim$\" 3wf#3iEOxT*F^s7$F]jm$\"3H(4)p\\,7yCF=7$Fbjm$\"3XV=PQf+!4%F=7$Fgjm$\"3O $HA\"*\\(4'e&F=7$F\\[n$\"3jE4&4RhK#pF=7$Fa[n$\"3q9<)QL0H1)F=7$Ff[n$\"3 ***p?Wmp;c*F=7$F[\\n$\"3#p%*=i'=w45Ff_o7$F`\\n$\"3i.1aBXYs&*F=7$Fe\\n$ \"3iB4nN7%)**zF=7$Fj\\n$\"3Y*GJ[)H\")**oF=7$F_]n$\"3vEdLlGd?cF=7$Fd]n$ \"3AAaR!z(p'>%F=7$Fi]n$\"3S#z)>K7[mEF=7$F^^n$\"36W/19$p4&**F^s7$Fc^n$! 3%pnG\"*3 !**F=7$Fdbn$!3%*4[>H4l)*)*F=7$Fibn$!3=NnsB`iy)*F=7$F^cn$!3+iQmRk$>%)*F =7$Fccn$!3uxuG#fn')y*F=7$Fhcn$!3k_(R>SHkz)F=7$F]dn$!3!eD6Ky\"4%)oF=7$F bdn$!3^V#H`Xyuu&F=7$Fgdn$!3hY'[_+n.[%F=7$F\\en$!3\"Q(*)z_Y64JF=7$Faen$ !3Lu*ydOYAm\"F=7$Ffen$\"3NdR!pVwA)\\F]bo7$F[fn$\"3'yTNW=)HtVh4\"*4oCvF=7$Fdgn$\"3[zsWI:&Q])F=7$Fign$\"31pL!4cI,F*F=7$F^hn$\"31U+ !y5(oB)*F=7$Fchn$\"34(H!R8sM35Ff_o7$Fhhn$\"3$o\"=X)[S*45Ff_o7$F]in$\"3 !4WKZ38R+\"Ff_o7$Fbin$\"3/A9L8qu.**F=7$Fgin$\"3w54YZcW&p*F=7$F\\jn$\"3 =i:\\N5J1\"*F=7$Fajn$\"3s;StJ%[sH)F=7$Ffjn$\"3!z^6\\cfQJ(F=7$F[[o$\"3h ,433sx/iF=7$F`[o$\"3[Pv))oA@7\\F=7$Fe[o$\"3\\=QJkkq,NF=7$$\"3;+D1k2/P) *F=$\"3()>2!Hl@nr#F=7$Fj[o$\"3Sa$R*o*Qd$=F=7$$\"30+voa-oX**F=$\"3ZO\"=d8Feeq7$FA$\"3qv\\)*p.v$3\"Feeq7 $FF$\"3s\"*Relu@J')!#87$FK$\"3:&>\"\\&f8b&oF_fq7$FU$\"3%4Dd&G0ChXF_fq7 $Fin$\"3I)=! y)F=7$Fjbl$\"3YJGULsT\\**F=7$F^dl$\"3&o*4IS)>\"zpF=7$Fbel$\"3')GQN@+&H a\"F=7$Fffl$!3%*>Us`_z\"*\\F=7$Fjgl$!3#f\\T4*H6*>*F=7$Fhil$!348/o>v,P' *F=7$F\\[m$!3['\\U!>y,1jF=7$F`\\m$!3b%**\\iE&f]:F^s7$Fe]m$\"3%yvwT-=cE 'F=7$Fi^m$\"3cqfdThQ!e*F=7$Fg`m$\"37\")yn!otsN*F=7$F[bm$\"3%3'[0*oc/7& F=7$F_cm$!32r.Wbz)H?\"F=7$Fcdm$!3p38-JXFqoF=7$Fgem$!3k[Y#F=7$Fa[n$ \"3c6I)>cIJz(F=7$F[\\n$\"3TeebSqJ^*)F=7$Fe\\n$\"3l36<5+)RO#F=7$Fi]n$!3 ?@%4#y(*o\"H#Ff_o7$F]_n$!3wDLC;9*>;\"Ffhq7$Fa`n$!3_uqZ'z^nJ%Ffhq7$Fccn $!3TK[0$yPt^\"F_hq7$F]dn$!3nQ-8A5&o$\\F_hq7$Faen$!3Kf1Mm%=1Q\"Fbgq7$F[ fn$!3UGyKmc'GV#Fbgq7$Fefn$!3+Lg$yhc-A%Fbgq7$F_gn$!3!*yc,0\"p*>oFbgq7$F ign$!3$=')356d&*3\"F_fq7$Fchn$!3RZFS(3%*Rx\"F_fq7$Fgin$!3M<'[^\"[EaGF_ fq7$Fajn$!36@\\_aq5_WF_fq7$F[[o$!3,FW " 0 "" {MPLTEXT 1 0 69 "err1 := plot(fu nc(x) - Lapprox(n,x), x=-1..1,y=-1..1, color=green):\n" }{MPLTEXT 1 0 68 "err2 := plot(func(x) - Tapprox(n,x), x=-1..1,y=-1..1, color=blue): \n" }{MPLTEXT 1 0 26 "display(\{err1, err2\});\n" }{MPLTEXT 1 0 76 "pr int(\"The error in the 3rd degre Legandre and Taylor approximations.\" ):\n" }{MPLTEXT 1 0 1 " " }}{PARA 13 "" 1 "" {TEXT 213 0 "" } {GLPLOT2D 400 400 400 {PLOTDATA 2 "6'-%+AXESLABELSG6'Q\"x6\"Q\"yF'-%%F ONTG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%'CURVESG6$7gs7$$!\"\"\"\"!$\"3` '*****)>.hN*!#=7$$!3%pmm\"HU,\"*)*F9$\"3?aZHXz'y*yF97$$!3!RLL$e%G?y*F9 $\"3/\"zs'f*>8Y'F97$$!3t****\\(oUIn*F9$\"3Qe`UAU.d]F97$$!3ommm;p0k&*F9 $\"3;KBcM!f`p$F97$$!3#HL3-)*G#p%*F9$\"3)e,rCrUIb#F97$$!3E++vV5Su$*F9$ \"3MZj*Qp'*oX\"F97$$!3^m;H2Jdz#*F9$\"3x)[MYT2%HT!#>7$$!3vKL$3A)F9$!3ruu/& )HZjjF97$$!3NnT5l^]o\")F9$!3SO$3+0ss%HaF9$\"3QRwmf,%GO)F97$$!3]***\\7)*G; K&F9$\"3ya^V#QCij)F97$$!3]LL$3x&y8_F9$\"3?!orqvFo\"))F97$$!30,]ilT')f^ F9$\"3z8:ZC!*)F97$$!3%RL3_&4-_]F9$ \"3!4vJ'RI23*)F97$$!3Q+++]$*4)*\\F9$\"3Wc]y`)*z*)))F97$$!3E++++t_\"y%F 9$\"3W=\\3>XRq&)F97$$!38+++]_&\\c%F9$\"35^!)o7NMlyF97$$!33+++]zCcVF9$ \"3Z(o6mgcR%oF97$$!30+++]1aZTF9$\"3HVn/-IIBbF97$$!3RLL3FW,eRF9$\"3[-0/ Mo00TF97$$!3umm;/#)[oPF9$\"3ZU**R%zhK_#F97$$!3<++v$>BJa$F9$\"3nVbpe`Zy \\Fgn7$$!3hLLL$=exJ$F9$!3%3fdWW#>*f\"F97$$!3!QLL$eW%o7$F9$!3(\\\")Ged^ ?O$F97$$!3)RLLLtIf$HF9$!3>knaJs\">0&F97$$!3CnmT&o_Qr#F9$!3eQYR!RlH&oF9 7$$!3]++]PYx\"\\#F9$!3a@S=,W,,%)F97$$!3QnmTNz>&H#F9$!3SSnTY?A/&*F97$$! 3EMLLL7i)4#F9$!3U++1#zE:.\"!#<7$$!3g+]PMLy!*>F9$!3zpguyo\\i5Fi\\l7$$!3 #pm;aVXH)=F9$!3I!zo8l4M3\"Fi\\l7$$!3%**\\Pf[E!H=F9$!3v'yWLe6+4\"Fi\\l7 $$!3CL$ek`2^x\"F9$!3]M2;$f7S4\"Fi\\l7$$!3am\"zpe)=@@J\"4\"Fi\\l7 $$\"3g)***\\7JFn=F9$\"3wN%Rwc&f&3\"Fi\\l7$$\"3F)**\\(==-s>F9$\"3#>0f6N mo1\"Fi\\l7$$\"3#z****\\_qn2#F9$\"3=\">vb!*)fQ5Fi\\l7$$\"3=)**\\P/q%zA F9$\"3I'yWtKB.e*F97$$\"3U)***\\i&p@[#F9$\"3D%*fJ\\([6Y)F97$$\"3L)**\\( =GB2FF9$\"3#\\!4#=lUJ!pF97$$\"3A)****\\2'HKHF9$\"3sV*=J)o,$3&F97$$\"3u JL$3UDX8$F9$\"35Q*4Cce@H$F97$$\"3ElmmmZvOLF9$\"3MW#Rhpl>U\"F97$$\"3VKL Lexn_NF9$!3$pd&f#*GCeeFgn7$$\"3h******\\2goPF9$!3-CEfS,CCDF97$$\"3,mm \"H2fU'RF9$!3l9MZ#oZY:%F97$$\"3UKL$eR<*fTF9$!39(pTph]\"4cF97$$\"3.mm\" HiBQP%F9$!3c;%e#>x4UpF97$$\"3l******\\)Hxe%F9$!3vDH(oiJq&zF97$$\"36KLe R*)**)y%F9$!323Dqk+$ze)F97$$\"3ckm;H!o-*\\F9$!3PQs\"3p5^)))F97$$\"31JL 3xS'G/&F9$!3]_)RI2%p1*)F97$$\"3c(****\\7ga4&F9$!3x&)=#*3=-0*)F97$$\"31 km\"Haj^mF97$$\"3WK LeRBzom@F97$$\"3%QLLL$*zym'F9$!36%f]([L#)>^Fgn7$$\"3ILL $3sr*zoF9$\"3LSZ!R/!p^6F97$$\"3wKLL3N1#4(F9$\"3:M6caT=2FF97$$\"3c*** \\(o!*R-tF9$\"3Kvlh?cKnSF97$$\"3Mmm;HYt7vF9$\"3kn&>)frlw^F97$$\"3\"HL$ ek6,1xF9$\"3#)3v?w'RI#fF97$$\"3Y*******p(G**yF9$\"3'*H&fZ[N!ojF97$$\"3 7L$3-)omazF9$\"3hr[.WcfNkF97$$\"3ymmTgg/5!)F9$\"3>$H&*)4uPvkF97$$\"3K* *\\iS_Ul!)F9$\"3Ee&fQT.p['F97$$\"3)HLL3U/37)F9$\"3b#\\0&[qwpkF97$$\"3j m;/,O=w\")F9$\"36\"[d?`LOU'F97$$\"3=***\\7yi:B)F9$\"3*Qm_txS#[jF97$$\" 3%GLe9'>%pG)F9$\"3oc!>G1+MC'F97$$\"3]mmmT6KU$)F9$\"3!*4#QO]'**3hF97$$ \"3a****\\P$[/a)F9$\"3r8A!\\(e<'Q&F97$$\"3fKLLLbdQ()F9$\"3U+dYQZC\"H%F 97$$\"3`mm\"zW?)\\*)F9$\"3WK4?5!4Tt#F97$$\"3Z++]i`1h\"*F9$\"3=tU7H<\\( 4)Fgn7$$\"3Y++DJ&f@E*F9$!3C_O_9!=OF#Fgn7$$\"3X++++PDj$*F9$!3a!QjXKc8L \"F97$$\"3X++voyMk%*F9$!3YEU\"\\VSa\\#F97$$\"3W++]P?Wl&*F9$!3cwQ1iaN7P F97$$\"3K+]7G:3u'*F9$!3&f<.ayM-2&F97$$\"3A++v=5s#y*F9$!33#yW#))yNqkF97 $$\"35+]P40O\"*)*F9$!3\"z?(e!\\sC!zF97$$\"\"\"F6$!3`'*****)>.hN*F9-%&C OLORG6&%$RGBG$F6F5$F-F5F^hm-F06$7S7$F4$!3&)***z8nM5I\"!#:7$FJ$!3_s^'4r (\\M6Ffhm7$Fin$!3DSn'\\Sz9+\"Ffhm7$Fco$!3(*pd\\&pg,k)!#;7$F]p$!3]*GH[i 8jP(F`im7$Fer$!3TAdYpA4IiF`im7$F_s$!3Y'Hb,PqAE&F`im7$Fis$!3BC@Y\"4bPN% F`im7$Fct$!3M@\">H;CK^$F`im7$F]u$!3])*oEx9QvFF`im7$Fgu$!3lM=))*y;.7#F` im7$Fav$!37NNMHFgH;F`im7$F_x$!3A_vSc_0r6F`im7$Fix$!3zjP$e*zf`!)Fi\\l7$ Fcy$!3f%p_A3C;N&Fi\\l7$F]z$!3W6#>'**3'o^$Fi\\l7$Fgz$!3%*pKn-pP$)>Fi\\l 7$Fa[l$!3!G,[tr7)H6Fi\\l7$F[\\l$!3TY080f0G_F97$Fe\\l$!3'>hLY9O**H#F97$ Fd^l$!3)*)QO\"yr1IvFgn7$Fb`l$!3Y*)R]#Febl7$Fjdl$\"3n'ok2Nijo\"Fgn7$Fdel$\"3$QXKO(4%HK(F gn7$Fbgl$\"3)z)*zg\"F`im7$F`^m$\"3SZExpN'Q8#F`im7$Fj^m$\"3B_\"3LulTy#F`im7$Fd_m$ \"3 " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 206 0 "" }{TEXT 200 43 "Animating the approximations and the errors" } }{EXCHG {PARA 0 "" 0 "" {TEXT 209 136 "We can also compare the results with an animation. To prepare for the animation we create a procedur e that will produce a single frame." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "framer := proc(n,x, lowy, highy)\n" }{MPLTEXT 1 0 31 "local pl1, pl2, pl3, B, C, D:\n" }{MPLTEXT 1 0 57 "pl1 := plot(func(x ), x=-1..1,y=lowy..highy, color=red):\n" }{MPLTEXT 1 0 64 "pl2 := plot (Lapprox(n,x), x=-1..1,y=lowy..highy, color=green):\n" }{MPLTEXT 1 0 63 "pl3 := plot(Tapprox(n,x), x=-1..1,y=lowy..highy, color=blue):\n" } {MPLTEXT 1 0 47 "B := textplot(\{[-0.8,highy-(highy-lowy)/20, \n" } {MPLTEXT 1 0 64 " `The function and the `||n||` term approxim ations`],\n" }{MPLTEXT 1 0 77 " [0-0.8,highy-(highy-lowy)/9, `Tay lor (blue) vs Legandre (green)`]\},\n" }{MPLTEXT 1 0 46 " align=RI GHT, font = [TIMES, BOLD, 12] ):\n" }{MPLTEXT 1 0 58 "display(\{pl1, p l2, pl3, B\},view=[-1..1, lowy..highy]);\n" }{MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 209 102 "Using that procedure, it is easy to produce an animation of the approximations of degree 1 through 20. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "highy := 1.6: lowy :=-1 .2:\n" }{MPLTEXT 1 0 62 "display([seq(framer(counta,x, lowy, highy), c ounta=1..20)], \n" }{MPLTEXT 1 0 27 " insequence = true);" } {TEXT 209 0 "" }}{PARA 13 "" 1 "" {TEXT 213 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6%-%+AXESLABELSG6'Q\"x6\"Q\"yF'-%%FONTG6$%*HELVETICAG \"#5%+HORIZONTALGF.-%(ANIMATEG667'-%%TEXTG6&7$$!\")!\"\"$\"2-++++++Y\" !#;QLThe~function~and~the~1~term~~approximationsF'%+ALIGNRIGHTG-F*6%I& TIMESGF'I%BOLDGF'\"#7-F46&7$F7$\"+*))))))G\"!\"*QCTaylor~(blue)~vs~Leg andre~~(green)F'F>F?-%'CURVESG6$7S7$$F9\"\"!$!30+++Fjzq:F<7$$!3ommm;p0 k&*!#=$!3\\%*=fZ&=B]\"F<7$$!3vKL$3y/\"F<7$$!3[++](y$p ZiFW$!3H>di`W&Q\")*!#<7$$!33LLL$yaE\"eFW$!3L!)*yPy'\\I\"*F\\q7$$!3hmmm \">s%HaFW$!3;S'=i(\\fG&)F\\q7$$!3Q+++]$*4)*\\FW$!3o)3h45'*4&yF\\q7$$!3 8+++]_&\\c%FW$!3zm$>'R\\hqrF\\q7$$!30+++]1aZTFW$!3-$>.\">;%\\^'F\\q7$$ !3umm;/#)[oPFW$!3fF\\q7$$!3hLLL$=exJ$FW$!3!H>a#oB_6_F\\q7$$!3) RLLLtIf$HFW$!3VF\\q7$$!3)3LLL3x%z#)!#>$!3'o1)>>s`+8F\\q7 $$!3KMLL3s$QM%F\\u$!3GNf$>`$GBoFW7$$!3\\^omm;zr)*!#@$!3\"*\\444ul]:F \\u7$$\"3%pJL$ezw5VF\\u$\"3Pj#*\\v%Q8x'FW7$$\"3s*)***\\PQ#\\\")F\\u$\" 3JV(H2Pz+G\"F\\q7$$\"3GKLLe\"*[H7FW$\"3w>j*R0x7$>F\\q7$$\"3H*******pvx l\"FW$\"3u_)\\0)z-/EF\\q7$$\"3#z****\\_qn2#FW$\"3ee=p7N=iKF\\q7$$\"3U) ***\\i&p@[#FW$\"3u%>m<$G)*)*QF\\q7$$\"3A)****\\2'HKHFW$\"369lG/*Rgg%F \\q7$$\"3ElmmmZvOLFW$\"3iR(z:8i8C&F\\q7$$\"3h******\\2goPFW$\"3TW%Hg@/ (>fF\\q7$$\"3UKL$eR<*fTFW$\"3%QS)*f'HQMlF\\q7$$\"3l******\\)Hxe%FW$\"3 -h#[w>*Q1sF\\q7$$\"3ckm;H!o-*\\FW$\"3]D0'4p%pQyF\\q7$$\"3y)***\\7k.6aF W$\"3ob#=?7O'*\\)F\\q7$$\"3#emmmT9C#eFW$\"3-Y=(*zr#e9*F\\q7$$\"32**** \\i!*3`iFW$\"3os)y#G#y;!R9F <7$$\"3W++]P?Wl&*FW$\"3\")RO'=7OD]\"F<7$$\"\"\"FQ$\"30+++Fjzq:F<-%&COL ORG6&%$RGBG$FQF9Ff]l$F-F9-FL6$7iw7$FP$!3Mint&*RBBh!#L7$$!3%pmm\"HU,\"* )*FW$!3i=kW$e%f.*>$)FW7$$!3^m;H2Jdz#*FW$!3+bI:-O7^!*FW7$ Fen$!3))yL/T^(=e*FW7$$!3!))\\(o/KUJ\"*FW$!3Go1F*>qwy*FW7$$!3&fmT&Q75y! *FW$!3i5!Q\"G5%[#**FW7$$!35LeRs#zZ-*FW$!35,=p#)eU#***FW7$$!39***\\iId9 (*)FW$!349Vn04&**)**FW7$$!3YL$eRP8['))FW$!3([Dqk&>Qv(*FW7$Fjn$!3$[w*Gc 79(G*FW7$$!3qlm;/'=3l)FW$!3s&>&)\\>8J`)FW7$$!3%emmmwnMa)FW$!3u\")o6<>- PvFW7$$!3+mm;Hp6O%)FW$!3Zr?&p0CrK'FW7$F_o$!3'=m)=59uP\\FW7$$!3iL$eRZD> A)FW$!3nN&eC'3#eT$FW7$$!3)****\\i&[3:\")FW$!3Ol)3/q=zz\"FW7$$!3Mm;aQUC 3!)FW$!3)fB'>md'\\H\"F\\u7$Fdo$\"3+KI1*GkDa\"FW7$$!3Mnmm\"p[B!yFW$\"3l ./g5:1bIFW7$$!3'3++Dw$H.xFW$\"3!fmhnFXP\\%FW7$$!3RMLLL)QUg(FW$\"3;&4&G &\\bQ#eFW7$Fio$\"3)Hb_MKbK,(FW7$$!3tKL$3i_+I(FW$\"3wHAGe**o4*)FW7$F^p$ \"3zses:'[!*))*FW7$$!3(fmmT&HSoqFW$\"31%zIu5JB%**FW7$$!3ILL$3d%)=/(FW$ \"3%)z5-%yk$y**FW7$$!3i++](=m`,(FW$\"3gO4;Jr3(***FW7$$!3$ommT!y%)))pFW $\"33gZqncY)***FW7$$!31LL$3UHB'pFW$\"3+R#[?+)\\#)**FW7$$!3Q++]P5\"e$pF W$\"3%R]xi$=@\\**FW7$$!3rnm;aEH4pFW$\"3qv\"4\"=\\m)*)*FW7$$!3#RLL3FuF) oFW$\"3X.YqS\\%4$)*FW7$$!3,,+]P2qwnFW$\"3Gf9YGk7\"R*FW7$Fcp$\"3UJ&3jB# >\"p)FW7$$!3\")******\\Q*[c'FW$\"3PIN31Y!Rv(FW7$$!3tLL$e\\g\"fkFW$\"3: v-1m7A.mFW7$$!3mnmmTrU`jFW$\"3i)33oC53F&FW7$Fhp$\"3OC95=4M$z$FW7$$!3YM $ek.M*QhFW$\"3%4y))**\\)3l@FW7$$!3MnmT&Gu,.'FW$\"3Jt&y<>\"*zt%F\\u7$$! 3?+]PMXT@fFW$!310T$H,&GJ7FW7$F^q$!3^pp9[+_+HFW7$$!3YmmTN\"for&FW$!3kt_ X#*>Q-VFW7$$!3%)****\\([j5i&FW$!3g?[)pP4qg&FW7$$!3ALLeRyEDbFW$!3g4.Z-u \"\\y'FW7$Fcq$!3\")[gc[^[4yFW7$$!3]LL$3x&y8_FW$!3c,#>Y5C9W*FW7$Fhq$!3; Gc'HVb*****FW7$$!3E++++t_\"y%FW$!3!)e/a*=2pT*FW7$F]r$!3mFo)[h8Xv(FW7$$ !3R++++;ggWFW$!39y>-#=/-i'FW7$$!33+++]zCcVFW$!3LcdsnHT3`FW7$$!3z****** *H%*=D%FW$!3([>>o#zIaQFW7$Fbr$!3qY-9@B(oH#FW7$$!3Xm;aQvx_SFW$!3%**zk6I $y!G)F\\u7$$!3RLL3FW,eRFW$\"35y1_A&)H!f'F\\u7$$!3M+]i:8DjQFW$\"3we_J.E cJ@FW7$Fgr$\"3!\\&e*yl_pb$FW7$$!3=L$e*)p0el$FW$\"3)>ugEL7q9&FW7$$!3<++ v$>BJa$FW$\"36Xeo5&Rid'FW7$$!3(HjF[y)**FW7$$!3zLLek\"fO)HFW$\"3WJJ&y\"fq'***FW7$Fas$\"3cYb`84S\\** FW7$$!3M+]P4<*[#GFW$\"3\"zpFW7$$!3%RLekG '[$R#FW$\"3uT>kn*pZz&FW7$$!3QnmTNz>&H#FW$\"3SR30vscsWFW7$$!3%3+vVe4p># FW$\"36Fpn)\\lR/$FW7$F[t$\"3/3sOB+&Ha\"FW7$$!3g+]PMLy!*>FW$!3\\h]8x)*p Z9F\\u7$$!3#pm;aVXH)=FW$!3cIV?tfMG=FW7$$!3CL$ek`2^x\"FW$!3!R$pa[defMFW 7$F`t$!3SxUmZ_z\"*\\FW7$$!3k***\\iX#ek:FW$!3Pgr4))Gh=jFW7$$!3r*****\\F &*=Y\"FW$!3/4Y*yi*Q\"[(FW7$$!3z***\\P43#f8FW$!3$*Ho+'*z$*\\%)FW7$Fet$! 3%eL\"R))H6*>*FW7$$!3:))F\\u$!3X;rxwlJF)*FW7$Fjt$!3;faF=v,P'*FW 7$$!3uJLe9rc&H(F\\u$!3m1[r;^;6\"*FW7$$!3gKL$e9d;J'F\\u$!3rR*)*Qv7\"o$) FW7$$!3XLL3xruF`F\\u$!30XC2lRdDuFW7$F`u$!3sP'=#=y,1jFW7$$!3'ym;aQdDG$F \\u$!3Mv#)\\g'o2$\\FW7$$!3T,+]ivF@AF\\u$!3$>Yxd,2)=MFW7$$!3'\\L$eRx**f 6F\\u$!3'pT(RWT07=FW7$Feu$!3fHv/m_f]:F\\u7$$\"3&yK$3_Nl.5F\\u$\"3)e4CA s7+d\"FW7$$\"3@CL$3-Dg5#F\\u$\"3Y'\\%3el7[KFW7$$\"3e?Le*['R3KF\\u$\"3w >Em4.4H[FW7$F[v$\"3'eIeL-=cE'FW7$$\"38&)**\\ibQq_F\\u$\"3RT-esO#\\O(FW 7$$\"3K`mmmJ+IiF\\u$\"3aAADVT?(H)FW7$$\"3_@L$3x?'*=(F\\u$\"3+hZZVnJT!* FW7$F`v$\"3Md4>ShQ!e*FW7$$\"3Nc;/E]Wn')F\\u$\"3tD*=i8IqH=**FW7$$\"3h*)\\7G$eQq*F\\u$\"34G#o1T#=*)**FW7$$ \"3ilm\"z\\1A-\"FW$\"3#G%p?()o\"R***FW7$$\"3%*)*\\7Gy%e7\"FW$\"3OQC)>' fC0)*FW7$Fev$\"3Q&o[#yOFd$*FW7$$\"3.***\\PzglL\"FW$\"3aLiESWzM')FW7$$ \"3ylm;HCjV9FW$\"3/==n@qjowFW7$$\"3aKLekSq]:FW$\"30e6)Q'o1'['FW7$Fjv$ \"3Xep6$oc/7&FW7$$\"3&*)**\\iSCDw\"FW$\"3C[c$R*zMWOFW7$$\"3g)***\\7JFn =FW$\"3IY'\\R5)zp?FW7$$\"3F)**\\(==-s>FW$\"3YzWPiyQ$R%F\\u7$F_w$!3z))R afz)H?\"FW7$$\"3/)*\\P%G?\"y@FW$!3qRK$[#[ahFFW7$$\"3=)**\\P/q%zAFW$!3$ eO/$y\"f-D%FW7$$\"3I)*\\7.)>3Q#FW$!3CbzS$zz9j&FW7$Fdw$!3V%3&pySFqoFW7$ $\"3Q)*\\i!>,Zf#FW$!3M\\Bj[o'4/)FW7$$\"3L)**\\(=GB2FFW$!3d*3*HV#o5'*)F W7$$\"3F)*\\(oWk(>GFW$!3cv[m:R!>g*FW7$Fiw$!3]\\#**)[E]V**FW7$$\"3fJ$e9 T`G)HFW$!3[.3\"=2tj***FW7$$\"3)\\m;zu5M.$FW$!3ziv&=jJi)**FW7$$\"3P)* \\P%3oR3$FW$!3r\"\\F`DUJ\"**FW7$$\"3uJL$3UDX8$FW$!3Uuj4fccx(*FW7$$\"3] )**\\P4ScB$FW$!3M$oB/sdFK*FW7$F^x$!3/'pj>AdKj)FW7$$\"3%))***\\iirWMFW$ !3(4,)e<#*pdwFW7$$\"3VKLLexn_NFW$!3(*zubK)QCY'FW7$$\"3-mm;a#R1m$FW$!3( y-KSbo<3&FW7$Fcx$!3(\\%HT^.IbNFW7$$\"3\"GLe9\"*Hk'QFW$!3[N`iQWv#3#FW7$ $\"3,mm\"H2fU'RFW$!3U1?y^HA6cF\\u7$$\"3A**\\PM#)3iSFW$\"3iv/*eWVtt*F \\u7$Fhx$\"3duu?s?k&[#FW7$$\"3B**\\P40(oE%FW$\"3P_6`_8HqSFW7$$\"3.mm\" HiBQP%FW$\"3W,*yDbG.a&FW7$$\"3%GLektw2[%FW$\"3xH*G+2gV&oFW7$F]y$\"3[Y) f=9&QvzFW7$$\"36KLeR*)**)y%FW$\"3&yha`T[dX*FW7$Fby$\"3y:XYp:$))***FW7$ $\"3nJL$3A_1?&FW$\"3WzkPuUR2&*FW7$Fgy$\"3X/))*H>hq)zFW7$$\"3al;a84)Q^& FW$\"3lZ4[_5A:pFW7$$\"3IKLe9as;cFW$\"3^`y9L=IjcFW7$$\"31**\\i:*p&>dFW$ \"3Eo*>&GZ!RE%FW7$F\\z$\"3cP4i*RrMv#FW7$$\"37**\\7yI3IfFW$\"3k:fJ%*f/' 4\"FW7$$\"3WKLeR$[1pFW$!3-Oqx5>I#*)*FW7$$\"3;L$3xm%*H$pFW$!3(3Qv9[h Y%**FW7$$\"3bKe9Th]fpFW$!3b!\\4G=x(z**FW7$$\"3-LLe9w,')pFW$!3M8kwC\")e (***FW7$$\"3_L3-)3HD,(FW$!3i!z$[BM1)***FW7$$\"3!HLe9cS!RqFW$!3XZ>![D-7 )**FW7$$\"3GKe*[._b1(FW$!30mHpbQ.Z**FW7$F[[l$!3cwUctuh&*)*FW7$$\"3c*** \\(o!*R-tFW$!3s@X*zh$*G*))FW7$F`[l$!3ew&oaT;#GpFW7$$\"3i**\\(o*GP4wFW$ !3C*yBP#*3\"edFW7$$\"3\"HL$ek6,1xFW$!3fP'*G@AdbWFW7$$\"3>m;HK%\\E!yFW$ !3!p/FA+j00$FW7$Fe[l$!32SQm)p%Rv:FW7$$\"3ymmTgg/5!)FW$\"3oRD\"y)f'zd\" F\\u7$$\"3)HLL3U/37)FW$\"3&3TqXkBi)=FW7$$\"3=***\\7yi:B)FW$\"3*zxtx9,x b$FW7$Fj[l$\"34q6`eKw@^FW7$$\"3-LLeRZQT%)FW$\"35\\_jwX)4R'FW7$$\"3a*** *\\P$[/a)FW$\"3NU\"zS5md](FW7$$\"31mmTN>^R')FW$\"3eIssd)o\"R%)FW7$F_ \\l$\"3L9RH>pjo\"*FW7$$\"36+]i!*z>W))FW$\"3BsyZ')*>?q*FW7$$\"3`mm\"zW? )\\*)FW$\"3H3%))\\l^*o**FW7$$\"3I+Dcw;j-!*FW$\"3[iqXdX\"*****FW7$$\"3& HL3_!HWb!*FW$\"3+SkpZ55i**FW7$$\"3rmT&Q8a#3\"*FW$\"3V^@o&>rd&)*FW7$Fd \\l$\"3j,hB1jl\"o*FW7$$\"3Y++DJ&f@E*FW$\"3JS^*Q\\@S;*FW7$$\"3X++++PDj$ *FW$\"36ZQ1GNy:%)FW7$$\"3X++voyMk%*FW$\"3O?kxX5xbuFW7$Fi\\l$\"3;X.Xr<9 3jFW7$$\"3K+]7G:3u'*FW$\"3-8\")e1^z)*[FW7$$\"3A++v=5s#y*FW$\"3_N*GadLr M$FW7$$\"35+]P40O\"*)*FW$\"3k=I]([M#)p\"FW7$F^]l$\"3Mint&*RBBhF^^l-Fc] l6&Fe]lFg]lFf]lFf]l-FL6$7S7$FP$!3#******fJf)4>FW7$FU$!3%o+0,K+m#=FW7$F en$!3:0-'H6dTv\"FW7$Fjn$!3\"FW7$F^q$!3y7T'))GN,6\"FW7$Fcq$!3-_\"*)[\">7#zF\\u7$Fg r$!3q=Q'RI#G(>(F\\u7$F\\s$!3]-Wnu8XOjF\\u7$Fas$!3=zt=iY@2cF\\u7$Ffs$!3 p)*=![+R*eZF\\u7$F[t$!3!y2PKJr!3SF\\u7$F`t$!3%oJj%\\/D%=$F\\u7$Fet$!3Q Xp[q\"y(*R#F\\u7$Fjt$!37n7#Rkj7e\"F\\u7$F`u$!3]]G_fz6'H)!#?7$Feu$!3J&[ >!GLP&)=Fgu7$F[v$\"3u%=PVMgHB)Fh\\p7$F`v$\"31V)zG))*Qc:F\\u7$Fev$\"3m8 R'HK^\"[BF\\u7$Fjv$\"3b>M[k$=h;$F\\u7$F_w$\"3scaOM&Rj'RF\\u7$Fdw$\"3Am A$Gm%fSZF\\u7$Fiw$\"3.\")*3h(HF+cF\\u7$F^x$\"3NOdKw@tsjF\\u7$Fcx$\"3!e 3s1D(\\(>(F\\u7$Fhx$\"3b[FA#*p&[%zF\\u7$F]y$\"3$>y8Lf=>w)F\\u7$Fby$\"3 994%[))42`*F\\u7$Fgy$\"35MG;I=VL5FW7$F\\z$\"3v`$G:C**>6\"FW7$Faz$\"3%G Lz*R?D%>\"FW7$Ffz$\"3f7Y'3ErMF\"FW7$F[[l$\"3'fS0hN%[a8FW7$F`[l$\"3Nw \\,Am#[V\"FW7$Fe[l$\"3>@4O?Gl3:FW7$Fj[l$\"3)QeZ\\(fE$f\"FW7$F_\\l$\"3E L14*)\\%*o;FW7$Fd\\l$\"3Eabq-Yj\\FW-Fc]l6&Fe]lFf]lFg]lFf]l7'FKFh]lF]ho-F46&F6QLThe~function ~and~the~2~term~~approximationsF'F>F?FD7'FD-F46&F6QLThe~function~and~t he~3~term~~approximationsF'F>F?-FL6$7S7$FP$!3A+++D[G&4'FW7$FU$!3<^1f<9 0w\\FW7$Fen$!3s\\?xP^B&4%FW7$Fjn$!3v>#4T!*4O?$FW7$F_o$!3xCuRL]62CFW7$F do$!3N$3d(>=95+X6FW7$F^p$!3aY#Q(pEMxjF\\u7$Fcp$!3_ [)oiXz#>>F\\u7$Fhp$\"3*\\x/zfFTx\"F\\u7$F^q$\"3Q8Psy[qT[F\\u7$Fcq$\"3& )pQ.BN&)ppF\\u7$Fhq$\"3()**F\\u7$Fbr$ \"3*Hbxp.C_1\"FW7$Fgr$\"3,Ph)\\p*='3\"FW7$F\\s$\"3Vg#e!op:n5FW7$Fas$\" 31t:N;qp<5FW7$Ffs$\"3nvS)4K>fE*F\\u7$F[t$\"3&RJ-4uR-?)F\\u7$F`t$\"3kYJ %=m[\")z'F\\u7$Fet$\"3E%\\,-H_7G&F\\u7$Fjt$\"3Ox$*Gl*>tb$F\\u7$F`u$\"3 Qqs[gH$*))=F\\u7$Feu$\"3G0yJ7RE7VFgu7$F[v$!3Tq%4Ux\"ou=F\\u7$F`v$!3t( \\J(=n=.NF\\u7$Fev$!3%[S'H86Gw^F\\u7$Fjv$!3I:_vpP\"\\w'F\\u7$F_w$!3NeN ^$=&oM\")F\\u7$Fdw$!3%em!Ry,hU#*F\\u7$Fiw$!3m5;zP:4<5FW7$F^x$!3kl6w\"f `)o5FW7$Fcx$!3-plF%o)='3\"FW7$Fhx$!3#y[PnJFR1\"FW7$F]y$!3f0&*ogP&p$**F \\u7$Fby$!3K4Fh`Gl&z)F\\u7$Fgy$!3D+z/BKL/sb>#Fj[ q7$Fcp$\"3%fPf(4*)f7=Fj[q7$Fhp$\"3nd-#o5xrZ\"Fj[q7$F^q$\"3#eMhlD;t<\"F j[q7$Fcq$\"3Nmml=W?'[*F<7$Fhq$\"3t!*QGspC!G(F<7$F]r$\"3/cOceG)yU&F<7$F br$\"38==8`fAdRF<7$Fgr$\"3$fFTTpI^'GF<7$F\\s$\"3qvv_:s#z$=F<7$Fas$\"3U yZu)=aN<\"F<7$Ffs$\"3oCE'*G=#)zgF\\q7$F[t$\"3!opEEj\"*Rn#F\\q7$F`t$\"3 *G$QF-\\!)[PFW7$Fet$!3%**fKi\"eTApFW7$Fjt$!3<8p!y9k\"R$*FW7$F`u$!3C*yM HfGQH'FW7$Feu$!3jlG]l_f]:F\\u7$F[v$\"3*y8ybz$)QD'FW7$F`v$\"3oq!*[H,*[I *FW7$Fev$\"3[ibwU'3sI(FW7$Fjv$!3\"p')QAH5%*Q$FW7$F_w$!3'p%z3UjvBDF\\q7 $Fdw$!3Jr_t=OyzfF\\q7$Fiw$!3TFfu)Gh!o6F<7$F^x$!3hgV%*[!)pv=F<7$Fcx$!3; v*R>wAa'GF<7$Fhx$!3![)[#*RRm'*RF<7$F]y$!3+,K@zlt;bF<7$Fby$!3128H$oCOC( F<7$Fgy$!30/X+*fQTQ*F<7$F\\z$!3Y'GfZ&Qc$=\"Fj[q7$Faz$!3ivdlMq<\"[\"Fj[ q7$Ffz$!3@m(RE/u-\"=Fj[q7$F[[l$!3[F[Fg-$G>#Fj[q7$F`[l$!3i\")*znUc5i#Fj [q7$Fe[l$!3'*>xo+=\"*fIFj[q7$Fj[l$!3-7e;`dG>OFj[q7$F_\\l$!3?m%HaAhK<%F j[q7$Fd\\l$!3QWx-y7bA[Fj[q7$Fi\\l$!3e&=GR-EL]&Fj[q7$F^]l$!34++IV8c-jFj [qFb]l7'FD-F46&F6QLThe~function~and~the~4~term~~approximationsF'F>F?F^ bpFh]lFd[q7'FD-F46&F6QLThe~function~and~the~5~term~~approximationsF'F> F?-FL6$7W7$FP$!3D***pm\\1!Rt!#97$Fe^l$!3PQ@>h7][lFieq7$FU$!3l@nNuS4FeF ieq7$Fg_l$!3us)ymZP?D&Fieq7$Fen$!3;m;?VH\"Hs%Fieq7$Fjn$!35Yn)ox@ko$Fie q7$F_o$!3\"p\"G()*RwP$GFieq7$Fdo$!3]#=,))[A\"[@Fieq7$Fio$!3oV<'>j&RO;F ieq7$F^p$!3Mln3Ua987Fieq7$Fcp$!3,\"eGIG8Jr)Fj[q7$Fhp$!39$pSwS#*)3hFj[q 7$F^q$!3FQKe)yv16%Fj[q7$Fcq$!3L+Eg=&Q:\"GFj[q7$Fhq$!3D/&QP5Swv\"Fj[q7$ F]r$!3or*)z7o+P5Fj[q7$Fbr$!3O-IQqo]BeF<7$Fgr$!3_WdbN'==>$F<7$F\\s$!38& 48-g)pl8F<7$Fas$!3bKT$)>hN[cF\\q7$Ffs$!3`$G&f6MZv:F\\q7$F[t$!3*f#z.'*= l+dFW7$F`t$!3qD]$zb%G=lFW7$Fet$!3dDY>*>V&=%*FW7$Fjt$!3o\"3@I#[@\\'*FW7 $F`u$!3!G%[>iN:1jFW7$Feu$!3*))**\\iE&f]:F\\u7$F[v$\"3o&z]#4nuliFW7$F`v $\"3EdoEo+J\"f*FW7$Fev$\"3GA22EX9Y&*FW7$Fjv$\"3#H&4/;)p')e'FW7$F_w$\"3 dwM#)eL'*[bFW7$Fdw$\"3EuSe<**4H:F\\q7$Fiw$\"3wLvG*fXff&F\\q7$F^x$\"3VA G7N&*p?9F<7$Fcx$\"33\"y-_2JC>$F<7$Fhx$\"3IkO4.n(3$fF<7$F]y$\"3\"Rd\\ln Cz1\"Fj[q7$Fby$\"3o&G>/#)*)=u\"Fj[q7$Fgy$\"3*[T`^.R$eFFj[q7$F\\z$\"3Ak 8Ic5(*[TFj[q7$Faz$\"3!>a7DL/x8'Fj[q7$Ffz$\"3GXvBO6y$p)Fj[q7$F[[l$\"3Mc ]/'[O0@\"Fieq7$F`[l$\"38hlK!pZ^k\"Fieq7$Fe[l$\"3nJB([d!4X@Fieq7$Fj[l$ \"3`'H4ehZ!eGFieq7$F_\\l$\"3_6$39:lNk$Fieq7$Fd\\l$\"3]96JHS$*fYFieq7$F ]fo$\"3PPFetWp>_Fieq7$Fi\\l$\"3-i(3G#*p9$eFieq7$F_go$\"36(3p&>5!4b'Fie q7$F^]l$\"3D***pm\\1!RtFieqFb]l-FL6$7jn7$FP$!3?++qTfnA5F\\q7$Fe^l$!3P( *>WpI$=S)FW7$FU$!3iWKUO;t%z'FW7$Fg_l$!3Z^OAe[6gbFW7$Fen$!3)eBtB&yQmWFW 7$Fcal$!3U1I*\\M]lm)=.x;FW7$F_o$! 3uQI!eeP2\"FW7$Fcil$\"3Mq@Q0T7#3\"FW7$Fhp$\"3lk6nN%\\l0\"FW7$F^q$\"3Yp t#eAk:A*F\\u7$Fcq$\"3B&3&=gL7?uF\\u7$Fhq$\"3/a3'Qj'>R]F\\u7$F]r$\"3XGt UwnEkDF\\u7$Fbr$\"3)=j6[*>r@KFh\\p7$Fgr$!3?2w2&*G_c9F\\u7$F\\s$!3eFK*z tw^8$F\\u7$Fas$!3)fP,\"pIYFTF\\u7$Ffs$!3,b'4Q,biv%F\\u7$F[t$!39`;w1S!z %[F\\u7$F`t$!3f%f5w'*\\Q[%F\\u7$Fet$!3p[skFgUXPF\\u7$Fjt$!3E7]#[H\"[_E F\\u7$F`u$!3#G_DF;ZkW\"F\\u7$Feu$!3OsZ.'R**[L$Fgu7$F[v$\"3MMCG&*3vN9F \\u7$F`v$\"3eXW-*Gf^h#F\\u7$Fev$\"3H:`%R_U_o$F\\u7$Fjv$\"3+\")[y`)=3Z% F\\u7$F_w$\"3/QV1(=C2%[F\\u7$Fdw$\"3w=%eCJ4Ow%F\\u7$Fiw$\"3kg[t!)\\\" \\8%F\\u7$F^x$\"3Y\\[9bZ;vIF\\u7$Fcx$\"3f#4#yl1/c9F\\u7$Fhx$!3bC%H\" \\F!)\\QFh\\p7$F]y$!3+1^:59Q#p#F\\u7$Fby$!3)*)y'o#p&Q%*\\F\\u7$Fgy$!3x om#QJ%)QK(F\\u7$F\\z$!3U)\\Lo_v7E*F\\u7$Faz$!3w(pz:wtv0\"FW7$Fa\\o$!3t -$y5$Q=#3\"FW7$Ffz$!3Xb&Q`\">6u5FW7$Fc]o$!3xn&G[slc-\"FW7$F[[l$!3?A]An x\\&H*F\\u7$F^`o$!3IPrZSdX'z(F\\u7$F`[l$!3JEK%R`bQn&F\\u7$Fe[l$\"30ad' zAvzT\"Fh\\p7$Fj[l$\"3g@%R+#*440\"FW7$F_\\l$\"3Sq#Q![!e-R#FW7$Fado$\"3 empNj^R\"H$FW7$Fd\\l$\"3;M>sy\"*>RVFW7$F]fo$\"33_*zv(f.#\\&FW7$Fi\\l$ \"3tPMn5KH/oFW7$F_go$\"3?mJ&e0xsS)FW7$F^]l$\"3?++qTfnA5F\\qFdapFh]l7'- F46&F6QLThe~function~and~the~6~term~~approximationsF'F>F?FDFceqFf_rFh] l7'-FL6$7fo7$FP$!3M-++*)oKyxFW7$FU$!32Mfwi!>eb'FW7$Fg_l$!3+Ox&Qq%HyeFW 7$Fen$!3tqCzG(fU:&FW7$Fcal$!3)p6>g*z59VFW7$Fjn$!3#o'>_^v0uMFW7$Febl$!3 yAef+7R_EFW7$F_o$!3.qPphX)R(=FW7$Fgcl$!3qiP&>2*3d6FW7$Fdo$!3\"ehGgqH*) 3&F\\u7$Fio$\"3hwIiL)*)H([F\\u7$F^p$\"3EuVj=7FW7$Fahl$\"3Sep!**yv(y 9FW7$Fcp$\"3*R$\\e(R+1m\"FW7$F^il$\"35R_StgdB5,qn#o,\"=FW7$Fejl$\"38\"oAF7\\*)y\"FW 7$F^q$\"3b8I==Ns5'f%)*3RT>$F\\u7$F\\s$!3&37;'[)[MF(F\\u7$Fas$!3M'=))e\"FW7$F[t$!30`* \\Xee_?\"FW7$F_em$!3(yfJVLhN=\"FW7$F`t$!3j=`&fJ'QJ6FW7$Fet$!3+PG(or*)= b*F\\u7$Fjt$!3\\$4*>C5X*z\"\\/7 FW7$Fgan$\"3e$*)p'HQ&*)>\"FW7$Fdw$\"3*y*z)fyGZ;\"FW7$Fibn$\"3]6Lg*fvH4 \"FW7$Fiw$\"3G0@j$*)[K')*F\\u7$F^x$\"3gjx%4)emArF\\u7$Fcx$\"3oC0hIP+$> $F\\u7$Fhx$!3+Z$>$=V[45F\\u7$F]y$!3m_o:&z'f@fF\\u7$Fby$!3[*y(Gc#p5/\"F W7$Fgy$!3c/D$*>GbV9FW7$F]jn$!3g,H5Ck0)f\"FW7$F\\z$!3i)4CcsW`r\"FW7$F_[ o$!3s8a6jm=R(RRs\"FW7$Ffz$!3,Gxtv;[\"FW7$F[[l$!3#oL3ZdnLA\"FW7$F`[l$!3bPWpp[U4ZF\\u7$Fe[l$\"3!y 7')eUh%G]F\\u7$Fj[l$\"3Y$R%*p(Ga@>FW7$F_co$\"3)oH'**zV5TEFW7$F_\\l$\"3 3jlPtKv(R$FW7$Fado$\"3chrd/8XGUFW7$Fd\\l$\"3xa?]GV#=1&FW7$F]fo$\"3L!HU TKtn$eFW7$Fi\\l$\"3h%yOL=I0c'FW7$F^]l$\"3M-++*)oKyxFWFdapFD-F46&F6QLTh e~function~and~the~7~term~~approximationsF'F>F?Fh]l-FL6$7X7$FP$\"3u**H L*[`y%R!#87$Fe^l$\"3\"G[7G-8wN$Fbjs7$FU$\"3#)RL0K#3U%GFbjs7$Fg_l$\"3NO 28TRS`CFbjs7$Fen$\"3\"[?&)p(eC4@Fbjs7$Fjn$\"3+h![Z&G\">[\"Fbjs7$F_o$\" 3Eop![`4#=5Fbjs7$Fdo$\"3OcLj)\\jQ&oFieq7$Fio$\"39J\"[e+\"HVYFieq7$F^p$ \"3]nqkt4pBIFieq7$Fcp$\"3n([P!y_A!)=Fieq7$Fhp$\"3#\\4B4;+(G6Fieq7$F^q$ \"3)3%\\!R@\"[&Q'Fj[q7$Fcq$\"3ylL7USU+PFj[q7$Fhq$\"3Qw(H0\")e-*=Fj[q7$ F]r$\"3e?2Z\\%z-(*)F<7$Fbr$\"31MLd#RI21%F<7$Fgr$\"3^,_w]P_h=F<7$F\\s$ \"3q0uarywfqF\\q7$Fas$\"3$Q!G.MUhaJF\\q7$Ffs$\"3?Q!zs3xo@\"F\\q7$F[t$ \"3WnFm#QCHp#FW7$F`t$!3)=^atA2;%[FW7$Fet$!3\\NIhh)=q=*FW7$Fjt$!30#f=6m Gnj*FW7$F`u$!3lP\"H$Rp,1jFW7$Feu$!3b%**\\iE&f]:F\\u7$F[v$\"3Tpx\\FW7$F _w$!3]9xBk(R;D#FW7$Fdw$!3[REFc.x)=\"F\\q7$Fiw$!3I[C/c9-JJF\\q7$F^x$!3` 4'3rs([atF\\q7$Fcx$!3DZ!Q;$y'>'=F<7$Fhx$!3Gn*RM?)ohTF<7$F]y$!3S@,Cs%=n M*F<7$Fby$!3r2/:_k=m=Fj[q7$Fgy$!3z@-:h;T+OFj[q7$F\\z$!3<.\"[kws6Z'Fj[q 7$Faz$!3ZNTjh>OO6Fieq7$Ffz$!3*>1(HAeBu=Fieq7$F[[l$!3s`,G$>oV,$Fieq7$F` [l$!3#)Qj_!>(*)yYFieq7$Fe[l$!3rG%p'QW.SoFieq7$Fj[l$!3#\\yhDK$oI5Fbjs7$ F_\\l$!3aMvdu;Td9Fbjs7$Fado$!3pc,._\">1u\"Fbjs7$Fd\\l$!39.KkHJJp?Fbjs7 $F]fo$!3IhS3OH$>V#Fbjs7$Fi\\l$!3#*>M@'HYs%GFbjs7$F_go$!3eQzXD?OfLFbjs7 $F^]l$!3u**HL*[`y%RFbjsFb]l7'-F46&F6QLThe~function~and~the~8~term~~app roximationsF'F>F?FDFh[sFh]lF\\js7'FD-F46&F6QLThe~function~and~the~9~te rm~~approximationsF'F>F?Fh]l-FL6$7_t7$FP$\"3mb'****\\$R0bFW7$$!3,n;HdN vs**FW$\"3O\\)HA5Z@)RFW7$$!3.MLe9r]X**FW$\"3:,QAEx#fb#FW7$$!3/,](=ng#= **FW$\"3=xN!p#4/B7FW7$F`^l$!3:f[vE^/6?Fh\\p7$$!3&GLekynP')*FW$!3Ps>m> \"Gq<\"FW7$$!3')***\\PM@l$)*FW$!3^p_ul,7^AFW7$$!3)omT5!\\F4)*FW$!3.vnV [GqXKFW7$Fe^l$!3iO:#)oB+kTFW7$$!3#om;HdNvs*FW$!3\"\\qc8*o@%y&FW7$Fj^l$ !36@gETz$e8(FW7$$!3wLL3-)\\&='*FW$!3:e@xn!3;C)FW7$FU$!3Ik*48i!*H7*FW7$ Fb_l$!3ci=(\\A;FW7$Fbcl$!3!4#=_4_zJWF\\u7$Fgcl$\"3obX)otEOW'F\\u7$F\\dl$\"3!p%=H ,acD;FW7$Fdo$\"3/Xszde,!\\#FW7$Fidl$\"3!e=]I?%HkPFW7$Fio$\"3L5*zTDkXg% FW7$$!3K++]i#=ES(FW$\"3+4ms5o$R([FW7$Ffel$\"3P/8HP!pi.&FW7$$!3#*)***** *zp([sFW$\"3)>^4\"=V=z]FW7$$!3Cmm;zp[(>(FW$\"3Q-*=*G?u(4&FW7$$!3bLLLeT ?YrFW$\"3gD[-73(G4&FW7$F^p$\"3o.=+ezal]FW7$Fahl$\"35,E0A')oOZFW7$Fcp$ \"3wK6jg$*))=TFW7$Fhp$\"3%[3jq]AcM#FW7$F^q$\"3[Aw!)3lESHF\\u7$Fcq$!3q3 &HF=x!)G\"FW7$Fd\\m$!3oJ&4*ok0')>FW7$Fhq$!3#[.-)*oa6^#FW7$$!3K+++DL\") *)[FW$!3QwBrd.40FFW7$F\\]m$!3XL$RBLV9&GFW7$$!3>+++v7CtYFW$!3f67([p#H]H FW7$F]r$!3[(HU'z+L-IFW7$$!3E+++D%yF^%FW$!3>bI&e)H/6IFW7$Fd]m$!3Jue'G\" oN4IFW7$$!3_+++vZU3WFW$!3sp7\"G5$[(*HFW7$Fi]m$!3'=!Grd(ec(HFW7$F^^m$!3 ucj(H`CK!HFW7$Fbr$!3GQF#fR0Wz#FW7$F[_m$!33e`VO\"*e7DFW7$Fgr$!3KP]8'R** y8#FW7$F\\s$!3>$*[&>!GC^**F\\u7$Fas$\"3;(Q5JjMio(Fh\\p7$Ffs$\"3O9N\") \\E/27FW7$F]dm$\"3j8X.j)fAh\"FW7$F[t$\"3+dQB&Rqz$>FW7$F_em$\"3]z/pSzc \">#FW7$F`t$\"3Sd(>o[?$GBFW7$F\\fm$\"3\\+x*Q3X<7b?#FW7$F_w$!3*H$y7tixo>FW7$F gan$!3c0k2FqST;FW7$Fdw$!3n/VJaA\\G7FW7$Fiw$!3?W.YfT3\"p)Fh\\p7$F^x$\"3 GH,VANsZ5FW7$Fcx$\"3_qz&Q_X\"Q@FW7$Fifn$\"3%)*[ABagM_#FW7$Fhx$\"3`7p1: (G\"4GFW7$Ffgn$\"3222C(y9f\"HFW7$F[hn$\"3Ol&\\.(f5%)HFW7$$\"3W*\\(oz,I FWFW$\"3]Q^,3-%H+$FW7$F`hn$\"3nV))p*)*G7,$FW7$$\"3Em\"HKH`U`%FW$\"36av UBts3IFW7$F]y$\"3_!**))\\w=_*HFW7$$\"3*em\"z%Rk$)o%FW$\"3.f3SQ=KRHFW7$ Fhhn$\"39P+Ir_(G%GFW7$$\"3M)*\\P%[L'*)[FW$\"3\"*Qs$H$HP0FFW7$Fby$\"31. `YKzwEDFW7$F`in$\"3c`\"f(H3;B?FW7$Fgy$\"3t6Ri!HfRN\"FW7$F\\z$!3*)3['RJ #G&Q$F\\u7$Faz$!3P@_wm)Q3P#FW7$Ffz$!3YgMes(\\$4TFW7$Fc]o$!3e`X4fRBIZFW 7$F[[l$!3qrXwl_Qj]FW7$$\"3=*\\P%)RZY9(FW$!3WK'ecwmB4&FW7$$\"3gl;a)GJs> (FW$!3OspLk]x(4&FW7$$\"38Leky^\")\\sFW$!3^n@1qdby]FW7$F^`o$!3K3***HB3P .&FW7$$\"3]L$e*[oc2uFW$!3dMw0)fUM'[FW7$F`[l$!3szbMu*f.e%FW7$F[ao$!3>p( )z84w\\PFW7$Fe[l$!3=t%Q#yl!f]#FW7$Fhao$!3cf$y3>i*4;FW7$F]bo$!3z%*eGujC ()eF\\u7$Fbbo$\"3ZI/iKbNgaF\\u7$Fj[l$\"3M6]M4]Ox% HFW7$F_co$\"3N8^A(e$z*F W7$Fd\\l$\"3%)H%*[*QPn/\"F\\q7$$\"3Z+](oW7;@*FW$\"35&3*et(3y1\"F\\q7$F heo$\"3E&HaDO..3\"F\\q7$$\"3Y+]i:mq7$*FW$\"3ZcTI\"*>.\"*FW7$$\"3 Q+D\"Gyh(>'*FW$\"3HMrN\"fJ&>#)FW7$Fjfo$\"3?ypVn)RC6(FW7$$\"3G+vVt7SG(* FW$\"3!oTCM$*Q1w&FW7$F_go$\"3+DJQt(z:9%FW7$$\"3v]iS\"*3))4)*FW$\"3][zD '[BWA$FW7$$\"3;+D1k2/P)*FW$\"3#)GyHBxRJAFW7$$\"3e\\(=nj+U')*FW$\"3+->K 9^If6FW7$Fdgo$\"3jFj.atLf[Fgu7$$\"3k]7.#Q?&=**FW$!3;*obXi;`B\"FW7$$\"3 0+voa-oX**FW$!3r,'*p]ApkDFW7$$\"3Z\\PMF,%G(**FW$!3u@(4XFJo)RFW7$F^]l$! 3mb'****\\$R0bFWFdap-FL6$7Y7$FP$!3++n1,li47!#77$Fe^l$!3-b^\"y>r**z*Fbj s7$FU$!3)G[=A6T!)*yFbjs7$Fg_l$!3Hq')3w&4p^'Fbjs7$Fen$!3uimKxW'QN&Fbjs7 $Fcal$!3J:j4Cy`oUFbjs7$Fjn$!38#G3U%=c#Q$Fbjs7$F_o$!3&>g2J#R%f2#Fbjs7$F do$!3Y'ol:!RgS7Fbjs7$Fio$!3Ufy%R8J&yuFieq7$F^p$!3kiGFOh2&G%Fieq7$Fcp$! 36LHO#\\&e:BFieq7$Fhp$!3%fUgdH%G)>\"Fieq7$F^q$!3DmBvYmgndFj[q7$Fcq$!3l i.%oc*=yGFj[q7$Fhq$!3kE=QMPlK7Fj[q7$F]r$!3TVn$))HR8%[F<7$Fbr$!3b:_?:74 m*FW7$Fjt$!3+C>O\")>-P'*FW7$F`u$!3F.0xAy,1jFWF^_t7$F[v$\"3%\\M/w -=cE'FW7$F`v$\"3y+HW!*)*Q!e*FW7$Fev$\"3ungKDZhd$*FW7$Fjv$\"3d.o))[eTH^ FW7$F_w$!3wTEn#[%o)4\"FW7$Fdw$!3S'ypf7(>\\hFW7$Fiw$!3ed[S)4K_f&FW7$F^x $\"3cKFWQ4zB()FW7$Fcx$\"3%\\my'p2V!)fF\\q7$Fhx$\"3?d8Opc]B=F<7$F]y$\"3 q\">nq#yd(4&F<7$Fby$\"3m'=P,giH@\"Fj[q7$Fgy$\"3r\"=#*zsr)zFFj[q7$F\\z$ \"3`94Cd@!o'eFj[q7$Faz$\"3uBqgk0x37Fieq7$Ffz$\"3\"z'ow%H[gI#Fieq7$F[[l $\"3x]-`re%zE%Fieq7$F`[l$\"3U\\/Hwg7`vFieq7$Fe[l$\"3!o*\\tB\"Fbjs7 $Fj[l$\"3!R@S!ey34@Fbjs7$F_\\l$\"3.%z3\\3*)*4LFbjs7$Fado$\"3Urw!GUg,<% Fbjs7$Fd\\l$\"377o/d?SA_Fbjs7$F]fo$\"3)\\6)*GXMGW'Fbjs7$Fi\\l$\"33,r+U H,4zFbjs7$F_go$\"3*oO\"R&\\3m!)*Fbjs7$F^]l$\"3++n1,li47FhevFb]l7'FD-F4 6&F6QMThe~function~and~the~10~term~~approximationsF'F>F?Fh]lFjdtFbev7' -FL6$7gn7$FP$\"3U+L$*Rn@*Q#Fhev7$F`^l$\"3d66Kf^o+@Fhev7$Fe^l$\"33'*\\f (Q4T%=Fhev7$Fj^l$\"3)3bfu&*yih\"Fhev7$FU$\"3+#4'\\s?F99Fhev7$Fg_l$\"3# eFo\\B^l6\"Fhev7$Fen$\"3q0(y+U'Ho()Fbjs7$Fcal$\"3cv2'*o0APmFbjs7$Fjn$ \"3!3tXDaMr)\\Fbjs7$F_o$\"3K9[1LcaQFFbjs7$Fdo$\"3%*y,MTVdc9Fbjs7$Fio$ \"3Xn%=$or>PyFieq7$F^p$\"3w[lz;'Ru'RFieq7$Fcp$\"3Pkzc3XJs=Fieq7$Fhp$\" 3S&pClgJ8R)Fj[q7$F^q$\"3!4')*[[S)GW$Fj[q7$Fcq$\"3i=RmGZ)=Z\"Fj[q7$Fhq$ \"3tD9MM3gs^F<7$F]r$\"3I1g&Q@yYh\"F<7$Fbr$\"3YsEfA)*QA[F\\q7$Fgr$\"3H( 4(3Fl1c=F\\q7$F\\s$\"3s`/'>1.X<\"F\\q7$Fas$\"3%[wn'4M0d5F\\q7$Ffs$\"3P 8oSCLxaqFW7$F[t$\"3'*)\\naTM7b\"FW7$F`t$!3Yan1B8P\"*\\FW7$Fet$!3'*ox& \\6-\"*>*FW7$Fjt$!3MW5y,1jFWF^_t7$F[v$\"3!fnmT-= cE'FW7$F`v$\"3DuA4-hQ!e*FW7$Fev$\"3!HkInZlsN*FW7$Fjv$\"3eQooYH1?^FW7$F _w$!3[67v,gA57FW7$Fdw$!3$*4Z?J9MEqk&=F\\q7$Fhx$!3[t:U]pP(*\\F\\q7$F]y$!3>#G))> r6>I-c[Fbjs7$Fado$!3Dv4H<3u\\kFbjs7$Fd\\l$!3')frf;(zV])F bjs7$F]fo$!3]t(*>'[m45\"Fhev7$Fi\\l$!3z&)z%yq)o;9Fhev7$Fjfo$!3'o.=zGC$ =;Fhev7$F_go$!3_!eny#ekX=Fhev7$Fdgo$!3k,YEC&\\:5#Fhev7$F^]l$!3U+L$*Rn@ *Q#FhevFb]l-FL6$7]t7$FP$\"3\"*))***))*R(33\"F\\q7$$!3]LekynP')**FW$\"3 I4b^)\\o$o&*FW7$Faet$\"3y!R^OzQKP)FW7$$!3_+v$fLI\"f**FW$\"3!QLA\")4%HA sFW7$Ffet$\"3Yo&>O!e\\9hFW7$F[ft$\"3PDBIxLFCSFW7$F`^l$\"3!*=54!*oh%4#F W7$Fcft$\"3:^*f\\.s%yJF\\u7$Fhft$!3[\\%4#3nW88FW7$F]gt$!3,(G@k(\\T1GFW 7$Fe^l$!3J$3GU!R&z;%FW7$Fegt$!3#fD)))e65BlFW7$Fj^l$!3U$Qa$y9CH%)FW7$F] ht$!3a85&>8%4L**FW7$FU$!3+N)*yRXz26F\\q7$Fb_l$!3+&)3Q/6;L7F\\q7$Fg_l$! 3p;/%)f`s!G\"F\\q7$F[it$!37IB4l\\V#G\"F\\q7$F`it$!3]LJ1&4_0G\"F\\q7$Fe it$!3cHxWJ9Iv7F\\q7$F\\`l$!36&pNcC**oE\"F\\q7$F]jt$!3_=NO'pk9C\"F\\q7$ Fen$!3U7aX2*=e?\"F\\q7$Fi`l$!3?4/cg?l&4\"F\\q7$Fcal$!3xVR:.[Ja&*FW7$Fh al$!3zV@\"3,f0(zFW7$Fjn$!3->Z&p@2=I'FW7$F`bl$!35wKDct49YFW7$Febl$!3%34 (>pugzHFW7$Fjbl$!33M=hD.vW9FW7$F_o$!3,U8&G&R9CVFh\\p7$Fgcl$\"3iIC&)*z4 uE#FW7$Fdo$\"34#o,\\^WL)QFW7$Fddl$\"3Y]Pan`k(R%FW7$Fidl$\"3M1&R[A93x%F W7$$!3inm\"zHmPl(FW$\"3`ot#R6Xq!\\FW7$F^el$\"3zd=;;')H6]FW7$$!3/++vo8r avFW$\"3Ztu&fsy[3&FW7$Fio$\"3]-m.+0;H^FW7$$!37MLL$3,RX(FW$\"3x0=KsdmX^ FW7$F_]u$\"3kS@sm\"QR8&FW7$$!3`mmmTaL^tFW$\"3;s]Zg$fc4&FW7$Ffel$\"3V7< %)f!QD.&FW7$F\\^u$\"3Eg&*>38nQ[FW7$F^p$\"3<\"[F@pVhc%FW7$Fahl$\"3r0$zt A<4\"QFW7$Fcp$\"3Mr5e$\\%yi@FW7$Ff`u$!3E^dC G8s5@FW7$F]r$!3;^Rt_aFI?FW7$Fi]m$!3qm\\w#e2v!=FW7$Fbr$!3W1[/>J*f^\"FW7 $Fgr$!3&H))>`j!pr))F\\u7$F\\s$!3!p*>\"\\=0i=\"F\\u7$Fas$\"3yh!za9$H)R% F\\u7$Ffs$\"3'H?J113T-*F\\u7$F]dm$\"3_d&4dO)=L5FW7$F[t$\"3;.UOb%Rz6\"F W7$F_em$\"3K7XjCN\")f6FW7$F`t$\"3M*3Rc%pn^6FW7$Fafm$\"3G-vp@\"*R-:\"FW7$Fe`n$!3Mm \"o$*Gc3;\"FW7$F_w$!3Sh\">@$=dC6FW7$Fgan$!3qnn%oD^;/\"FW7$Fdw$!3ux!))= \\\"y)4*F\\u7$Fiw$!3+zne#39]W%F\\u7$F^x$\"3!*[6Iyby$\\\"F\\u7$Fcx$\"3. #yoGOZO())F\\u7$Fhx$\"3>qpJzFwM:FW7$F[hn$\"3^%o')f)oDH=FW7$F]y$\"3+9g1 O#)Q\\?FW7$F`ju$\"3aD@%>Cs(>@FW7$Fhhn$\"3V20y%pZ_;#FW7$Fhju$\"3wA.mF2E $=#FW7$Fby$\"3_2tb'Q49<#FW7$$\"3c(****\\7ga4&FW$\"33O/$3*\\rC@FW7$F`in $\"3bgu)\\4#yS?FW7$$\"3ylmm;V%eI&FW$\"3nG$*z5Ky<>FW7$Fgy$\"3fFc@rkMa9*=)G:8FW7$F\\z$\"3m')=m&Qm()>(F\\u7$Faz$!3aX(RZpndj*F \\u7$Ffz$!3d:.#)3UpxGFW7$Fc]o$!3Q)Q+[v'f*z$FW7$F[[l$!3x&3'G1jbdXFW7$Fj \\v$!35MO4)*R3Q[FW7$F^`o$!3t*oU.-Wf.&FW7$$\"3)f;a)eH)\\N(FW$!3[r\"f*eG A*4&FW7$Fg]v$!3e>%\\nHai8&FW7$$\"3#**\\i!R2:guFW$!3\"\\)He(G)>X^FW7$F` [l$!3_shULfCC^FW7$F[ao$!3))o*Gps*QiZFW7$Fe[l$!3!*R!*o+N&e*QFW7$Fhao$!3 e*)[-'*ze\\JFW7$F]bo$!32)*oyUN[9AFW7$Fbbo$!33qu`^mO\"4\"FW7$Fj[l$\"3C \\qwaem>@F\\u7$Fjbo$\"3B#oBeQ&=<:FW7$F_co$\"31,kyu)*)[$HFW7$Fdco$\"3U! ymO'z^QWFW7$F_\\l$\"3G+%o(GkM#*fFW7$F\\do$\"30(3(f;M&>l(FW7$Fado$\"3xZ tkV8BW#*FW7$F[eo$\"3J6IW?`$z1\"F\\q7$Fd\\l$\"3_eAn2*\\Y=\"F\\q7$Fi`v$ \"3%y^#*3o$=F7F\\q7$Fheo$\"34kQTh5%)e7F\\q7$$\"3,,vVtIV(G*FW$\"3NYTY'> @+F\"F\\q7$Faav$\"3(Q;b6ayxF\"F\\q7$$\"3!**\\7y:!)zL*FW$\"3b$*fG#Qd=G \"F\\q7$F]fo$\"3AGg,?8*>G\"F\\q7$Fbfo$\"3Jd!GF=etB\"F\\q7$Fi\\l$\"3*H0 'fweH06F\\q7$F_bv$\"3I%=o'=Yq.**FW7$Fjfo$\"3Cb?$Rb2oR)FW7$Fgbv$\"3=R?< iWK*['FW7$F_go$\"3hX\\Ax*=\\8%FW7$F_cv$\"37*GB*)*QmuFFW7$Fdcv$\"30[i<[ Pm$G\"FW7$Ficv$!3(H,(o!pt\"\\MF\\u7$Fdgo$!3X6H,'4m\"=@FW7$Fadv$!3o5j3# yEM/%FW7$Ffdv$!3eW()e*e3$GhFW7$$\"3@Cc,\">g#f**FW$!3%4j<+ZfIB(FW7$F[ev $!3?,yN_Fp!Q)FW7$$\"3tu=nj+U')**FW$!3W\"QU#=tBs&*FW7$F^]l$!3\"*))***)) *R(33\"F\\qFdapFh]lFD-F46&F6QMThe~function~and~the~11~term~~approximat ionsF'F>F?7'F]`wF^[xFh]lFD-F46&F6QMThe~function~and~the~12~term~~appro ximationsF'F>F?7'-FL6$7cx7$FP$!3W%>)**[j[`7F\\q7$Fe[x$!3%)Hv:tT)o4\"F \\q7$Faet$!3;&pmpx#**pFW 7$F[ft$!3Lbs!H8PZ!\\FW7$F`^l$!3BHl3&=iN?$FW7$Fhft$!3m#=%*=xn&*H)F\\u7$ Fe^l$\"3[[LD-e$*)*RF\\u7$$!3Sn\"zpB0%o(*FW$\"3g')eMyU#yd&F\\u7$$!3\"4+ Dc,#ya(*FW$\"3s#)QM#Rp%HmF\\u7$$!3UM3F%ze6u*FW$\"3i\"*f/q]>(=(F\\u7$Fe gt$\"3whR,:=U*FW$!3d*391L!oh dFW7$Fg_l$!3Au(=x()Hm9(FW7$F\\`l$!3WjRMF199(*FW7$Fen$!3x8*o.#=t\"=\"F \\q7$Fi`l$!3Eo7Aze(\\M\"F\\q7$Fcal$!3i8+)4^xCU\"F\\q7$$!3<#3xJK'zW*)FW $!3uZI=_BOG9F\\q7$$!3JmT5S`8=*)FW$!3`$4(\\7u(*G9F\\q7$$!3W]7.dVZ\"*))F W$!3LqfD&))GWU\"F\\q7$Fhal$!3eyu_\"yV[T\"F\\q7$$!3^*\\7yS\"\\6))FW$!3Z k;xvJ;\"Q\"F\\q7$Fjn$!3%*p&QF3*GH8F\\q7$F`bl$!3a)>G=jdh<\"F\\q7$Febl$! 3xs+))[N+,(*FW7$$!3#fm;zM#z*[)FW$!3-`BzByO?&)FW7$Fjbl$!3`WGYJ;[lsFW7$$ !32mmT5:W#Q)FW$!3>')GOu)*\\bfFW7$F_o$!3i(e508=!4YFW7$$!3))*\\7GyX`F)FW $!3pfEhA&y.D$FW7$Fbcl$!3Pmt,2*G**)=FW7$$!3NnT5l^]o\")FW$!3xQp_`q3LaF \\u7$Fgcl$\"3*[5Y!G!4)\\xF\\u7$F\\dl$\"3h)z25!QpuKFW7$Fdo$\"3\"><%4v@M =bFW7$Fddl$\"3(oa3C\"G\\3tFW7$Fidl$\"3Q0,#3rl&y()FW7$F^el$\"3l`ddMhe.* *FW7$Fio$\"3%*Q>rajAn5F\\q7$F_cx$\"3q\\%)zPI,$4\"F\\q7$F_]u$\"3)\\X?7@ $R46F\\q7$Fgcx$\"3-\"oh!)psl6\"F\\q7$Ffel$\"3Qfw'f^B[6\"F\\q7$F\\^u$\" 38\\DGw6#f3\"F\\q7$F^p$\"37I$4?p^f-\"F\\q7$F]gl$\"36\"f!*)**>Ra$*FW7$F ahl$\"3KPi\\MW(y?)FW7$Ffhl$\"3#*[)3SFB@(oFW7$Fcp$\"3AjKK9Hq+aFW7$F^il$ \"3bUVl!)H!>&QFW7$Fcil$\"3#z9.*>!z?F#FW7$Fhil$\"3))*G(>2r6!4(F\\u7$Fhp $!31jK_KkpOzF\\u7$F`jl$!3)**=a[?$oNAFW7$Fejl$!3C>n[]N-PNFW7$Fjjl$!3jvv GI+[oYFW7$F^q$!3K3U$[Zqxg&FW7$Fg[m$!3)z(4xGE?_nFW7$Fcq$!3J#[Y.b-IA(FW7 $$!30L$eke]bP&FW$!3/jGDr#3iB(FW7$Fgex$!3O(FW7$$!3%fmTgP2xE&FW$ !3!Gu(e='yK6(FW7$Fd\\m$!3ApXa(=]1)pFW7$F_fx$!3O%z(4M\"HPe'FW7$Fhq$!3p) )oMiW+GgFW7$F\\]m$!3#GTPE)[_DXFW7$F]r$!3&G*HIx:/!o#FW7$Fi]m$!3I**z5*p# \\(y(F\\u7$Fbr$\"3*o-KAQ r$FW7$Fh_m$\"3\")*)QS^()>aUFW7$F]`m$\"3'*3X\"QNW#fYFW7$$!3RLe9T>y'[$FW $\"3&35?(3J74[FW7$Fb`m$\"3kdz')\\XPB\\FW7$$!3R+v$fV*4uLFW$\"3w&pus\"[* =+&FW7$F\\s$\"31I&QtD5[/&FW7$$!3zLL3_(H+F$FW$\"36()pk4))4rF*=$\\FW7 $Fiam$\"3tDA8nGALZFW7$Fas$\"3I$Q6L9UwW%FW7$F[cm$\"3M3\\$FW7$Ffs$ \"3)p=\"\\O=lGAFW7$F[t$!3)ec$zEy$oNFW7$Fet$!3-+@3!3#\\GSFW7$$!3_m\"z>E\\H?\"FW$!3A*GW/'=u*3%FW7$ F^gm$!3zO\"\\oN/h7%FW7$$!3#)*\\P4'f!e4\"FW$!3-LjJt7_PTFW7$Fcgm$!3\\\"G 'f$>rS7%FW7$Fhgm$!3/+&4'G#of3%FW7$F]hm$!3BuLl<+cBSFW7$Fbhm$!3W>`\\(4Dt $RFW7$Fjt$!3%HBFflpy#QFW7$F_im$!3QQ_j=jjRKFW7$F`u$!3]x6-J#)z)R#FW7$Feu $!3zd(\\8w:,\"eFh\\p7$F[v$\"3#G,#R?J(GQ#FW7$Fe\\n$\"3b+p!3Ch$4KFW7$F`v $\"3aDLE44$yz$FW7$Fg]n$\"3c6j#e\"QZ**RFW7$Fa^n$\"3Gh*z%ogp7TFW7$$\"3GK 3-jr-u5FW$\"3oAZt>S/NTFW7$Ff^n$\"3)*y:HE&*=MTFW7$$\"3hl\"HK\\ow<\"FW$ \"3T\"GL>R4+6%FW7$Fev$\"31\"e`$>)*\\iSFW7$Fc_n$\"3__/X@>MBOFW7$Fjv$\"3 mi'o/xt.#GFW7$Fe`n$\"3w%3')zZSLu\"FW7$F_w$\"3\\M0r\"\\bb+R%\\FW7$$\"37l;HdF3& =$FW$!3J-VGZ#ew+&FW7$Fjdn$!3;*zqKYpV/&FW7$$\"3)=L3-V(>'G$FW$!3qp*GUM8M 0&FW7$F^x$!3e.#\\(3>IM]FW7$Fgen$!3e^#*o.\\JIYFW7$Fcx$!3G2D3`w@8PFW7$Fi fn$!3%fV:H9FK[#FW7$Fhx$!3%4_)4:$)p>&*F\\u7$F[hn$\"39m0K(oluQ*F\\u7$F]y $\"3hRIXjm#Q)GFW7$Fhhn$\"3yC*RA*RI%e%FW7$Fby$\"3V'\\rt8c?)fFW7$Fi]y$\" 3+uOR+xLOlFW7$F`in$\"333z02O\\TpFW7$$\"3G***\\(o#[KD&FW$\"3]`LC!HBA3(F W7$Fa^y$\"3Yj=.\"QN*yrFW7$$\"3GKLek.We`FW$\"3OzqVD.\")HsFW7$Fgy$\"3!4G WO\\nKB(FW7$$\"3rK3-j'eCY&FW$\"3aIH&3KM&*=(FW7$Fhin$\"3oS#o<()G#)4(FW7 $$\"3O)\\iS;.`c&FW$\"3(>C%e&[O'epFW7$F]jn$\"3(o\\RV-C.x'FW7$Fbjn$\"3X, 092HHZiFW7$F\\z$\"3og_9R%y<`&FW7$Fjjn$\"3giT<'y'4&e%FW7$F_[o$\"3UtvdSP c^MFW7$Fd[o$\"3$[kwi!HI`@FW7$Faz$\"3_l=O,M%3>(F\\u7$F\\\\o$!3fMSr(Rt'z vF\\u7$Fa\\o$!3S,nT)*Q&=H#FW7$Ff\\o$!3oZl*Q9=8%QFW7$Ffz$!3?X%e6fr7O&FW 7$F^]o$!3@uC;DY7NoFW7$Fc]o$!3ci:**>Byu\")FW7$Fg^o$!3QJ=Y$[4nK*FW7$F[[l $!3v&>A1isQ-\"F\\q7$Fj\\v$!3UPy/#z3e3\"F\\q7$F^`o$!3qy))3dR4:6F\\q7$Fa `y$!3fVOMh0O;6F\\q7$Fg]v$!3@$*fkpu@36F\\q7$Fi`y$!3=L$Q-?x.4\"F\\q7$F`[ l$!3r.KS'4EE1\"F\\q7$Ff`o$!3QGjof*))R&)*FW7$F[ao$!3Y#\\98sdFu)FW7$F`ao $!3ybl+we_.tFW7$Fe[l$!3*[2LoT!ofbFW7$Fhao$!3Z\\ORXPVMKFW7$F]bo$!3@7Q9 \\r+cjF\\u7$$\"3jm;/,O=w\")FW$\"3a2-1+v]atF\\u7$Fbbo$\"3iJXiE#zZ8#FW7$ $\"3%GLe9'>%pG)FW$\"3&*R=s&)y,YNFW7$Fj[l$\"330\"*G&p/:&\\FW7$Fjbo$\"3o QlSnNQ\"R(FW7$F_co$\"3K2X\"f$Q*oj*FW7$Fdco$\"3X^*Hg5&oc6F\\q7$F_\\l$\" 3!)=H!HAgfI\"F\\q7$$\"3Mm\"z>w'Q\"z)FW$\"3TS)H,#4mj8F\\q7$F\\do$\"3ri@ #3&42/9F\\q7$$\"3W;z%\\g.1())FW$\"3'pO!=DlM<9F\\q7$$\"3yK3F>#4q*))FW$ \"3*>fi*G>zD9F\\q7$$\"3?]PfL[TB*)FW$\"3k6=l4$p#H9F\\q7$Fado$\"3MyZ5&>f wU\"F\\q7$F[eo$\"3Le\\S_qro8F\\q7$Fd\\l$\"3O)4_p)*3^A\"F\\q7$Fi`v$\"3 \"\\:bC8ry7\"F\\q7$Fheo$\"3)*eu_*\\qS,\"F\\q7$Faav$\"3!\\JRv5A&f))FW7$ F]fo$\"3MLkNHE^luFW7$$\"3X+]P%y+QT*FW$\"3;LV+z9*z*fFW7$Fbfo$\"3#)*4>?@ (R1XFW7$$\"3W+]7`\\*[^*FW$\"3.^N!)e*[:0$FW7$Fi\\l$\"3:0pB_w+2#eGx$F\\u7$$\"3'3D\"y+9C,(*FW$!3Yw.p= t.wiF\\u7$Fgbv$!3K=)3;8i,H(F\\u7$$\"3UCcw47)>u*FW$!3]yYmG)Rp;(F\\u7$$ \"3p\\P4Y6cb(*FW$!3g)**[j[`7F\\qFdapFD-F46&F6QMThe~function~and~the~13~term~~approximatio nsF'F>F?Fh]l-FL6$7[o7$FP$!3C*pmS>cHI$Fhev7$F`^l$!35*3%*4m(pNGFhev7$Fe^ l$!3@1Y!z/$3ICFhev7$Fj^l$!3*H2)Q7\\ey?Fhev7$FU$!3Qm#y?(*=Xx\"Fhev7$Fg_ l$!3ipzd#=M7M\"Fhev7$Fen$!3jfhplCZ25Fhev7$Fcal$!3cr%eZ=%pP-VOFbjs7$F_o$!3WKc%4/*zVDFbjs7$Fgcl$!3Q 2&e=iH9w\"Fbjs7$Fdo$!3\"=`f>m1o?\"Fbjs7$Fio$!34sk@%)R'y!eFieq7$F^p$!3$ 3nLh?)o?4=*\\FW7$Fet$!3k\")pa$>8\"*>*FW7$Fjt$!3lO=2?v,P'*FW7 $F`u$!3!4_U!>y,1jFWF^_t7$F[v$\"3/!ywT-=cE'FW7$F`v$\"3o.Y)=9'Q!e*FW7$Fe v$\"3g#*e(p#QFd$*FW7$Fjv$\"3x!Gle\\p/7&FW7$F_w$!3^\"39;)ph-7FW7$Fdw$!3 <>:c*py\\'oFW7$Fiw$!3,[Hqy;O!))*FW7$F^x$!3`0[XLB8/#)FW7$Fcx$!3sgz/ZPh. '*F\\u7$Fhx$\"3mr]Z>K[h8F\\q7$F]y$\"3n6Rx&pUT[&F\\q7$Fby$\"3E9E:rgT/F?F[g\\l7'-FL6$7[y7$FP$ \"3!Qj1++)4MjFW7$Fe[x$\"3dp%fdE3$Q\\FW7$Faet$\"3gS=oD!oVl$FW7$F]\\x$\" 341uawzZvCFW7$Ffet$\"3Kqc\"3HY^R\"FW7$F[ft$!3#>4\")4KWL%\\F\\u7$F`^l$! 3W(>l11]-1#FW7$Fhft$!3fi>(eS'*\\Q%FW7$Fe^l$!3Ez[(FW7$$!3vm;zWizX'*FW$!3G]*pRSt`J(FW7$F]ht $!3(FW7$F`it$!3%eGI/qZ#3sFW7$F\\`l$!3 ]`%[\"oS6osFW7$F]jt$!3lfk>\"zhTP(FW7$Fen$!3o\"f\\#=UJAvFW7$Fcal$!3kf!3 %[esy%)FW7$Fjn$!3]b&3fV%)H<*FW7$$!3umm\"H-%\\/()FW$!3G*\\4#Q0O'>*FW7$F `bl$!3![#G7(4xp8*FW7$$!3wlmT&=Vrf)FW$!3Qb'eAlyv)*)FW7$Febl$!3!>P!ppk>V ()FW7$Fjbl$!3()\\=<2'))3'zFW7$F_o$!3ej*)=[e<%z'FW7$Fbcl$!30[cIy>,'G&FW 7$Fgcl$!3]L$=!)FW7$F_]u$\"3NbNp:'HsW*FW7$Ffel$\"3Z*yBik&ec5F\\q7$F \\^u$\"31Lo%H)=(Q8\"F\\q7$F^p$\"35%)G*zr\\W<\"F\\q7$Fcfl$\"3\\_Jr+Au!= \"F\\q7$F]gl$\"3&)**3rZ$\\q<\"F\\q7$Fggl$\"3F.L$Gi4N;\"F\\q7$Fahl$\"3A kkVa3NS6F\\q7$Ffhl$\"3$\\#R()fQ]m5F\\q7$Fcp$\"3&ysv'H&\\\"*e*FW7$F^il$ \"3ybbl\\2GD#)FW7$Fcil$\"3%Rp7*\\P(Hi'FW7$Fhil$\"3'GlIITc1%[FW7$Fhp$\" 3'z!\\elARSHFW7$F`jl$\"3_AOd^/#\\H*F\\u7$Fejl$!31/LyGq#42\"FW7$Fjjl$!3 %f#>YOc\"e*HFW7$F^q$!3jY&H9^C_y%FW7$Fb[m$!3M>h9tFk1iFW7$Fg[m$!3$*GuY&[ U/X(FW7$F\\\\m$!3EX1.CM#4\\)FW7$Fcq$!3qk^pF>@3$*FW7$Fgex$!3E'feGcDU%** FW7$Fd\\m$!3WomQ\"\\cp-\"F\\q7$$!30,]ilT')f^FW$!3ja!3m26:.\"F\\q7$F_fx $!3%e:,%=%*** FW7$F^`u$!3u8kJFJ$fT*FW7$F\\]m$!3S(yAJ<#=u&)FW7$Ff`u$!3$G%f1O;m+vFW7$F ]r$!3)ePu\\qdEB'FW7$Fd]m$!3P\\D&4P6b'[FW7$Fi]m$!3%z!='3/ahR$FW7$F^^m$! 3asem/b#e'=FW7$Fbr$!3.0cvf?'3;$F\\u7$Ff^m$\"3!=ahmkJP2\"FW7$F[_m$\"3)[ (*=ao QFW7$F]dm$\"3TFpJ.NG^EFW7$Fbdm$\"3#zE*>.IlP9FW7$F[t$\"3\"QCRvajA0#F\\u 7$Fjdm$!3=l4]z(>o8\"FW7$F_em$!3O.5H*3&zMCFW7$Fdem$!3A+L]]')zbOFW7$F`t$ !3(G*HPq*3Dwj)e(FW7$F[^[l$!3B# z^&\\(Hhv(FW7$F^gm$!3'Hl&[cH+tyFW7$Fc^[l$!36CN0^zqQzFW7$Fcgm$!3S@jQ)=7 I&zFW7$Fhgm$!3'zGUJq**f\"zFW7$F]hm$!3)zshE8p!GyFW7$Fbhm$!3+4utb*H**o(F W7$Fjt$!3o$o1!ypf-vFW7$F_im$!3)e;C6BamT'FW7$F`u$!3%pCbj%**[&y%FW7$F\\j m$!3)f`x7Vyts$FW7$Fajm$!346hrTsJxDFW7$Ffjm$!3[*GAK;ePO\"FW7$Feu$!3qV57 #**fi;\"F\\u7$F^[n$\"3&)f>*HE69=\"FW7$Fc[n$\"39&z'GQd2[CFW7$Fh[n$\"3G> !4_[#o\\OFW7$F[v$\"3%=#=N]Wpwm^%zFW7$F_a[l$\"35I^c\"4E2&zFW7$Ff^n$\"3@5&[4_ %=3zFW7$Fga[l$\"3MmKPU7jI<#pHT()FW7$ Fjdn$!3(e$z\"4'y(zm)FW7$Fed[l$!3))3))ok!H])f>T7w3#FW7$F[hn$\"3JW[(HqL!\\OFW7$F`hn$\"3V\"emRkf!R^FW7$F]y$\"3Cv EuRr[8lFW7$Fhhn$\"3)=*>nEJ5S')FW7$Fby$\"3ev-,)f\"zh**FW7$$\"31JL3xS'G/ &FW$\"3^CLb^N,:5F\\q7$Fi]y$\"3k:>@z#Ro-\"F\\q7$$\"31km\"HFb vsohFW7$F\\z$\"3=f&)zC1iJYFW7$Fjjn$\"3(f14?_6m%GFW7$F_[o$\"3#G\"yF!*=v K$*F\\u7$Fd[o$!3H5hAp8z\\5FW7$Faz$!3'Q3O=g:#RIFW7$F\\\\o$!3#e*4fouV**[ FW7$Fa\\o$!3\\)GvqcxUk'FW7$Ff\\o$!3q)pb,vI_@)FW7$Ffz$!3czQ!e(==d&*FW7$ F^]o$!3X7'\\#[<5k5F\\q7$Fc]o$!3SAO-7^')Q6F\\q7$F]^o$!3`t!)zs#>D;\"F\\q 7$Fg^o$!3%*\\Jc&*Qdw6F\\q7$Fa_o$!3bO;65nz!=\"F\\q7$F[[l$!3D8eyoW/v6F \\q7$Fj\\v$!3!=JjCF>S8\"F\\q7$F^`o$!3\"eKhI4-W0\"F\\q7$Fg]v$!3B3OPF*p] Q*FW7$F`[l$!3!33v)Qw8-zFW7$Ff`o$!3oJ9Gvut*H'FW7$F[ao$!3m[t$pQ(*3_%FW7$ F`ao$!31-b%fV>ci#FW7$Fe[l$!3kCily`8+oF\\u7$Fhao$\"3idh/hr)=_\"FW7$F]bo $\"3iHMk:5=$f$FW7$Fbbo$\"3A37^E_&[V&FW7$Fj[l$\"3F%zm^\"RwhpFW7$Fjbo$\" 3KzHN:2T3!)FW7$F_co$\"3)=v0fbhls)FW7$$\"3zK$ek8!)**e)FW$\"3()\\u9-&R0' *)FW7$Fdco$\"3GO\"\\gB\">8\"*FW7$$\"3K**\\PMP/*o)FW$\"3l^22&y6\")=*FW7 $F_\\l$\"39V$\\^ec0>*FW7$Fado$\"36PrCh1+y&)FW7$Fd\\l$\"31.8zw1'*4wFW7$ Fi`v$\"36(y24P\\NV(FW7$Fheo$\"3`gP$pzO=I(FW7$Faav$\"3NGe8\">C8A(FW7$F] fo$\"3UVe+X!oQ>(FW7$Fbfo$\"38\"ffS\\+JF(FW7$Fi\\l$\"3_\"QC)eeZ)R(FW7$$ \"3'4Dc,\">g#f*FW$\"3r7*4I]\"Q-uFW7$F_bv$\"3&Q9ZOx&>xtFW7$$\"3!)\\(oal @pk*FW$\"3b(*z^.mp6tFW7$Fjfo$\"3[6(FW7$Fgbv$\"30xG*4V*fLnFW7$F_ go$\"3QWS0Rdd`eFW7$Fdcv$\"3P#o<)pGAnVFW7$Fdgo$\"3%R>hr4+A/#FW7$Fadv$\" 3P)pBLs8#zZF\\u7$Ffdv$!3niW/WTG39FW7$Fegy$!3D#))4(3(Ri[#FW7$F[ev$!3UP3 ua#)=iOFW7$F]hy$!3C$*o5'HiD%\\FW7$F^]l$!3!Qj1++)4MjFWFdapFh]lFD-F46&F6 QMThe~function~and~the~15~term~~approximationsF'F>F?-FL6$7[o7$FP$\"3O, L$*G*y]Q$Fhev7$F`^l$\"3%)fvt?ZeQGFhev7$Fe^l$\"3W0)R6F[`P#Fhev7$Fj^l$\" 3w\\xATOY$)>Fhev7$FU$\"3PD&)3rHh_;Fhev7$Fg_l$\"3K1_Un7`'>\"Fhev7$Fen$ \"3dU)e1^$>-')Fbjs7$Fcal$\"3)pG?<3SJ)eFbjs7$Fjn$\"3)fjTB=LW)RFbjs7$Feb l$\"3_(y5)>\"4Qm#Fbjs7$F_o$\"30.m\"4e^:w\"Fbjs7$Fgcl$\"3c*3C(eg%R:\"Fb js7$Fdo$\"395CTz,8puFieq7$Fio$\"3(Qgij%zxAKFieq7$F^p$\"3ctM2250$G\"Fie q7$Fcp$\"34E'Qw`8'eYFj[q7$Fhp$\"3rFkp)e%e\"e\"Fj[q7$F^q$\"3^fZ0b)oyu%F <7$Fcq$\"3El)4&oDNh9F<7$Fhq$\"3>&=8Pre#yGF\\q7$F]r$\"3F->3&>d.y(F\\u7$ Fbr$!3M`q65hmYeF\\u7$Fgr$\"3#>EU)>Z\\**QFW7$F\\s$\"3?**4Dvp7?))FW7$Fas $\"3`(>DYx2X&**FW7$Ffs$\"3nA,g%yQ%zpFW7$F[t$\"39xp`su'Ha\"FW7$F`t$!3gF 6K+\\z\"*\\FW7$Fet$!3%peJ!))H6*>*FW7$Fjt$!3:wzn>v,P'*FW7$F`u$!3['\\U!> y,1jFWF^_t7$F[v$\"3%yvwT-=cE'FW7$F`v$\"3r3TdThQ!e*FW7$Fev$\"31\\jmyOFd $*FW7$Fjv$\"3-@GFojX?^FW7$F_w$!3m&GQ][**FW7$F^x$!3OQE%y3hun)FW7$Fcx$!3?7ZI)R9!)*QFW7$Fhx$\"3*e2by &y1doF\\u7$F]y$!3M4Ijmkv(H\"FW7$Fby$!3.fS+BtDyFF\\q7$Fgy$!3(yA\"RtX# \\P\"F<7$F\\z$!3g(f'*3\\xS)[F<7$Faz$!3U-ZiyDL/;Fj[q7$Ffz$!3))yqBST9FYF j[q7$F[[l$!3n'=kfe)fu7Fieq7$F`[l$!3**)3%HHcBwKFieq7$Fe[l$!3cJ$*Qm_cOuF ieq7$F]bo$!3oW^K@\\En6Fbjs7$Fj[l$!3JCxgF\\v3=Fbjs7$F_co$!3li)fU;l&[EFb js7$F_\\l$!3eIk0wpOUQFbjs7$Fado$!3Zv$=Fhev7$F_go$!3]lf+f\\0yBFhev7$Fdgo$!3n;Ok)[#=SGFhev7$F^]l$!3O,L$*G*y ]Q$FhevFb]l7'FD-F46&F6QMThe~function~and~the~16~term~~approximationsF' F>F?-FL6$7[yF`c]lFcc]lFfc]lFic]lF\\d]lF_d]lFbd]lFed]l7$Fe^l$!3o>@5!=r \"oeFW7$Fegt$!3?AEnuByVnFWF^e]l7$Fbe]l$!3')4F//MP:tFWFfe]l7$Fje]l$!3\" *>(4@buFS(FWF^f]l7$Fb_l$!3cveA4,\"*zsFWFdf]lFgf]lFjf]lF]g]lF`g]lFcg]lF fg]lFig]l7$F`bl$!3p959(4xp8*FWFah]lFfh]l7$Fjbl$!3)*fO:2'))3'zFWF\\i]l7 $Fbcl$!3+VZJy>,'G&FWFbi]lFei]lFhi]lF[j]lF^j]lFaj]lFdj]lFgj]lFjj]lF][^l 7$F^p$\"3&)QL*zr\\W<\"F\\qFc[^l7$F]gl$\"3[s1rZ$\\q<\"F\\qFi[^lF\\\\^lF _\\^lFb\\^lFe\\^lFh\\^lF[]^lF^]^lFa]^l7$Fejl$!3\\sQyGq#42\"FWFg]^lFj]^ lF]^^lF`^^lFc^^l7$Fcq$!3[!)[pF>@3$*FWFi^^lF\\_^lF__^l7$F_fx$!31q6S=p;1w&*4^)FWF`^_l7$F `dn$!3kSp^P*)oH()FWFf^_lFi^_lF\\__l7$Fed[l$!3kE))ok!H/* y&[*zi$)FWFe__l7$Fcx$!3#pBo1W6:&[FW7$Fdfn$!3?;P&)))[hVOFW7$Fifn$!3'*Q2 nBx.IBFW7$F^gn$!3;sFKz&zzQ*F\\u7$Fhx$\"3IF!4Y5cb*\\F\\uFg`_lFj`_lF]a_l F`a_lFca_lFfa_lFia_lF^b_l7$Fbb_l$\"3?34()ePZJ5F\\q7$F`in$\"3(3P&oDDyG5 F\\qFib_lF\\c_lF_c_lFbc_l7$Fbjn$\"3\\3DFbvsohFWFhc_lF[d_lF^d_lFad_lFdd _lFgd_l7$Fa\\o$!3ODk2nvFWmFWF]e_lF`e_lFce_l7$Fc]o$!3-&RB?6l)Q6F\\qFie_ lF\\f_lF_f_lFbf_lFef_lFhf_lF[g_lF^g_lFag_lFdg_lFgg_l7$Fe[l$!36uruy`8+o F\\uF]h_lF`h_lFch_lFfh_lFih_l7$F_co$\"3+iv)ebhls)FW7$F`i_l$\"3wRc;-&R0 '*)FWFdi_l7$Fhi_l$\"3whD0&y6\")=*FW7$F_\\l$\"3/Lv;&ec0>*FW7$Fado$\"3*o ^$Gh1+y&)FW7$Fd\\l$\"3FB\\vw1'*4wFWFej_lFhj_lF[[`l7$F]fo$\"3ABA/X!oQ>( FW7$Fbfo$\"3M6K-%\\+JF(FW7$Fi\\l$\"3Ih2')eeZ)R(FW7$Fh[`l$\"3]#HYI]\"Q- uFW7$F_bv$\"31k2htd>xtFWF_\\`l7$Fjfo$\"31rW^+[*G>(FWFg\\`l7$F_go$\"3&R !o7Rdd`eFWF]]`lF`]`lFc]`l7$Ffdv$!35.<(R9%G39FWFi]`lF\\^`lF_^`lFb^`lFda pFh]lFh^`l7'-FL6$7ex7$FP$!3%)\\lU+%f6,#FW7$Faet$!31kn,?g$fb\"FW7$Ffet$ !3u?ujn*e!38FW7$F[ft$!3wC_-D_'RB\"FW7$F`^l$!3Fv@C*e'*QI\"FW7$Fhft$!3kJ EqAx!Qx\"FW7$Fe^l$!3wIFK@y[VDFW7$Fj^l$!3I@bccey%[%FW7$FU$!3!Q\\?eN_CV' FW7$Fb_l$!3`[_8s!*=gyFW7$Fg_l$!3mQ8)4TI?%*)FW7$F\\`l$!37`J4R28j'*FW7$F en$!3R4rk\"R\"*[+\"F\\q7$Fd`l$!3**f9w&Q^M,\"F\\q7$Fi`l$!3-M?$[JAO,\"F \\q7$F^al$!3AJ')z5cB15F\\q7$Fcal$!3kpp\"o+$p?**FW7$Fhal$!3:9*R_TM7Y*FW 7$Fjn$!3//lw>dc.))FW7$F`bl$!3[)=7h_NQ(zFW7$Febl$!3gN_]o^E(*pFW7$Fjbl$! 3#>nYe;*4))eFW7$F_o$!3*RPq8j-xl%FW7$Fbcl$!31\\[dG[/DLFW7$Fgcl$!3\"*z/. !3r1!>FW7$F\\dl$!3$oP#f5+rkSF\\u7$Fdo$\"3y%3_%*o8\"H6FW7$Fddl$\"3Kuyv/ )Q&fDFW7$Fidl$\"3[x)>![s&Q'RFW7$F^el$\"3%z4h'o4W2`FW7$Fio$\"3\"y]?wy!G alFW7$F_]u$\"3x'=%p='z^q(FW7$Ffel$\"3Fmt(>FTin)FW7$F\\^u$\"3X+*o\"))3k L%*FW7$F^p$\"3P'fGqi)4\\**FW7$Fcfl$\"3/>E<43M65F\\q7$F]gl$\"3iNvD_]]?5 F\\q7$Fggl$\"3z6-p(e\\A-\"F\\q7$Fahl$\"3=rc3;S[;5F\\q7$Ffhl$\"3sJYeg2# Q#)*FW7$Fcp$\"3SUgE;%ew=*FW7$F^il$\"3ne&Gam&ev#)FW7$Fcil$\"3u>MlHwJ6rF W7$Fhil$\"37+^TS`yGdFW7$Fhp$\"3p&=giRG$pTFW7$F`jl$\"3WM&3C2&FW7$ F^^m$!3]geA([')e^$FW7$Fbr$!3s%eapF%ot=FW7$Ff^m$!3FnMv#z\")pY$F\\u7$F[_ m$\"3m[9[YUPy6FW7$F`_m$\"3k&)foq?EnEFW7$Fgr$\"3YTny10#p3%FW7$Fh_m$\"3a yd@$ph?k&FW7$F]`m$\"3-)H6tBs!3qFW7$Fb`m$\"3#)\\Jue?8W\")FW7$F\\s$\"32J [H()>9^C\">eOl)*FW7$$!3b70F#)*FW7$ Fas$\"3,L\"43z6kw*FW7$Ffbm$\"3.RGFx[vG$*FW7$F[cm$\"311HHfCk9')FW7$F`cm $\"3\\4m!peyxk(FW7$Ffs$\"3R#)[S,0/fkFW7$Fhcm$\"3V.F@S]H^_FW7$F]dm$\"3B @&\\i9#pFRFW7$Fbdm$\"3.w)=\"*=\\'>DFW7$F[t$\"3U$yC:&*Q-1\"FW7$Fjdm$!3A qU-v$=rf&F\\u7$F_em$!3^:YUe:jb@FW7$Fdem$!34Mk#p-#e$o$FW7$F`t$!3F\"[dsh K=5&FW7$F\\fm$!3kBw?&R%=:jFW7$Fafm$!3kv))FW7$F^gm$!3l)R>zo3PJ*FW7$Fcgm$!3-5z=?QT'\\*FW7$Fhgm$ !3/i1J?rK!\\*FW7$F]hm$!3>\")4LAZM>%*FW7$Fbhm$!3IWd.Oh0%G*FW7$Fjt$!3kZ. 7V(ya3*FW7$F_im$!3e4Kv3edSyFW7$F`u$!3t%3Oq9&G$)eFW7$F\\jm$!3wMa-Y\"fGf %FW7$Fajm$!3qIxMhx(3=$FW7$Ffjm$!3&338rhsZo\"FW7$Feu$!3'fzMv628W\"F\\u7 $F^[n$\"3YL$f%=rjf9FW7$Fc[n$\"3)yM)\\\"e#y@IFW7$Fh[n$\"3U`\\n@Ms(\\%FW 7$F[v$\"3+_swH)f_%eFW7$Fe\\n$\"3u5\">@U\"[sxFW7$F`v$\"38O#G3_#zF!*FW7$ Fb]n$\"3M6poMibN#*FW7$Fg]n$\"3'[#)oNpIWQ*FW7$F\\^n$\"3oO:1iIQt%*FW7$Fa ^n$\"3Te?`5Lv,&*FW7$$\"3&*[(o/$o6[5FW$\"3i?7A%4e\\bQFW7$Fe`n$\"3q&R@:OKHQ#FW7$Fj`n$\"3M;SwcQ].%)F\\u7$F_w$ !3Oj0]PzDAtF\\u7$Fban$!3t-hSBsfVAFW7$Fgan$!3T.(=OX6sq$FW7$F\\bn$!3ccpM'FW7$Fdbn$!3LKT.Dt-ovFW7$Fibn$!3[OL'3W7Qc)F W7$F^cn$!37_3^8`r,$*FW7$Fiw$!3@n))[+\"[mv*FW7$$\"3\"\\;HKuuv&HFW$!3o.# G-bT!=)*FW7$Ffcn$!31Z)\\rx[T')*FW7$$\"3I)\\(oz?83IFW$!3%))fj\\3l[*)*FW 7$F[dn$!32(=cLQ4,\"**FW7$F`dn$!3cMS&o%o(R*)*FW7$Fedn$!3%fl#Qq;e:)*FW7$ Fjdn$!3dR)*o.UEt%*FW7$F^x$!3,ZL$*)=h&*)))FW7$Fben$!3%Q\"RiV5:9!)FW7$Fg en$!3eQZ\\CUk+pFW7$F\\fn$!3VYt\"*3(=*ybFW7$Fcx$!3DBmtF!)G&3%FW7$Fdfn$! 3xQvUUJE=EFW7$Fifn$!3>yj>e'>'y5FW7$F^gn$\"3K)f]r$f0s\\F\\u7$Fhx$\"3T=t yh#R92#FW7$Ffgn$\"3BIP\"=P%fXPFW7$F[hn$\"3%=uY\"GJ(HK&FW7$F`hn$\"3(f \\6u0Clv'FW7$F]y$\"3[`vio'*y-!)FW7$Fhhn$\"3=2K&4rV*4(*FW7$Fby$\"3ilfNW )))H/\"F\\q7$Fja_l$\"3wj^@o%QT/\"F\\q7$Fi]y$\"3rQ=S8>#y.\"F\\q7$Fbb_l$ \"3^7!GZicS-\"F\\q7$F`in$\"34U`moa\"H+\"F\\q7$Fa^y$\"3o6tLwZt!R*FW7$Fg y$\"3Yf9PV7vz%)FW7$Fhin$\"3v7vobkr\\tFW7$F]jn$\"3&4Yq6p5R,'FW7$Fbjn$\" 3RCp$3XP$4XFW7$F\\z$\"38Ey-XkEyGFW7$Fjjn$\"3sUWWoIf&3\"FW7$F_[o$!3RE$4 uX_oQ(F\\u7$Fd[o$!3;tDQkG+PDFW7$Faz$!3C,Ag7RY_UFW7$F\\\\o$!3/#**Q$R@nv dFW7$Fa\\o$!3TGRL-_KFrFW7$Ff\\o$!3\\GLJ8\"f&o#)FW7$Ffz$!3\"[zg&=;Qn\"* FW7$F^]o$!3X2()4-1*3\")*FW7$Fc]o$!3wof/`'pf,\"F\\q7$F]^o$!3\\%*zaXF8A5 F\\q7$Fg^o$!3QLtnCmy?5F\\q7$Fa_o$!3RK2bT#=?,\"F\\q7$F[[l$!3_t!H&R=xf** FW7$Fj\\v$!3%zYs$HrBN%*FW7$F^`o$!3.?uRrUHc')FW7$Fg]v$!34[N/[**[`wFW7$F `[l$!3$)4N-2!fMY'FW7$Ff`o$!3A:$[cOn)R_FW7$F[ao$!3'=Zp.qzf#RFW7$F`ao$!3 vye`c%=_b#FW7$Fe[l$!3QB))=/jrf6FW7$Fhao$\"3=+6PEE-@VF\\u7$F]bo$\"3+9M6 rX\"*y>FW7$Fbbo$\"3JJ6Ke\\H\\MFW7$Fj[l$\"3E[FW7$Fjbo$\"3#oPFR_Y a%fFW7$F_co$\"3Br\\.BV#y'pFW7$Fdco$\"3p#>Siy0w(yFW7$F_\\l$\"3v]ydGw\"R m)FW7$F\\do$\"34%yq!*)zP[$*FW7$Fado$\"37uK$pN4b%)*FW7$Ffdo$\"3;W#zEdK6 +\"F\\q7$F[eo$\"3UXYq&>g8,\"F\\q7$F`eo$\"3JZ(*Gnl]95F\\q7$Fd\\l$\"3y^- nXGy45F\\q7$Fheo$\"3%RE#HR^'zv*FW7$F]fo$\"3V3*eeJz_/*FW7$Fbfo$\"3g)4hV e&oCzFW7$Fi\\l$\"3+<[0mTN4kFW7$Fjfo$\"3:_+$*Q'yaY%FW7$F_go$\"30!pS.a:C `#FW7$Fdcv$\"3(3&42/@nnF?FD-FL6$ 7[o7$FP$!31*pm]\"o&=o#Fhev7$F`^l$!3KY\"f%3q:(>#Fhev7$Fe^l$!3eETUaA&ez \"Fhev7$Fj^l$!3:H&e;=PVY\"Fhev7$FU$!3mwPlS95\">\"Fhev7$Fg_l$!3eMk!QX;` E)Fbjs7$Fen$!3yym7-_P!p&Fbjs7$Fcal$!339^s6N#Hq$Fbjs7$Fjn$!3?]@@>nY$Q#F bjs7$Febl$!3y&*odZ_47:Fbjs7$F_o$!3hm5\\*phkZ*Fieq7$Fgcl$!3M8F#>Rjm(eFi eq7$Fdo$!3s:V&fb>cf$Fieq7$Fio$!3Kz:3L#*e\"R\"Fieq7$F^p$!3V&[0M@&e9\\Fj [q7$Fcp$!3*)y&RP8k:c\"Fj[q7$Fhp$!3UbnIe)o6h%F<7$F^q$!3>nC%>P%)3C\"F<7$ Fcq$!3N$31:c,J<%F\\q7$Fhq$!3&GAcM0D1s\"F\\q7$F]r$!3CKl.))G#42*FW7$Fbr$ !3xYOh`T,9DFW7$Fgr$\"3lM$*f*o;7_$FW7$F\\s$\"3@e.^bHuw()FW7$Fas$\"3`1ee r'z!\\**FW7$Ffs$\"3Lc%GJ)[5zpFW7$F[t$\"3K1)\\'[%\\Ha\"FW7$F`t$!3!G\"o& 4E&z\"*\\FW7$Fet$!3aB](4*H6*>*FW7$Fjt$!3ID/o>v,P'*FWFed`lF^_tFhd`l7$F` v$\"3WzfdThQ!e*FW7$Fev$\"3:h+q!otsN*FW7$Fjv$\"3o%yVbpc/7&FW7$F_w$!3pWp ,'[()H?\"FW7$Fdw$!37U'=@jg-(oFW7$Fiw$!3!y\"=9S))=V**FW7$F^x$!3'y[1x4b' H')FW7$Fcx$!3U?jjoUa>NFW7$Fhx$\"3wz>K9>I:FFW7$F]y$\"3m;V)RF\\q7$F\\z$\"3kBG[z9$zF\"F<7$F az$\"3dQ!Q@x\\co%F<7$Ffz$\"3QgXU()oi\\:Fj[q7$F[[l$\"3xsg3)4*)z([Fj[q7$ F`[l$\"3y]@f(=!p<9Fieq7$Fe[l$\"33Z4/W1#zd$Fieq7$F]bo$\"3\\i^(G[IL&fFie q7$Fj[l$\"3+1vA.Q$Qw*Fieq7$F_co$\"3xbd@B1K-:Fbjs7$F_\\l$\"3ZE/r(GJwG#F bjs7$Fado$\"3kU9Ek_'Ga$Fbjs7$Fd\\l$\"3=5uCWY9GaFbjs7$F]fo$\"3tQTE4^t(3 )Fbjs7$Fi\\l$\"334fK70D%>\"Fhev7$Fjfo$\"3#Q))y2E0sY\"Fhev7$F_go$\"3k,^ 'e\")o\")z\"Fhev7$Fdgo$\"39!zoh;d&)>#Fhev7$F^]l$\"31*pm]\"o&=o#FhevFb] lFh]l7'FgfalFD-F46&F6QMThe~function~and~the~18~term~~approximationsF'F >F?Fd`dlFh]l7'FD-F46&F6QMThe~function~and~the~19~term~~approximationsF 'F>F?Fh]l-FL6$7]x7$FP$\"39l'>$**4d^XF\\u7$Ffet$!3?#HVt==93)F\\u7$F`^l$ !3qa%4yyh<%=FW7$Fhft$!3+H4_IDhCFFW7$Fe^l$!3BL__(=m7^$FW7$Fj^l$!33^X`(G Q]#\\FW7$FU$!3t(=)FW7$Fen$!3q()3hmm?z&*FW7$ Fd`l$!3f\")[_-+<@)*FW7$Fi`l$!3qR+/*eU,***FW7$$!33](oaDS90*FW$!3hF)32+N Y+\"F\\q7$F^al$!3#\\Ws!)oO$35F\\q7$$!37;HK*G=\")**)FW$!3\\DQO-H555F\\q 7$Fcal$!3>[/I[T#*45F\\q7$Fhal$!3'*o,s:/)z*)*FW7$Fjn$!3UA2I3m7%R*FW7$F` bl$!3'eW1ZH4Eg)FW7$Febl$!3W-[(=o#QdvFW7$Fjbl$!3CX#>Q%)\\rH'FW7$F_o$!3& )=7'HG*4l[FW7$Fbcl$!3Ku()R!z'\\9LFW7$Fgcl$!3Q%yT<%e![o\"FW7$F\\dl$!3NB *)QtHku@Fh\\p7$Fdo$\"35G6JRm%*H;FW7$Fddl$\"3?S&Hlh9N6$FW7$Fidl$\"3CGg# 45\\v^%FW7$F^el$\"3APhyRqO6eFW7$Fio$\"3*QdwF#)*ompFW7$Ffel$\"3Y\")z>y- 'G\"))FW7$F^p$\"3u;\\I>)R>y*FW7$F^fl$\"3_-%)*)Q?+P)*FW7$Fcfl$\"39mz:yh [v)*FW7$Fhfl$\"3OclcLgI(*)*FW7$F]gl$\"3sO2!\\%=S-**FW7$Fbgl$\"3w_B_51u !*)*FW7$Fggl$\"3(HLh#ondB%FW7$Fg[m$!3bdVVco/&o'FW7$Fi]m$!3!z/d&ee0]`FW7$F^^m$!3^0*pV8R-(QFW7$Fbr$!3h`=4WL?'G#FW 7$Ff^m$!3[T>Z\"4d'QzF\\u7$F[_m$\"3le>J(Rx)\\rF\\u7$F`_m$\"3()\\5Q-_S1A FW7$Fgr$\"3aRkIN!=pk$FW7$Fh_m$\"3?1$y`13\"\\_FW7$F]`m$\"3FaUMx%4Mo'FW7 $Fb`m$\"3UDe$*ey*[!zFW7$F\\s$\"3K<[;^ccv))FW7$Fj`m$\"3;tYoJ=9z%*FW7$F_ am$\"3W0vv^q\"z')*FW7$Fdam$\"3d>*=JFt*y**FW7$Fiam$\"3taBQEAO.5F\\q7$F^ bm$\"3^ilF!RiJ+\"F\\q7$Fas$\"3$\\@vpveI(**FW7$Ffbm$\"3@U`-l5u?'*FW7$F[ cm$\"33o!HV$zjw*)FW7$F`cm$\"3GC7DF&e61)FW7$Ffs$\"3A#eT_EjG!pFW7$Fhcm$ \"3qFqI=Cc.dFW7$F]dm$\"3jx%[24]3P%FW7$Fbdm$\"3S[>!QWim$HFW7$F[t$\"3hi* H<'pAN9FW7$Fjdm$!3:Y\"40o[-Z#F\\u7$F_em$!3=:1;k=F>>FW7$Fdem$!3IQZpxp*R `$FW7$F`t$!3\"[H'y4*oa/&FW7$F\\fm$!3g/'[%3Qq\\jFW7$Fafm$!3-'yfkN<$)[(F W7$Fffm$!3-A`mHEOK%)FW7$Fet$!3d!=F_![%z:*FW7$F^gm$!3gNBxGdXi'*FW7$Fcgm $!3]+i=I*[c*)*FW7$Fhgm$!37;u!=;'H3**FW7$F]hm$!3qzp?r5V^)*FW7$Fbhm$!37g Rgm%)\\D(*FW7$Fjt$!3aAgN,HUJ&*FW7$Fjhm$!3R@U*)\\1X-!*FW7$F_im$!3&[l&Hx LJh#)FW7$Fdim$!34;G$zvGcK(FW7$F`u$!3bxILlKa$FW7$Fh[n$\"3]EKQ@M))eZFW7$F[v$\"3G Y4)RY.w<'FW7$F`\\n$\"3#\\4DEpCbE(FW7$Fe\\n$\"3E!ox)>')y!>)FW7$Fj\\n$\" 3u]$**oNmD$*)FW7$F`v$\"3Rcucc>4u%*FW7$Fb]n$\"3JJ?p#fF(y'*FW7$Fg]n$\"3< >G[yj\"*>)*FW7$F\\^n$\"3MOgCJcp'*)*FW7$Fa^n$\"3?:4`e6_3**FW7$Ff^n$\"3) >c,E6[st*FW7$Fev$\"3cchA&[)>5$*FW7$F^_n$\"3k$eBnS%*=h)FW7$Fc_n$\"3vEYA ?c>rwFW7$Fh_n$\"3/*RO][dR^'FW7$Fjv$\"3Irt0529s^FW7$F``n$\"3%)zV7Ag`;PF W7$Fe`n$\"34(37i.E'e@FW7$Fj`n$\"3w94\"yZ)G+aF\\u7$F_w$!3'G4drlxe4\"FW7 $Fban$!3gFS[I)ePl#FW7$Fgan$!3p-h6>!3t9%FW7$F\\bn$!3)yfIR2c'QbFW7$Fdw$! 3CSJv>\"zBz'FW7$Fdbn$!3?1hfK?O%)zFW7$Fibn$!3WjMOq7UH*)FW7$F^cn$!3)oX&= '\\+tf*FW7$Fiw$!3:(>&oE-Hm**FW7$Ffcn$!3&fkOVR5J+\"F\\q7$F[dn$!3!z1I')[ XK+\"F\\q7$F`dn$!3BlXjv@Dq**FW7$Fedn$!3mNMn^'3[%)*FW7$Fjdn$!3:R4D]7Y2% *FW7$F^x$!3mJ$HO*>*3t)FW7$Fben$!3E`WQf#4Lw(FW7$Fgen$!3>%[%FW7$F\\z$\"3P9%)>K7[mEFW7$Fjjn$\" 3giL19$p4&**F\\u7$F_[o$!3#H#)>\"*3 q/!*fyixFW7$Ffz$!3myP*)\\f'Ql)FW7$F^]o$!3CM+apoeG$*FW7$Fc]o$!3$oml;d\" yZ(*FW7$Fh]o$!3QQsp#H'Q6)*FW7$F]^o$!3'\\d&3:NJe)*FW7$Fb^o$!3E:#44h6&)) )*FW7$Fg^o$!3G#R4ARc>!**FW7$F\\_o$!3=?U?H4l)*)*FW7$Fa_o$!3\">hLPKD'y)* FW7$Ff_o$!3;Q.mRk$>%)*FW7$F[[l$!33d8G#fn')y*FW7$F^`o$!3u`(R>SHkz)FW7$F `[l$!35fo@$y\"4%)oFW7$Ff`o$!3`G'R`Xyuu&FW7$F[ao$!3\")RyD0qO![%FW7$F`ao $!3S5u!Gl9\"4JFW7$Fe[l$!3%o?&yljCi;FW7$Fhao$\"3Lb=MPkF#)\\Fh\\p7$F]bo$ \"3j0rZ%=)HtfL2\"*4oCvFW7$Fdco$\"3m4ESI:&Q])FW7$F_\\l$\"3=.& Q4cI,F*FW7$F\\do$\"3T[a#z5(oB)*FW7$Fado$\"3!GF*Q8sM35F\\q7$Ffdo$\"3].# e%)[S*45F\\q7$F[eo$\"3$)e3s%38R+\"F\\q7$F`eo$\"3D&pUL,ZP!**FW7$Fd\\l$ \"3Gsq_ZcW&p*FW7$Fheo$\"3\\0F]N5J1\"*FW7$F]fo$\"3MB#)oJ%[sH)FW7$Fbfo$ \"3Mdp4l&fQJ(FW7$Fi\\l$\"3T'))**z?xZ?'FW7$Fjfo$\"3dkV!)oA@7\\FW7$F_go$ \"31\")p=kkq,NFW7$Fdcv$\"3M1O$Hl@nr#FW7$Fdgo$\"3t,V2p*Qd$=FW7$Ffdv$\"3 **oTfD\"4a/)F\\u7$F^]l$!39l'>$**4d^XF\\uFdap-FL6$7[o7$FP$\"3+,L$Rm3_p \"Fhev7$F`^l$\"3,h$*e&>\"=d8Fhev7$Fe^l$\"3qv\\)*p.v$3\"Fhev7$Fj^l$\"3s \"*Relu@J')Fbjs7$FU$\"3:&>\"\\&f8b&oFbjs7$Fg_l$\"3%4Dd&G0ChXFbjs7$Fen$ \"3I)=!y)FW7$Fas$ \"3YJGULsT\\**FW7$Ffs$\"3&o*4IS)>\"zpFW7$F[t$\"3')GQN@+&Ha\"FW7$F`t$!3 %*>Us`_z\"*\\FW7$Fet$!3#f\\T4*H6*>*FW7$Fjt$!348/o>v,P'*FWFed`lF^_tFhd` l7$F`v$\"3cqfdThQ!e*FW7$Fev$\"37\")yn!otsN*FW7$Fjv$\"3%3'[0*oc/7&FW7$F _w$!32r.Wbz)H?\"FW7$Fdw$!3p38-JXFqoFW7$Fiw$!3k[Y#FW7$F]y$\"3c6I)>cI Jz(FW7$Fby$\"3TeebSqJ^*)FW7$Fgy$\"3l36<5+)RO#FW7$F\\z$!3?@%4#y(*o\"H#F \\q7$Faz$!3wDLC;9*>;\"F<7$Ffz$!3_uqZ'z^nJ%F<7$F[[l$!3TK[0$yPt^\"Fj[q7$ F`[l$!3nQ-8A5&o$\\Fj[q7$Fe[l$!3Kf1Mm%=1Q\"Fieq7$F]bo$!3UGyKmc'GV#Fieq7 $Fj[l$!3+Lg$yhc-A%Fieq7$F_co$!3!*yc,0\"p*>oFieq7$F_\\l$!3$=')356d&*3\" Fbjs7$Fado$!3RZFS(3%*Rx\"Fbjs7$Fd\\l$!3M<'[^\"[EaGFbjs7$F]fo$!36@\\_aq 5_WFbjs7$Fi\\l$!3,FWF?F]dgl -%%VIEWG6$;$!#5F9Fg]l;$!2-++++++?\"F<$\"#;F9" 1 2 2 1 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "C urve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Cu rve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve \+ 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Cur ve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 4 4" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" " Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curv e 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63 " "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "C urve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Cu rve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve \+ 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" }}{TEXT 213 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 226 "If you step through \+ the frames, it is noteworthy that the Legandre approximation becomes r easonable at degree 7, and that the graph of the function we are appro ximating has 6 bumps, and thus looks like a 7th degree polynomial." }} {PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 82 "In our \+ example the Legandre approximation becomes very good when the degree i s 13." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 206 "Since we are looking a t approximations it is worthwhile to to also look at the error between the approximations and the original function. Once again we start wi th a procedure for producing a single frame." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "framererr := proc(n,x, lowy, highy)\n" }{MPLTEXT 1 0 20 "local pl1, pl2, B:\n" }{MPLTEXT 1 0 60 "pl1 := plot(func(x) - \+ Lapprox(n,x), x=-1..1, color=green):\n" }{MPLTEXT 1 0 59 "pl2 := plot( func(x) - Tapprox(n,x), x=-1..1, color=blue):\n" }{MPLTEXT 1 0 18 "B : = textplot(\{\n" }{MPLTEXT 1 0 63 " [-0.8,highy-(highy-lowy)/20, \+ `The `||n||` term error`],\n" }{MPLTEXT 1 0 76 " [-0.8,highy-(hig hy-lowy)/9, `Taylor (blue) vs Legandre (green)`]\}, \n" }{MPLTEXT 1 0 52 " align=RIGHT, font = [TIMES, BOLD, 12] ):\n" }{MPLTEXT 1 0 54 "display(\{pl1, pl2, B\}, view=[-1..1, lowy..highy]);\n" } {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 105 "We now \+ look at the animation of the error terms. Initially we set the the wi ndow to see errors under .1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "highy := .1: lowy := -.1:\n" }{MPLTEXT 1 0 65 "display([seq(fra mererr(counta,x, lowy, highy), counta=1..20)], \n" }{MPLTEXT 1 0 25 " \+ insequence = true);" }}{PARA 13 "" 1 "" {TEXT 213 0 "" } {GLPLOT2D 400 400 400 {PLOTDATA 2 "6&-%+AXESLABELSG6'Q\"x6\"Q!F'-%%FON TG6$%*HELVETICAG\"#5%+HORIZONTALGF.-%(ANIMATEG667&-%%TEXTG6&7$$!\")!\" \"$\"+yxxxx!#6QBTaylor~(blue)~vs~Legandre~(green)F'%+ALIGNRIGHTG-F*6%I &TIMESGF'I%BOLDGF'\"#7-F46&7$F7$\"1-++++++!*!#F ?-%'CURVESG6$7]t7$$F9\"\"!$\"3)f*****f))4$=$!#=7$$!3!RLL$e%G?y*FT$\"3j ,WT5B!Q2\"FT7$$!3ommm;p0k&*FT$!3?MNfYQ1(\\*!#>7$$!3E++vV5Su$*FT$!3?9*= yeIkd#FT7$$!3vKL$3,:X4N(FT7$$!3iL $eRZD>A)FT$!3(*puhgg!yK(FT7$$!3NnT5l^]o\")FT$!3[&)eDQKYzsFT7$$!3)**** \\i&[3:\")FT$!3[1!QSH#31sFT7$$!3Mm;aQUC3!)FT$!3^)oMz<&=&)pFT7$$!3\"QLL 3i.9!zFT$!30M%\\\\8(pnmFT7$$!3'3++Dw$H.xFT$!3%obpq/!=OeFT7$$!3\"ommT!R =0vFT$!3d*\\%ymib;ZFT7$$!3tKL$3i_+I(FT$!33zX9O]c(H$FT7$$!3u****\\P8# \\4(FT$!3\\#3auS@\"p;FT7$$!3#RLL3FuF)oFT$\"3A^/%H&)>9o\"Fin7$$!3*pmmT? F1n'FT$\"3**\\]/&[**f3#FT7$$!3tLL$e\\g\"fkFT$\"3;w>%3Cr#**RFT7$$!3[++] (y$pZiFT$\"33'>(p')*3c$eFT7$$!3MnmT&Gu,.'FT$\"3R7NVdy*[c(FT7$$!33LLL$y aE\"eFT$\"3e'=KpH]x0*FT7$$!3%)****\\([j5i&FT$\"3Kg=]+\"\\C,\"FI7$$!3hm mm\">s%HaFT$\"3o#3x.D>?4\"FI7$$!3]***\\7)*G;K&FT$\"3v^WT*3+Q7\"FI7$$!3 ]LL$3x&y8_FT$\"38)4Z*f(Hd9\"FI7$$!30,]ilT')f^FT$\"3*p@#3YX\"H:\"FI7$$! 3]nmTgD%f5&FT$\"3D%*G(pZYv:\"FI7$$!3%RL3_&4-_]FT$\"3Z#y*fg\"4'f6FI7$$! 3Q+++]$*4)*\\FT$\"3D9\"FI7$$!3 E++++t_\"y%FT$\"3%H)ecfp2J6FI7$$!3>+++v7CtYFT$\"3\\o/XNpq,6FI7$$!38+++ ]_&\\c%FT$\"3P9D7#))>C1\"FI7$$!33+++]zCcVFT$\"3d,#[fT^=g*FT7$$!30+++]1 aZTFT$\"3\"RktB:nEE)FT7$$!3RLL3FW,eRFT$\"33%[o\"=\"yI\"oFT7$$!3umm;/#) [oPFT$\"3PhxoGdx'=&FT7$$!3<++v$>BJa$FT$\"3?F@,$>w@4$FT7$$!3hLLL$=exJ$F T$\"3s)e*)Rq\")G4*Fin7$$!3!QLL$eW%o7$FT$!3&y+z!*)=X&Q*Fin7$$!3)RLLLtIf $HFT$!3ny9$**)y$Rs#FT7$$!3CnmT&o_Qr#FT$!3=!*o'yFH&[YFT7$$!3]++]PYx\" \\#FT$!37(z$QsZ^KjFT7$$!3QnmTNz>&H#FT$!3dz'4muvbc(FT7$$!3EMLLL7i)4#FT$ !3MQJC(Q(o9&)FT7$$!3g+]PMLy!*>FT$!3ebaepFO.*)FT7$$!3#pm;aVXH)=FT$!3iYs @l'3O>*FT7$$!3%**\\Pf[E!H=FT$!36vF\"Gla4I*FT7$$!3CL$ek`2^x\"FT$!39UJy( y$y#Q*FT7$$!3am\"zpe)=@IS&FT7$$!3KMLL3s$QM%Fin$!3?#=$4 $z$HUQFT7$$!3T,+]ivF@AFin$!3fI>+u0a2?FT7$$!3\\^omm;zr)*!#@$!3wP$edL\"e *)*)!#?7$$\"3@CL$3-Dg5#Fin$\"3PY(e>OP[!>FT7$$\"3%pJL$ezw5VFin$\"3V\"=* H6dt9QFT7$$\"3K`mmmJ+IiFin$\"3Wfs`h\\rT`FT7$$\"3s*)***\\PQ#\\\")Fin$\" 32ICD'[Czo'FT7$$\"3ilm\"z\\1A-\"FT$\"376_&z2Hg)yFT7$$\"3GKLLe\"*[H7FT$ \"3'pS!4N'Ql2&f.K*FT7$$\"3;**\\(oCor\\\"FT$\"3Mb&4uv4hR*FT7$$\"3aKLekSq] :FT$\"3oEQ#44JnW*FT7$$\"3#fm\"H#))RUg\"FT$\"3M?/I(G^?Z*FT7$$\"3H****** *pvxl\"FT$\"3xCAjCD'>Z*FT7$$\"37**\\7`+:5J.([K*FT7$$ \"3g)***\\7JFn=FT$\"3VuZS6%GuA*FT7$$\"3F)**\\(==-s>FT$\"3&=9dOd>5'*)FT 7$$\"3#z****\\_qn2#FT$\"3$4%Roz>B,')FT7$$\"3=)**\\P/q%zAFT$\"3(>y/EfIC l(FT7$$\"3U)***\\i&p@[#FT$\"3[`xd8Dz)R'FT7$$\"3L)**\\(=GB2FFT$\"36%3C4 Y%e-ZFT7$$\"3A)****\\2'HKHFT$\"3J#[WjOapv#FT7$$\"3uJL$3UDX8$FT$\"3;SEQ @;$R[FT$!3 O0$z\\%\\fU6FI7$$\"3M)*\\P%[L'*)[FT$!3BK1csoB]6FI7$$\"3=J3xc2&*R\\FT$! 3,H^>UCkb6FI7$$\"3ckm;H!o-*\\FT$!3*z!Qs+J!)e6FI7$$\"31JL3xS'G/&FT$!30* )*4HW.(f6FI7$$\"3c(****\\7ga4&FT$!3=ZjeZ!\\\"e6FI7$$\"31km\"H#3vFT7$$\"32****\\i!*3`iFT$!3!H;ACjS.z&FT7$$\"3Xmm\"z\\%[gkFT$!3:V 7sB1X()RFT7$$\"3%QLLL$*zym'FT$!3\"eQ#*\\L<56#FT7$$\"3ILL$3sr*zoFT$!3Fe Fin7$$\"3wKLL3N1#4(FT$\"3%H\"\\ZtnCX;FT7$$\"3c***\\(o!*R-tFT$\" 3\"e\"o7UW4:LFT7$$\"3Mmm;HYt7vFT$\"3ePc`\"[YSw%FT7$$\"3\"HL$ek6,1xFT$ \"3@/$>c^<'\\eFT7$$\"3Y*******p(G**yFT$\"3)z7Y4mc/m'FT7$$\"3ymmTgg/5!) FT$\"3!z7B%)*)>(*)pFT7$$\"3)HLL3U/37)FT$\"3$Qd!*pZH^@(FT7$$\"3jm;/,O=w \")FT$\"3%RN/ %pG)FT$\"3%p=5)Qvi_tFT7$$\"3]mmmT6KU$)FT$\"3mONZPz>WtFT7$$\"3w**\\iSH& =R)FT$\"3-3+v(f%e8tFT7$$\"3-LLeRZQT%)FT$\"30&[#zvc@hsFT7$$\"3Gm;aQl\"4 \\)FT$\"3bnMlB4<(=(FT7$$\"3a****\\P$[/a)FT$\"3c&eIAKx:4(FT7$$\"31mmTN> ^R')FT$\"3nnj*p(e[OoFT7$$\"3fKLLLbdQ()FT$\"3U<0y*p))y\\'FT7$$\"3`mm\"z W?)\\*)FT$\"3AUt1Xi=5bFT7$$\"3Z++]i`1h\"*FT$\"3ilO*RMSA>%FT7$$\"3X++++ PDj$*FT$\"3'z'yU%o#*pm#FT7$$\"3W++]P?Wl&*FT$\"3I@pdx3%HP*Fin7$$\"3A++v =5s#y*FT$!37r'o8<8/3\"FT7$$\"\"\"FQ$!3)f*****f))4$=$FT-%&COLORG6&%$RGB G$FQF9$F-F9Ff[n-FL6$7S7$FP$\"3G+++izxC%*FI7$Fen$\"3pk7Sgs]9')FI7$F`o$ \"3:(3JOwh9'zFI7$Fjo$\"3-@A3NXZLtFI7$Fhq$\"3X8yevjo\\oFI7$Ffs$\"3'G,w$ p#='GlFI7$F`t$\"3fH=5Oh\"HO'FI7$Fjt$\"3c)3AaKbSH'FI7$Fdu$\"3.%fomJ&=$G 'FI7$F^v$\"3mZ*=6I.IF'FI7$Fhv$\"3e)e;=b]!*>'FI7$Fbw$\"3[38no2NOgFI7$F` y$\"3)Hlih0'f5dFI7$F^[l$\"3Gu=P2=[>_FI7$Fh[l$\"3]$4^vO6Kg%FI7$Fb\\l$\" 3I39ZD$R/&RFI7$F\\]l$\"3br63:ZB7JFI7$Ff]l$\"3lz:lg@?,CFI7$F`^l$\"35V'f ;9vej\"FI7$Fj^l$\"3\"=Esx)Rjf5FI7$Fh`l$\"3e0,(4vZOr&FT7$Ffbl$\"3]a.3!* 3H!e#FT7$F`cl$\"3[/i))3*z8o(Fin7$Fjcl$\"3!4l'4Mp2M6Fin7$Fddl$\"3q_l9Ty IU8!#C7$F`el$!3iMUh#o8&36Fin7$Fjel$!3dCs7VU`JtFin7$Fdfl$!3A$>g]8lXU#FT 7$Fbhl$!3]Ajek*=Xi&FT7$F`jl$!3&*H1u]73J5FI7$Fjjl$!3#ptam`+0i\"FI7$Fd[m $!3i5B+B1f%R#FI7$F^\\m$!3j$)oI)*=/[JFI7$Fh\\m$!3\"e\"\\tqEk]RFI7$Fb]m$ !3S66P_VABYFI7$F\\^m$!3_zCt43H\\_FI7$Fj_m$!3BvFb]Z<.dFI7$Fham$!3ls7Rc) yc-'FI7$Fbbm$!3Ys)z&RA%=?'FI7$F\\cm$!3cw@*3FBzFI7$Fdjm$!3qX k&f%)4qh)FI7$F^[n$!3G+++izxC%*FI-Fc[n6&Fe[nFf[nFf[nFg[n7&FKF3-F46&FFQ1 The~2~term~errorF'F>F?Fh[n7&-FL6$7S7$FP$!3&)***z8nM5I\"!#:7$Fen$!3_s^' 4r(\\M6F\\fn7$F`o$!3DSn'\\Sz9+\"F\\fn7$Fjo$!3(*pd\\&pg,k)!#;7$Fhq$!3]* GH[i8jP(Fffn7$Ffs$!3TAdYpA4IiFffn7$F`t$!3Y'Hb,PqAE&Fffn7$Fjt$!3BC@Y\"4 bPN%Fffn7$Fdu$!3M@\">H;CK^$Fffn7$F^v$!3])*oEx9QvFFffn7$Fhv$!3lM=))*y;. 7#Fffn7$Fbw$!37NNMHFgH;Fffn7$F`y$!3A_vSc_0r6Fffn7$F^[l$!3zjP$e*zf`!)FI 7$Fh[l$!3f%p_A3C;N&FI7$Fb\\l$!3W6#>'**3'o^$FI7$F\\]l$!3%*pKn-pP$)>FI7$ Ff]l$!3!G,[tr7)H6FI7$F`^l$!3TY080f0G_FT7$Fj^l$!3'>hLY9O**H#FT7$Fh`l$!3 )*)QO\"yr1IvFin7$Ffbl$!3Y*)R]#Fidl7$Fdfl$\"3n'ok2Nijo\"Fin7$Fbhl$\"3$QXKO(4%HK(Fin7$F`jl $\"3)z)*zg\"Fffn7$Fbbm$ \"3SZExpN'Q8#Fffn7$F\\cm$\"3B_\"3LulTy#Fffn7$Ffcm$\"3.hN*FT7$$!3%pmm\"HU,\"*)*FT$\"3?aZHXz'y*yFT7$FV$\"3 /\"zs'f*>8Y'FT7$$!3t****\\(oUIn*FT$\"3Qe`UAU.d]FT7$Fen$\"3;KBcM!f`p$FT 7$$!3#HL3-)*G#p%*FT$\"3)e,rCrUIb#FT7$F[o$\"3MZj*Qp'*oX\"FT7$$!3^m;H2Jd z#*FT$\"3x)[MYT2%HTFin7$F`o$!3C_K\"zq'fJdFin7$Feo$!3-Mc!ookJb#FT7$Fjo$ !3\"Q'QSoiJjTFT7$Fdp$!3Y[laG7Ds`FT7$Fhq$!3Q,(p%)3IY9'FT7$F]r$!3@$p'G:w zniFT7$Fbr$!3ruu/&)HZjjFT7$Fgr$!3SO$3+0s!*)FT7$F[y$\"3!4vJ'RI23*)FT7$F`y$\"3Wc]y`)*z*)))FT7$Fdz$ \"3W=\\3>XRq&)FT7$F^[l$\"35^!)o7NMlyFT7$Fc[l$\"3Z(o6mgcR%oFT7$Fh[l$\"3 HVn/-IIBbFT7$F]\\l$\"3[-0/Mo00TFT7$Fb\\l$\"3ZU**R%zhK_#FT7$Fg\\l$\"3nV bpe`Zy\\Fin7$F\\]l$!3%3fdWW#>*f\"FT7$Fa]l$!3(\\\")Ged^?O$FT7$Ff]l$!3>k naJs\">0&FT7$F[^l$!3eQYR!RlH&oFT7$F`^l$!3a@S=,W,,%)FT7$Fe^l$!3SSnTY?A/ &*FT7$Fj^l$!3U++1#zE:.\"FI7$F__l$!3zpguyo\\i5FI7$Fd_l$!3I!zo8l4M3\"FI7 $Fi_l$!3v'yWLe6+4\"FI7$F^`l$!3]M2;$f7S4\"FI7$Fc`l$!3WBIVOuR&4\"FI7$Fh` l$!3OsC'**\\dT4\"FI7$F]al$!3o7&\\X?N04\"FI7$Fbal$!3(=C(e()\\`%3\"FI7$F gal$!3iz@B`i;w5FI7$F\\bl$!3scUHhQWl5FI7$Fabl$!3&Gq[$=Y-P5FI7$Ffbl$!3!o oPABP]***FT7$F[cl$!3?OubR'*\\L*)FT7$F`cl$!3exN=)G@%HvFT7$Fecl$!3W!3.k1 nN)fFT7$Fjcl$!3eeq\")ynCVUFT7$F_dl$!3qjAzu208AFT7$Fddl$!3Q#)pCU!fO!**F idl7$$\"3&yK$3_Nl.5Fin$\"3Q'[%*fm\"[05FT7$F[el$\"3C<'=:t+(*4#FT7$$\"3e ?Le*['R3KFin$\"3B#H^gx;C<$FT7$F`el$\"3AVX3<`l7UFT7$Feel$\"3Of?cb3&[\"f FT7$Fjel$\"3(zQ$z(z^TV(FT7$F_fl$\"3dApqK#eg\"))FT7$Fdfl$\"3b(R*oOG]\") )*FT7$F^gl$\"3nW&eg5m51\"FI7$Fbhl$\"3#*H'oH:nO4\"FI7$Fghl$\"39&*32EuN& 4\"FI7$F\\il$\"3GbtSG-d%4\"FI7$Fail$\"33^@7>@J\"4\"FI7$Ffil$\"3wN%Rwc& f&3\"FI7$F[jl$\"3#>0f6Nmo1\"FI7$F`jl$\"3=\">vb!*)fQ5FI7$Fejl$\"3I'yWtK B.e*FT7$Fjjl$\"3D%*fJ\\([6Y)FT7$F_[m$\"3#\\!4#=lUJ!pFT7$Fd[m$\"3sV*=J) o,$3&FT7$Fi[m$\"35Q*4Cce@H$FT7$F^\\m$\"3MW#Rhpl>U\"FT7$Fc\\m$!3$pd&f#* GCeeFin7$Fh\\m$!3-CEfS,CCDFT7$F]]m$!3l9MZ#oZY:%FT7$Fb]m$!39(pTph]\"4cF T7$Fg]m$!3c;%e#>x4UpFT7$F\\^m$!3vDH(oiJq&zFT7$Ff^m$!323Dqk+$ze)FT7$Fj_ m$!3PQs\"3p5^)))FT7$F_`m$!3]_)RI2%p1*)FT7$Fd`m$!3x&)=#*3=-0*)FT7$Fi`m$ !3EV/!))o'=!)))FT7$F^am$!3)\\,;ZhTB$))FT7$Fcam$!3\"HrW#p:`o')FT7$Fham$ !3&=\"fKX/+;%)FT7$F]bm$!3^@kM(=q$zwFT7$Fbbm$!3.&HW$>aj^mFT7$Fgbm$!3s8A $on*H9`FT7$F\\cm$!3;/bm94etPFT7$Facm$!3))H!)>Bzom@FT7$Ffcm$!36%f]([L#) >^Fin7$F[dm$\"3LSZ!R/!p^6FT7$F`dm$\"3:M6caT=2FFT7$Fedm$\"3Kvlh?cKnSFT7 $Fjdm$\"3kn&>)frlw^FT7$F_em$\"3#)3v?w'RI#fFT7$Fdem$\"3'*H&fZ[N!ojFT7$$ \"37L$3-)omazFT$\"3hr[.WcfNkFT7$Fiem$\"3>$H&*)4uPvkFT7$$\"3K**\\iS_Ul! )FT$\"3Ee&fQT.p['FT7$F^fm$\"3b#\\0&[qwpkFT7$Fcfm$\"36\"[d?`LOU'FT7$Fhf m$\"3*Qm_txS#[jFT7$F]gm$\"3oc!>G1+MC'FT7$Fbgm$\"3!*4#QO]'**3hFT7$Ffhm$ \"3r8A!\\(e<'Q&FT7$F`im$\"3U+dYQZC\"H%FT7$Feim$\"3WK4?5!4Tt#FT7$Fjim$ \"3=tU7H<\\(4)Fin7$$\"3Y++DJ&f@E*FT$!3C_O_9!=OF#Fin7$F_jm$!3a!QjXKc8L \"FT7$$\"3X++voyMk%*FT$!3YEU\"\\VSa\\#FT7$Fdjm$!3cwQ1iaN7PFT7$$\"3K+]7 G:3u'*FT$!3&f<.ayM-2&FT7$Fijm$!33#yW#))yNqkFT7$$\"35+]P40O\"*)*FT$!3\" z?(e!\\sC!zFT7$F^[n$!3`'*****)>.hN*FTFb[nF3-F46&FFQ1The~3~term~errorF' F>F?7&FfenF`_oF3-F46&FFQ1The~4~term~errorF'F>F?7&F3-F46&FFQ1The~5~term ~errorF'F>F?-FL6$7cs7$FP$\"39*****zh_)G')FT7$Fg_o$\"3B>3C]x.&G(FT7$FV$ \"3jSZq2)HU&fFT7$F_`o$\"3^lrCkZQZYFT7$Fen$\"3'y=L(HCAvLFT7$F[o$\"3K$f4 A(*)pu7FT7$F`o$!3B23$>ED\\Q'Fin7$Feo$!3!*R\"zvgi+^#FT7$Fjo$!3pA?LZJ$Q. %FT7$Fdp$!3d:4')[3\"e<&FT7$Fhq$!3#*f?r>A8**eFT7$F]r$!3G3m%H(Rj7gFT7$Fb r$!3-n8U`Vf*4'FT7$Fgr$!3c'FT7$F_co$!3L4@ e2)3C?'FT7$Fas$!33t3]o]q%='FT7$Fgco$!3,QcuB%G:9'FT7$Ffs$!3T#HC&\\2GtgF T7$F[t$!3n^[0@0k4cFT7$F`t$!3&zoMF.7*Q[FT7$Fet$!33S5Z>khgPFT7$Fjt$!3x(R /zja9X#FT7$F_u$!3nP5)36[V@*Fin7$Fdu$\"3A^`$3cG(GrFin7$Fiu$\"3;'z=3,TtO #FT7$F^v$\"32vT$*[\"Q)oRFT7$Fcv$\"3X`T8w\\k\"[&FT7$Fhv$\"3`*>ToQOMy'FT 7$F]w$\"37%o6..lGq(FT7$Fbw$\"3)3r9(=hwq$)FT7$Fgw$\"3Z*Gk,=a^i)FT7$F\\x $\"3%pc,LS;ry)FT7$Fax$\"3cGa=EbcK))FT7$Ffx$\"3t;>%3_!*R&))FT7$F[y$\"3l UhTe6A^))FT7$F`y$\"3OHR9aK:C))FT7$Fdz$\"3WvC-#p(*3Z)FT7$F^[l$\"3OnH6)3 !oMxFT7$Fc[l$\"3D4c$*3'[ho'FT7$Fh[l$\"3QwA\"z,m9M&FT7$F]\\l$\"3WZ#3_g# G/RFT7$Fb\\l$\"3$[oxQ\\CkI#FT7$Fg\\l$\"3'QViI8?xl#Fin7$F\\]l$!3wb%eJHG A%=FT7$Fa]l$!3r+GBlS+6OFT7$Ff]l$!3q6e*eq8PI&FT7$F[^l$!3R^Eg\"GmU5(FT7$ F`^l$!3nM?@gB%yk)FT7$Fe^l$!33sA!4#=!Ru*FT7$Fj^l$!3w(p9B.&\\a5FI7$F__l$ !3;gER#[,[3\"FI7$Fd_l$!3C%fA=vs\\5\"FI7$Fi_l$!3k?s'frv66\"FI7$F^`l$!3] lw.%\\fZ6\"FI7$Fc`l$!3i?4>l#4d6\"FI7$Fh`l$!3KP&z+&o,96FI7$Fbal$!3;t8$z 1'[.6FI7$F\\bl$!3)z#zH<$GM3\"FI7$Fabl$!3qP'\\mP()R0\"FI7$Ffbl$!3oKiP.J R:5FI7$F[cl$!3i]TshV[o!*FT7$F`cl$!3l:c!>[I(QwFT7$Fecl$!34!4%HD#Q!ogFT7 $Fjcl$!3Ok4FM%e>I%FT7$F_dl$!3uH:#*)GqKC#FT7$Fddl$!3wE_:np#Q+\"Fin7$Fe] p$\"3Z(4]6+h\">5FT7$F[el$\"3.(GhJ'*e$G@FT7$F]^p$\"3RqNlOj&f@$FT7$F`el$ \"3!4#eHIy#4F%FT7$Feel$\"3%H)y]@&p#)*fFT7$Fjel$\"3Wm![K8^=a(FT7$F_fl$ \"3,V'>h%Gr[*)FT7$Fdfl$\"3]&*4>Z*[P+\"FI7$F^gl$\"324gA%ys)y5FI7$Fbhl$ \"3cj#zC1XM6\"FI7$Fghl$\"3)>hmR-yb6\"FI7$F\\il$\"3**)e\"4wr@:6FI7$Fail $\"3jZ,My'oB6\"FI7$Ffil$\"3'eg#f4Y/26FI7$F[jl$\"3DM#3sXZ!*3\"FI7$F`jl$ \"3%=hvP6R91\"FI7$Fejl$\"3fkG&y01$>)*FT7$Fjjl$\"3CEX?gep2()FT7$F_[m$\" 3zS!*zAqOarFT7$Fd[m$\"3/H8k]z$[L&FT7$Fi[m$\"36n]]WD$4a$FT7$F^\\m$\"3\" =76S#QCk;FT7$Fc\\m$!3F]?8QP3VNFin7$Fh\\m$!3%y?9l]6uI#FT7$F]]m$!3BB%e!3 AXaRFT7$Fb]m$!3'fDx*)\\Z'GaFT7$Fg]m$!3u_VTF*eky'FT7$F\\^m$!3y(y@.o2&Hy FT7$Ff^m$!38vh&3se&*[)FT7$Fj_m$!3a%)3#4<)>=))FT7$F_`m$!3X&fkF@R$[))FT7 $Fd`m$!3J*yga'3Nb))FT7$Fi`m$!3G<-xE'=$R))FT7$F^am$!3QO7vh0rl?%)FT7$F]bm$!3#fsjT'Q*3s(FT7$Fbbm$!3b?u \"*4k]InFT7$Fgbm$!31BwcW-3KaFT7$F\\cm$!3@CSuIFEHRFT7$Facm$!3\\;6h^j1dB FT7$Ffcm$!3I\")4o>UvVtFin7$F[dm$\"3tn+pxM`.!*Fin7$F`dm$\"3V1cPl[(=V#FT 7$Fedm$\"3Qln&***)oVx$FT7$Fjdm$\"3s&)**eR!fM([FT7$F_em$\"32&H#p\\!f\"= cFT7$Fdem$\"3p.Rh?w1qgFT7$Fifp$\"3?FKD;VOThFT7$Fiem$\"3o.)GBQ>d='FT7$F agp$\"3!z-:H:'o-iFT7$F^fm$\"3%o\\:4C*)=>'FT7$Fcfm$\"3,)>IigDI:'FT7$Fhf m$\"3sN9IGa'e3'FT7$F]gm$\"3Ma;n!y_-*fFT7$Fbgm$\"39?es0h5meFT7$Ffhm$\"3 C)QErTE*)=&FT7$F`im$\"3#\\))R?^_[:%FT7$Feim$\"3s+)H)ovF\"o#FT7$Fjim$\" 3w%zLF&>#)=')Fin7$Faip$!3Q#)[s<`dn6Fin7$F_jm$!3')\\v./Hfc6FT7$Fiip$!3W 1m8%\\h0D#FT7$Fdjm$!3!f^XXWH6R$FT7$Fajp$!3.orBRXpfYFT7$Fijm$!30mKa,Cii fFT7$Fijp$!3u>>U)p#H*G(FT7$F^[n$!39*****zh_)G')FTFb[n-FL6$7W7$FP$\"3U+ +i\\R'e*[F\\fn7$FV$\"3V'fC4$)oWL%F\\fn7$Fen$\"3%\\#Gc8+SCQF\\fn7$F[o$ \"39&yo$eFn>MF\\fn7$F`o$\"3\"p'p&*GG.\\IF\\fn7$Fjo$\"3M?xC?THHBF\\fn7$ Fhq$\"3[CHeXh%fu\"F\\fn7$Ffs$\"34&[$R))f]&G\"F\\fn7$F`t$\"3=??9:A:%\\* Fffn7$Fjt$\"3/8Na&oSpy'Fffn7$Fdu$\"3??&\\qjp:n%Fffn7$F^v$\"3(*eRNwhaBJ Fffn7$Fhv$\"3yE#[Q#ei\"*>Fffn7$Fbw$\"3;r\"3z(oH%H\"Fffn7$F`y$\"3S!=Vb# p(zh(FI7$F^[l$\"3g3Uf*o))3B%FI7$Fh[l$\"3et+DgP(QD#FI7$Fb\\l$\"3i-`\\)* G-$>\"FI7$F\\]l$\"3Jmq)HJDw2&FT7$Ff]l$\"3w)4%4:>f>AFT7$F`^l$\"3qey&4t; qC(Fin7$Fj^l$\"35hX!)*RCkA#Fin7$Fh`l$\"34m$o+&e@OXFidl7$Ffbl$\"3;MY&Ffdl7$Fbhl$!3x@9#[\"FI7$Fb]m$!3Mng3B\"f')H#FI7$F\\^m$!3`j(\\sPc0P%FI7$Fj_m$!3)fRQP &eofm7Fffn7$Fbbm$!3E4*)[wbr7?Fffn7$F\\cm$!3>*QupSw -9$Fffn7$Ffcm$!3.Sa(y1W)fYFffn7$F`dm$!3:GR;Z\">0x'Fffn7$Fjdm$!346)3yEB 4b*Fffn7$Fdem$!3<4?>EF\\$G\"F\\fn7$Fbgm$!3L.n\\F?F3F[\\qFcer7&-FL6$7it7$FP$!3*)4++]1dI6FI7 $$!3.MLe9r]X**FT$!3Yr@1%GIr/*FT7$Fg_o$!3?-GJIqX+qFT7$$!3')***\\PM@l$)* FT$!32J&pCQ\"fa^FT7$FV$!3+yRPa%Q()\\$FT7$$!3#om;HdNvs*FT$!3OYeMp:SA?FT 7$F_`o$!3?g([xS+Q:(Fin7$$!3wLL3-)\\&='*FT$\"3'=)>(p:IBK%Fin7$Fen$\"3WV \"fz%G0I9FT7$Fg`o$\"39@>>3-(o$GFT7$F[o$\"3fUI'eEW_'QFT7$F_ao$\"3\"f'HM 5-lfXFT7$F`o$\"3h\"R)oWz*>'\\FT7$$!3x:/w(=%3e\"*FT$\"37$pk*4Q(z-&FT7$$ !3!))\\(o/KUJ\"*FT$\"3\"Qiy#z6$[2&FT7$$!3#=e9;AiZ5*FT$\"3B>d+tSR.^FT7$ $!3&fmT&Q75y!*FT$\"38QD?x5Z9^FT7$$!35LeRs#zZ-*FT$\"3!R2*\\2]K(3&FT7$Fe o$\"3Mr**oPcb**\\FT7$$!3YL$eRP8['))FT$\"3+soLs!)elYFT7$Fjo$\"3))=!es=l q:%FT7$Fdp$\"3&G-IoQM`w#FT7$Fhq$\"3'*zG\"f0[_6\"FT7$F\\s$!3fH!oLj\"*za &Fin7$Ffs$!3w*z%)*y!zv2#FT7$F[t$!38v*3CahsC$FT7$F`t$!3KnsSN\"o_6%FT7$$ !3K++]i#=ES(FT$!3Al(FT$!3,L+G?U?(y%FT7$$!3bLLLeT?YrFT$!3t7W4 jsF<[FT7$Fjt$!3W(4T\")*fMC[FT7$$!3ILL$3d%)=/(FT$!3j$z&\\`h$y![FT7$$!3$ ommT!y%)))pFT$!3Ij+:)o%enZFT7$$!3Q++]P5\"e$pFT$!3r_dZ;P;/ZFT7$F_u$!3Q4 #fEPG#=YFT7$$!3,,+]P2qwnFT$!3YeCIm$==Q%FT7$Fdu$!3$o&Rq,Y0lSFT7$Fiu$!3' >7\">f09DKFT7$F^v$!3Qv=8x$[p;#FT7$Fcv$!3ARdwu%=[K*Fin7$Fhv$\"3Q=U)4\\$ 4JOFin7$F]w$\"3^E>9*Gtj[\"FT7$Fbw$\"3Cy=D-zONDFT7$F`y$\"3\"*R`UacN$Q%F T7$Fjy$\"3?88OjF9)p%FT7$Fdz$\"3?'y#>uo[V\\FT7$Fiz$\"3i2\\6MpO;^FT7$F^[ l$\"3KDD0VM#[@&FT7$$!3E+++D%yF^%FT$\"3_BHY*)*)QN_FT7$$!3R++++;ggWFT$\" 3F15!y%*Q%Q_FT7$$!3_+++vZU3WFT$\"3a]./]o,C_FT7$Fc[l$\"3IEp[z&FT7$$ !3z*******H%*=D%FT$\"3+yy&=m\"=x]FT7$Fh[l$\"3Kl8/Jq=&*[FT7$F]\\l$\"3P5 &zEW'o-WFT7$Fb\\l$\"3@k4#=b]6s$FT7$F\\]l$\"3zQ^g4^0F:FT7$Fa]l$\"3Sy\\A ?*)f0XFin7$Ff]l$!3qKSB!3&p+lFin7$F[^l$!3(p)*)p$ev7!>FT7$F`^l$!3H^@YfDO cIFT7$Fe^l$!3IW!)[@()[[RFT7$Fj^l$!3*[gz[ao!zYFT7$Fd_l$!3@n@-#[q*e_FT7$ Fh`l$!3p(ptW3L$zbFT7$F]al$!3=q!y*HcM:cFT7$Fbal$!3q!QZ]a.cj&FT7$Fgal$!3 kJ2B\"y`+k&FT7$F\\bl$!3#\\*RE_yoGcFT7$Fabl$!3:tY8aepebFT7$Ffbl$!3Ohj1Y aWEaFT7$F[cl$!3'4Y`Yr*Rd\\FT7$F`cl$!3$[LkU]5xC%FT7$Fecl$!3&yjGEPSNT$FT 7$Fjcl$!3Nwbg*4#yRCFT7$F_dl$!3(Q5,g%y$*y7FT7$Fddl$!3ntw'GAINt&Fidl7$F[ el$\"3CBT69-l87FT7$F`el$\"3wyw/kyWACFT7$Feel$\"3&y&*G!*>_cP$FT7$Fjel$ \"3a0*fqmJv>%FT7$F_fl$\"3!Gpj+f53!\\FT7$Fdfl$\"3>+Y2WPX\"Q&FT7$Fifl$\" 3`R)emaP[`&FT7$F^gl$\"3$*R.lLq$3i&FT7$Fcgl$\"3oznX.f>QcFT7$Fhgl$\"35u \"GTdm$QcFT7$F]hl$\"3Wb.fTRM@cFT7$Fbhl$\"3n?t@nb<(e&FT7$Ffil$\"3YHq$fI h7H&FT7$F`jl$\"3/d160n!*[ZFT7$Fejl$\"38u6xyrL8SFT7$Fjjl$\"3pxsY!QnJ5$F T7$F_[m$\"3f`/4*zBu$>FT7$Fd[m$\"3?fv-Wyj4nFin7$Fi[m$!3phr)y\"HPY\\Fin7 $F^\\m$!3O>Y&o)f8J;FT7$Fc\\m$!3'RO(fx\\2`FFT7$Fh\\m$!3KJp[,eg@PFT7$F]] m$!3w/m)o^p?U%FT7$Fb]m$!3Q6)))40\">?\\FT7$$\"3B**\\P40(oE%FT$!3%G_7%o$ fy4&FT7$Fg]m$!3L))y:$*Q'[?&FT7$$\"3W*\\(oz,IFWFT$!3zU<7dhCJ_FT7$$\"3%G Lektw2[%FT$!3oY+5[]LR_FT7$$\"3Em\"HKH`U`%FT$!3'yrX9=Y!H_FT7$F\\^m$!3&) >fF!yc.?&FT7$Ff^m$!3'*>8A+r&)G\\FT7$Fj_m$!3RVWYU!p$3WFT7$F^am$!3q!fcV- gHi$FT7$Fham$!3Em/'RiC0j#FT7$F]bm$!3C6AwFS96:FT7$Fbbm$!3sl@\"4'f')[IFi n7$Fgbm$\"3EX%fqrB-x*Fin7$F\\cm$\"3#*Q<^mc.'>#FT7$Facm$\"3m,gDDi:JKFT7 $Ffcm$\"3kI*G*[+&e0%FT7$$\"3dLL3Fe#Rx'FT$\"3D)RV`wBXP%FT7$F[dm$\"3=7g^ SZ28YFT7$$\"3;L$3xm%*H$pFT$\"3'R>PTDg,q%FT7$$\"3-LLe9w,')pFT$\"3%o?LM6 !ykZFT7$$\"3!HLe9cS!RqFT$\"3UqZgGaF1[FT7$F`dm$\"3%GcQ.'*pS#[FT7$$\"3=* \\P%)RZY9(FT$\"3i9T@G)Gy\"[FT7$$\"3gl;a)GJs>(FT$\"3bEni-$z3GDt%FT7$Fedm$\"3\"3,r$[x)Hl%FT7$$\"3]L$e*[oc2uF T$\"38TJw,sq>WFT7$Fjdm$\"32rfbUBC)3%FT7$F_em$\"3/1(y#=99LKFT7$Fdem$\"3 o,HeZ7Y\"4#FT7$F^fm$\"3Ev%G(\\aZ8^Fin7$Fbgm$!3\"=9lb&=QA7FT7$Ffhm$!3'= r'epUJVFFT7$F`im$!3/gz#otNz/%FT7$$\"36+]i!*z>W))FT$!3WGls6*z&zXFT7$Fei m$!3*=9u,J7![\\FT7$$\"3&GeRA1Ei(*)FT$!3j'3G>M=(4]FT7$$\"3I+Dcw;j-!*FT$ !39snL7c)y0&FT7$$\"3i;a)3HP!H!*FT$!3SNGfY**z\"4&FT7$$\"3&HL3_!HWb!*FT$ !3\"*GVn]Et5^FT7$$\"3rmT&Q8a#3\"*FT$!3.P&ymPl15&FT7$Fjim$!3!**oWm@h:-& FT7$Faip$!3=\\8\\cyAaYFT7$F_jm$!3hyxQ'4jL'RFT7$Fiip$!3=\")==x7l)*GFT7$ Fdjm$!3Z`4s9'okS\"FT7$$\"3Q+D\"Gyh(>'*FT$!3Zi91`os$3%Fin7$Fajp$\"3Z&3` K9`xQ(Fin7$$\"3G+vVt7SG(*FT$\"3GO!RR@0X/#FT7$Fijm$\"3K<9aJ[j=NFT7$$\"3 ;+D1k2/P)*FT$\"3E!Gf_5s7<&FT7$Fijp$\"3VV!))pCw<%F\\fn7$Fjo$!3E\\wr;c2^GF\\fn7$Fhq$!33iz+H\\m(* =F\\fn7$Ffs$!3K\\\"[;'RZM7F\\fn7$F`t$!3\")eN!fP^\\3)Fffn7$Fjt$!3vYjK1( *[t]Fffn7$Fdu$!3U&R-$z=(4.$Fffn7$F^v$!3Y0-$Fffn7$F` dm$\"3#3Ik!>+^c]Fffn7$Fjdm$\"3LnV^9]N_\")Fffn7$Fdem$\"3YM2EIswJ7F\\fn7 $Fbgm$\"3>y))pmi\"H#>F\\fn7$F`im$\"3Yk!)R*)*H+!GF\\fn7$Feim$\"3))=mw1Y %QR$F\\fn7$Fjim$\"3]4FlQ0>#4%F\\fn7$F_jm$\"3i#e7tcq?([F\\fn7$Fdjm$\"3B U^!)=#>ax&F\\fn7$Fijm$\"3%ee'3,ts,pF\\fn7$F^[n$\"3c++Q?805#)F\\fnF_enF 3-F46&FFQ1The~7~term~errorF'F>F?7&F[`s-F46&FFQ1The~8~term~errorF'F>F?F 3F]cu7&-FL6$7gn7$FP$\"3G++ifNsezF\\fn7$Fg_o$\"3l&f%Gebf?rF\\fn7$FV$\"3 &)zJ#=P`>O'F\\fn7$F_`o$\"3f,U$ouGhn&F\\fn7$Fen$\"3D:9Yqh$p0&F\\fn7$F[o $\"3!H5])*z'[@TF\\fn7$F`o$\"3WO'pimwMM$F\\fn7$Feo$\"3-`D7E*Rsi#F\\fn7$ Fjo$\"3#G`?R5%>^?F\\fn7$Fhq$\"3QSe,j)G0A\"F\\fn7$Ffs$\"3(fHlEj/[1(Fffn 7$F`t$\"3Wg%\\3Z_58%Fffn7$Fjt$\"3;J(>\"*fj?H#Fffn7$Fdu$\"3I`'\\;mQu>\" Fffn7$F^v$\"3T)fVB#Qt))fFI7$Fhv$\"3NQx,a'[7y#FI7$Fbw$\"3>Y/Gk!oLM\"FI7 $F`y$\"3]@SFQ%4P`&FT7$F^[l$\"3?+D#f)4A(3#FT7$Fh[l$\"3uv721&GHT(Fin7$Fb \\l$\"37u\"G.'RiDEFin7$F\\]l$\"3w#ewZ:*G\"e'Fidl7$Ff]l$\"3QR'[*=TnPE9r\"!#E7$Fjcl$\"3POLS)>K1< &!#G7$Fddl$\"3AmdyLX![@\"Fgjr7$F`el$!3Q.xz0Z&f7&Fgcv7$Fjel$!3;[Y<1*G+X \"Fccv7$Fdfl$!3[)fJ46JrD\"Ff`n7$Fbhl$!3+a\"GHk$zSLF\\cv7$F`jl$!3;SA=Hn 5\\RFcjn7$Fjjl$!3moYjYy&)yFFfdl7$Fd[m$!34bz\\B)fVr\"Fidl7$F^\\m$!3]`CX sC)H+(Fidl7$Fh\\m$!3r&=^KXuki#Fin7$Fb]m$!3H(y:jg^cl(Fin7$F\\^m$!3A1B)3 T+A?#FT7$Fj_m$!3i5Yp[eLTaFT7$Fham$!3Q^f.p=R&H\"FI7$Fbbm$!3wIvo'*fQJGFI 7$F\\cm$!3g$3I?_!zVgFI7$Ffcm$!3%ewF*4UC#>\"Fffn7$F`dm$!3#e2)p6ZQ#G#Fff n7$Fjdm$!3LtY\"z<1Z<%Fffn7$Fdem$!3@0:=*zU^/(Fffn7$Fbgm$!3'zth%z'\\7C\" F\\fn7$F`im$!3LQ(3hc6W+#F\\fn7$Feim$!3K*o9*f$fGc#F\\fn7$Fjim$!3!*=S%pN yhD$F\\fn7$F_jm$!3`*fhE@];2%F\\fn7$Fdjm$!3Zsf6\\=Tk]F\\fn7$Fajp$!3?EuS d>M#o&F\\fn7$Fijm$!3W*ezFT7$$!3&GLekynP')*FT$\"3jokW#py;X\"FT7$Fj`s$\"3gaO8DvR C)*Fin7$$!3)omT5!\\F4)*FT$\"3XO>8'Qbkd&Fin7$FV$\"3$\\XgC&H#*[E\"yo3)*Fin7$Fjas$!3_'>h^.sHO\"FT7$Fen$ !3sbTI^-6N;FT7$$!3D*\\P%[Hk;&*FT$!3#3&)pAMMLz\"FT7$Fg`o$!3YT?KUcV()=FT 7$$!3gm\"z>,:=U*FT$!33_M7G?ND>FT7$F[o$!3#f%p9#y_W\">FT7$F_ao$!36)Q\"*e p&*Hx\"FT7$F`o$!3e/rT'))4;^\"FT7$Feo$!3u*4t+q*))ooFin7$Fjo$\"3)oaf-sDb *>Fin7$Fdp$\"37eiJO*HLR*Fin7$Fhq$\"3;sL=1N)eT\"FT7$F]r$\"3Yy#))e[*Q)[ \"FT7$Fbr$\"3ud#)og.RU:FT7$Fgr$\"3)p>PNu_\"y:FT7$F\\s$\"3#GfjB_0hf\"FT 7$F_co$\"352j$QR@of\"FT7$Fas$\"3bo-bb$**4e\"FT7$Fgco$\"35\\$)*\\8[%\\: FT7$Ffs$\"3sZ#)eL02.:FT7$F[t$\"3d*etf$\\j=7FT7$F`t$\"3!)y$fS4:o+)Fin7$ Fet$\"3xVp\"eKWU\"HFin7$Fjt$!3k9alF^O?BFin7$F_u$!3h\"ROGA>LG(Fin7$Fdu$ !3wiFMIk*)H6FT7$Fiu$!3Bb7FDY(=S\"FT7$F^v$!3%)\\HiKXiF:FT7$$!3-o\"z>\"R J$>'FT$!39rB#pgMc`\"FT7$$!3YM$ek.M*QhFT$!3a^/kDA!Q`\"FT7$$!3*3]P4;aX3' FT$!3hjgxqFGA:FT7$Fcv$!35s\\:,_G,:FT7$$!3?+]PMXT@fFT$!3Ve%>J;U>V\"FT7$ Fhv$!3=!yd+W/%G8FT7$F]w$!3%p<@Ug/K2\"FT7$Fbw$!3D,UvLH%GW(Fin7$F`y$\"3B ;YMQNk@8Fin7$F^[l$\"3*RV*zN\"RQa*Fin7$Fc[l$\"3j\\Tk&=&Q^7FT7$Fh[l$\"3H !)GGlxza9FT7$$!3Xm;aQvx_SFT$\"3Zp\")fjrk7:FT7$F]\\l$\"3[PYUuT#za\"FT7$ $!3M+]i:8DjQFT$\"3n#p_2>L.c\"FT7$Fb\\l$\"3-t,%3`2*\\:FT7$Fg\\l$\"3:hje !p&3O9FT7$F\\]l$\"3foGWG2L27FT7$Ff]l$\"3^#RnDJ$pphFin7$F`^l$!3i,owE^'H F#Fin7$Fj^l$!3w[[D`n-h#*Fin7$Fd_l$!3TG7?9$>\"H7FT7$Fh`l$!3y\"QY7o(\\[9 FT7$Fbal$!3Ex:kK**>>:FT7$F\\bl$!3**eEgx%=mc\"FT7$$!3*)**\\P%o^0T\"FT$! 3@R`PWpP\"e\"FT7$Fabl$!3#QfE,P'4!f\"FT7$$!3p**\\7.X'yI\"FT$!3ksZ([N^Ff \"FT7$Ffbl$!3!fB)e9$R$*e\"FT7$$!3_m\"z>E\\H?\"FT$!3#R1jliL$z:FT7$$!39FT7$F_fl$\"3sS6'*G;'y\\\"FT7$$\"3%*)*\\7Gy%e7\"FT$\"3o*H198$[`: FT7$Fdfl$\"3MPR^s36&e\"FT7$$\"3kl;/w\\-$G\"FT$\"33J/.5v&=f\"FT7$Fifl$ \"3KrSyxc,#f\"FT7$$\"3UK$e9h'4!R\"FT$\"3ht64ljd&e\"FT7$F^gl$\"3'y5Am]d Dd\"FT7$Fhgl$\"3&=;Zml()p_\"FT7$Fbhl$\"3%3/87#o)fX\"FT7$Ffil$\"3_sZMMd 3[7FT7$F`jl$\"3#)3p?Cju+'*Fin7$Fjjl$\"3v\"y8:s\"*pX#Fin7$Fd[m$!3'FT$\"3sL@k(pS_` \"FT7$F\\cm$\"3!)Q=9Q))GE:FT7$Facm$\"3%4bo3'eh+9FT7$Ffcm$\"3oB*=qO2V8 \"FT7$F[dm$\"3m!H^u+pMM(Fin7$F`dm$\"3Q[@+Q]\\\"R#Fin7$Fedm$!3+;6Qr(pW( HFin7$Fjdm$!3.mt39cTz\"e\"FT7$Fagp$!3w(z0M&4K(f\"FT 7$F^fm$!3!QQx>a=]f\"FT7$$\"3D*\\P4,%\\[\")FT$!3b#Q4J?Ope\"FT7$Fcfm$!32 sNws+7u:FT7$$\"3\"H$e9\">tQ?)FT$!3_%3n\\d(\\c:FT7$Fhfm$!3_N$pKp1S`\"FT 7$F]gm$!3rW8`_'GUZ\"FT7$Fbgm$!3]:lU*fFXR\"FT7$Ffhm$!3%f(\\1,$\\,[*Fin7 $F`im$!3yFD.$ogtv#Fin7$Feim$\"3#*3[&[z[D&fFin7$Fjim$\"3+8-$)e.eK9FT7$F aip$\"3!\\c[#>gnK=v[._b!>FT7$$\"3X+]P%y+QT*FT$\"3E>FWGX uE>FT7$Fiip$\"3En^Y?P!Q*=FT7$$\"3W+]7`\\*[^*FT$\"3SUx;)\\\"*yz\"FT7$Fd jm$\"3%R%o2`(Q%H;FT7$F^au$\"3227>%=BdN\"FT7$Fajp$\"3[WsGSO>C(*Fin7$Ffa u$\"3Wnvo-:KXYFin7$Fijm$!3f\"3\\i='>T=Fin7$$\"3v]iS\"*3))4)*FT$!35>Ai; bFmcFin7$F^bu$!3D+PSV)='4**Fin7$$\"3e\\(=nj+U')*FT$!3'p7q1G0&f9FT7$Fij p$!3e7Nvgesu>FT7$$\"3k]7.#Q?&=**FT$!3M>&ov=z\"RDFT7$Ffbu$!3ZF?F37&F[^vFbivF3-F46&FFQ2The~10~term~errorF'F>F? 7&-FL6$7]v7$FP$!3aF'***R7$G,\"FT7$$!3]LekynP')**FT$!3*p*4_`+Ffdl7$Fa[w$\"3$G*Rg0$)HY'*Fidl7$FV$\"3)HOW$\\=T!z\"F in7$F_`o$\"3AN#e\"eE,qPFin7$Fen$\"3xM-w0A%G<%Fin7$Fg`o$\"3i\"*o#*eu\"3 q$Fin7$F[o$\"3vO'yR1.\"zFFin7$F_ao$\"3[j+x.lSO;Fin7$F`o$\"33.]&f!emLWF idl7$F]ds$!3<*Q_KNX(y!)Fidl7$Feo$!3T=%ebvXX&=Fin7$Fjds$!3\"R`xLl)=KEFi n7$Fjo$!3hcz!o%G.:JFin7$$!3En;HKs**=$Fin7$$!3umm\"H-% \\/()FT$!3&G*fHcTiYKFin7$$!3Am;a8jlx')FT$!36xH_K![`G$Fin7$F_p$!3+n>,'* 4d1LFin7$$!3wlmT&=Vrf)FT$!3?A%fCgA&)H$Fin7$Fdp$!3%ReL+]3qA$Fin7$F^q$!3 ^Y^$GQ!*\\\"HFin7$Fhq$!3Jb^\"3$fl;Fin7$F`t$\"3SpvbG[Q*\\#Fin7$Fdfs$\"3u \\fJ!y=wy#Fin7$Fet$\"3B%Qx)3hT`HFin7$F\\gs$\"3GARnJOC!*HFin7$Fags$\"3- foJLU'o*HFin7$Ffgs$\"3$)3f8zT*R(HFin7$Fjt$\"3OMH79RfAHFin7$F_u$\"3a)zJ Iv@eV#Fin7$Fdu$\"3Wu%p2j+@9PFFin7$$!3(fmTNmlJd&FT$!3%f@U7wAB!GFin7 $$!3ALLeRyEDbFT$!3qM0VnP0]GFin7$$!3Z+]i:+PxaFT$!3\\.fdO:D!)GFin7$Fbw$! 3Az)4z>OH*GFin7$$!30L$eke]bP&FT$!3O'HJ!fcT')GFin7$Fgw$!3&Rhu&[sfa\"Fin7$F^[l$!3^]%zHA-?h'F idl7$Fh[l$\"3K$oX$*>mm7\"Fin7$F]\\l$\"3sY(RXU+B#=Fin7$Fb\\l$\"3m$f\"3# \\!QoBFin7$$!3=L$e*)p0el$FT$\"3i,'p@q1dg#Fin7$Fg\\l$\"3%)R\"\\Z6g1x#Fi n7$$!3RLe9T>y'[$FT$\"3pGptgO\"[#GFin7$$!32JdyBvGFin7$F\\]l$\"3cwn)o\"4SrGFin7$$!3zLL3_(H +F$FT$\"3S&Gxbq*4`GFin7$$!3)RLL3K,BA$FT$\"3&p![6+C:@GFin7$$!3gLLe*)Gdu JFT$\"3yKkR[WxvFFin7$Fa]l$\"3^GPyU@C'H6fBFin7$F[^l$\"3mP`q/RHElFidl7$Fh`l$!3;&f\"H2h2K@Fin7$F\\bl$!3qZDS:cM$ f#Fin7$Ffbl$!3k`vf=\\6VGFin7$F^hw$!3;OV`EmwqGFin7$Fchw$!3'4^_@!*RD)GFi n7$Fhhw$!3rkcA#y&RyGFin7$F[cl$!3g!3#\\9;QeGFin7$F`iw$!3[_!*op-OrFFin7$ F`cl$!3yY4Zj'[Ni#Fin7$Fecl$!3OW`B_C\"e?#Fin7$Fjcl$!37'ea;fkki\"Fin7$Fd dl$!3f0&fSiW0$yGFin7$Fdfl$\"3Ah)y+cY!fGFin7$F^gl$\"3Y0m6%RBXi# Fin7$Fbhl$\"3tIlz,!3w:#Fin7$F`jl$\"3IW-\"3H;$ztFidl7$Fjjl$!3KM!3!>v`z) )Fidl7$F_[m$!3yzw\"4ZsWq\"Fin7$Fd[m$!3/dzsE5\\]BFin7$$\"3)\\m;zu5M.$FT $!3D`no&35fc#Fin7$Fi[m$!3@nD1En`FFFin7$$\"37l;HdF3&=$FT$!3am-`+n!py#Fi n7$$\"3])**\\P4ScB$FT$!3D3U.CMWJGFin7$$\"3)=L3-V(>'G$FT$!340\"*R3k$3'G Fin7$F^\\m$!3V\"\\*erB&[(GFin7$$\"3yJLe9bt!R$FT$!3>aC&e+rE(GFin7$$\"3% ))***\\iirWMFT$!3tOmc`(*p_GFin7$$\"3#fm;/,(p)\\$FT$!3'>m])>.'\\\"GFin7 $Fc\\m$!3mc0Q8GefFFin7$$\"3-mm;a#R1m$FT$!3-r/My%zpf#Fin7$Fh\\m$!3L*QSR '*3\"oBFin7$F]]m$!3Zr33p2^,=Fin7$Fb]m$!3+D\")44m@x5Fin7$F\\^m$\"3#4Ass nL*)e(Fidl7$Ff^m$\"3fJ^t^>6u:Fin7$Fj_m$\"3\"fJM4(pUYAFin7$Fd`m$\"34/Hc `!oi^#Fin7$F^am$\"3?Ek:CmX=FFin7$$\"3G***\\(o#[KD&FT$\"3ghKgUm)=z#Fin7 $Fcam$\"38w>/vg'e%GFin7$$\"3GKLek.We`FT$\"3M!p7w*zyzGFin7$Fham$\"3o_$y '4L<$*GFin7$$\"3rK3-j'eCY&FT$\"3JZ)*)y:ug)GFin7$$\"3al;a84)Q^&FT$\"3XP \"okQA)eGFin7$$\"3O)\\iS;.`c&FT$\"3'*p?#H[e8\"GFin7$F]bm$\"3\\tK:**)fP u#Fin7$$\"31**\\i:*p&>dFT$\"3-H_gKB<\\DFin7$Fbbm$\"3\"f\"[64U_yAFin7$F gbm$\"3=p,19\"=Q\\\"Fin7$F\\cm$\"3t\"oTDa_&))[Fidl7$Facm$!3\"fR::?(Q?d Fidl7$Ffcm$!3xLLvm#ote\"Fin7$F[dm$!33#o@'*GbnU#Fin7$F`dm$!3f6U/ZJ\"*=H Fin7$Fb[u$!3vAsx'H[G(HFin7$Fg[u$!39(owtlAo*HFin7$F\\\\u$!3c=r!>^'z*)HF in7$Fedm$!33/CwS***4&HFin7$Fd\\u$!3YfxGcP]wFFin7$Fjdm$!3f8H`-*HMZ#Fin7 $F_em$!3\\dv$)RYv/;Fin7$Fdem$!3y%35?_:<<%Fidl7$F^fm$\"3Jt$*GVZr<6Fin7$ Fbgm$\"3&G(=DqmG&)oT\"Fin7$F_jm$!3n(=q!HTXBFr\" Fin7$Fijp$\"3Om*4b![:TDFin7$F`jx$\"3J$[A'R-$*)3%Fin7$Ffbu$\"33#)>V\">U s&eFin7$$\"3@Cc,\">g#f**FT$\"3i\")yN*=u'HoFin7$Fhjx$\"3@;'4%*Q_S'yFin7 $$\"3tu=nj+U')**FT$\"3*fY`p'pxi*)Fin7$F^[n$\"3aF'***R7$G,\"FTFb[n-FL6$ 7gn7$FP$!3S***z.xtx4&F\\fn7$Fg_o$!3,sWk<#*>`WF\\fn7$FV$!3WXk\\'*=#Q)QF \\fn7$F_`o$!3+hMaG2z^52 `)*Fffn7$Fhq$!3\"Q0*>Vx`h_Fffn7$Ffs$!3o0m')G'y/s#Fffn7$F`t$!35)G)Qo'[a U\"Fffn7$Fjt$!3Keo`5#[#>qFI7$Fdu$!3wZJ'pH\">>KFI7$F^v$!3&*)ec)Gn&HS\"F I7$Fhv$!3:uDS\\D!Hg&FT7$Fbw$!3y)=Gq$3E[BFT7$F`y$!3)pbg2wj#\\\")Fin7$F^ [l$!3%piLXhJ+b#Fin7$Fh[l$!3%Ffdl7$Ff]l$!3Kc@%R0&)\\j)Fcjn7$F`^l$!3xxIc9'GS.\"Fcjn7$ Fj^l$!3;+K&R7(o<6F\\cv7$Fh`l$!3e(*RFs]&\\k&!#D7$Ffbl$!30jcC6[?Z8Fccv7$ F`cl$\"3/Q6gV@A9vFgcv7$Fjcl$\"3*zWB9Dx\\.&FgcvFhcv7$F`el$!3?;!=C4P7+&F gcv7$Fjel$!3dZx;\\^kivFgcv7$Fdfl$\"3cyB&oa6D$**F^hu7$Fbhl$\"3Y6d,pSi$z(\\)Fcjn7$F^\\m$\"35D6.ZV%4^%Ffdl7$Fh\\m$\"3^K2m+%=9FI7$Ffc m$\"3Y-hYHeW-KFI7$F`dm$\"3jqLtLlk$)pFI7$Fjdm$\"3)pS%R>wfV9Fffn7$Fdem$ \"37Zb)4ca8r#Fffn7$Fbgm$\"3F@wE.hep`Fffn7$F`im$\"3L'*Q$)[e$>e*Fffn7$Fe im$\"3(\\6vaa4+H\"F\\fn7$Fjim$\"3&*3w\\fw&Qs\"F\\fn7$F_jm$\"3X*4*Rf^1g AF\\fn7$Fdjm$\"3Sqm%yofY%HF\\fn7$Fajp$\"3QJ50\\Y7'Q$F\\fn7$Fijm$\"3Q#f P'y)>s)QF\\fn7$Fijp$\"3D\\C\"QoB^X%F\\fn7$F^[n$\"3S***z.xtx4&F\\fnF_en F3-F46&FFQ2The~11~term~errorF'F>F?7&Fg[yFab[lF3-F46&FFQ2The~12~term~er rorF'F>F?7&-FL6$7_w7$FP$\"3#HOz**4O!y9Fin7$F^\\y$\"3*)=C7-6\"p%Fidl7$Fg`o$!34^7PM'=,h#Fidl7$F[o$!3=ln$fn%Fidl7$Fjds$\"3%ej>SJmDF idl7$Fet$!3M5()fk3%zs)Ffdl7$Fjt$\"3rHz-27b(=*Ffdl7$F_u$\"3!pt5fb^n\\#F idl7$Fdu$\"3#RDw:h&o;NFidl7$$!3&RL$3F0w$Q^oy*ow$Fidl7$$!3xmm\"HP3_g$ Fidl7$Fjy$!3t;\"oAW\\`n$Fidl7$F_z$!3\\r;.n\"\\3r$Fidl7$Fdz$!3-6AUygw6P Fidl7$$!3]++](G%QFZFT$!3[uFw=*o%yOFidl7$Fiz$!33Ej)=YFT$!3&G@$p\"R$>7NFidl7$F^[l$!3NW*fY])R\"Q$Fidl7$Fc[l$!3IoTN>A 'Qh#Fidl7$Fh[l$!3YW1S:VQ?:Fidl7$Fb\\l$\"39)yv_$Fidl7$$!3*RL$3Fg6zIFT$\"31+HLV\"o`f$Fidl7$Fi[z$\"3'f\\.') Qo3k$Fidl7$$!3zLLek\"fO)HFT$\"3HsftC:\"Rm$Fidl7$Ff]l$\"31Je3ldYkOFidl7 $$!3:nTN@7T!)GFT$\"3s$pFFT$\"3S)**pNP#)G\\$Fidl7$F[^l$\"3#>]5F-$zxLFidl7$$!3:M$e 9m8Gg#FT$\"3&*)*pfjF)o1$Fidl7$F`^l$\"3yt5,af(yl#Fidl7$Fj^l$\"3\"QS5l@6 ;]'Ffdl7$Fh`l$!3;9j@@'4ky\"Fidl7$F\\bl$!3;*f'GVF$Qr#Fidl7$Ffbl$!396]sE .&HN$Fidl7$F^hw$!35b,[0:\\jMFidl7$Fchw$!3**4Wb5O6\\NFidl7$Fhhw$!3Q'eV$ eMD4OFidl7$F[cl$!3>hKD?)GNk$Fidl7$$!3IJ$e*)fEm))*Fin$!3$R$3CE.u^OFidl7 $F`iw$!3sv%fi8sQj$Fidl7$$!3Nk;a)e$>:))Fin$!3I&*\\X#Q\"4!f$Fidl7$F`cl$! 3EU\"*[#RY2_$Fidl7$Fecl$!3H%HVm-Aq0$Fidl7$Fjcl$!3I-)G&yQA.BFidl7$Fddl$ !3&)RuNF*fO=!\\9a' \\OFidl7$F]`z$\"3M!Hl@7Bji$Fidl7$F_[x$\"3qZ[JzKpyNFidl7$Fe`z$\"3\")*Gw KuSq]$Fidl7$Fdfl$\"3%HX.\"\\4!=T$Fidl7$F^gl$\"3#)*4qb-GNy#Fidl7$Fbhl$ \"3GhFw/,PM=Fidl7$F`jl$!3uT3yI`qN_Ffdl7$Fjjl$!3I;>\\\"fB$=EFidl7$$\"3Q )*\\i!>,Zf#FT$!3c3%>D=p,/$Fidl7$F_[m$!3jwru:GAiLFidl7$$\"3H)\\7Gj)\\jF FT$!3e[hfCQ.#[$Fidl7$$\"3F)*\\(oWk(>GFT$!3_k(32RnFd$Fidl7$$\"3D)\\P4EI g(GFT$!3E<:@1(4Nj$Fidl7$Fd[m$!3$f*3n9&)ejOFidl7$$\"3fJ$e9T`G)HFT$!37:t 'HS2Tm$Fidl7$F_bz$!3_/j#Gw%RROFidl7$$\"3P)*\\P%3oR3$FT$!3$o#G*[X!\\*e$ Fidl7$Fi[m$!3i'z4:*eg9NFidl7$F\\cz$!3o2On%y,;H$Fidl7$F^\\m$!3da%H$Q!* \\vHFidl7$Fc\\m$!3`-OG-X-G?Fidl7$Fh\\m$!3M18E\"38?2)Ffdl7$Fb]m$\"3_liM >#em;HK%\\E!yFT$\"3#*Qz*Q#y$H.%Fidl7$$\"3$G$e9 m&o4&yFT$\"3gF3)RZ'='4%Fidl7$Fdem$\"3*RYGH6G25%Fidl7$F^fm$\"3m>gQH\"yp K$Fidl7$Fbgm$\"3`3!*yJ*GXK\"Fidl7$Ffhm$!3Fidl7$F_jm$\"3_)\\*Q[/L@(*Fcjn7$Fiip$\"334$[,$R!R\\#Fidl7$Fdjm$\"3< n@9r*etr%Fidl7$F^au$\"3Fx]G-g]'e&Fidl7$Fajp$\"3X?)H!=w\"y-'Fidl7$$\"3f DJXk9m(o*FT$\"3#4_\"=da!)[gFidl7$$\"3'3D\"y+9C,(*FT$\"3L]&>(\\OGFgFidl 7$$\"37w$4rL@[r*FT$\"3$z'*3ALt&ffFidl7$Ffau$\"3Yq4%G%)>=%eFidl7$$\"3p \\P4Y6cb(*FT$\"3?'yF'G4&)RaFidl7$Fijm$\"3c6tnF2!oy%Fidl7$F`ix$\"3A6\\( p(f*[%QFidl7$F^bu$\"3\"pJThpgHd#Fidl7$Fhix$\"3H[#y*)R2?E*Ffdl7$Fijp$!3 aVR:9=-W6Fidl7$$\"3PDJqX/%\\!**FT$!30Nu#*R)RUN#Fidl7$F`jx$!3u(*p82[U!p $Fidl7$$\"3!fPf$=.5K**FT$!3nH\"4HYW)f^Fidl7$Ffbu$!3X7%p]/`+x'Fidl7$Fba [l$!3Cly%Qi**)G&)Fidl7$Fhjx$!3%eQ2Mm`W/\"Fin7$Fja[l$!3Viw-fGa_7Fin7$F^ [n$!3#HOz**4O!y9FinFb[n-FL6$7\\o7$FP$\"3C,+iE\"*fOBF\\fn7$Fg_o$\"3M5Z$ zLfS*>F\\fn7$FV$\"3O6V_f+d)p\"F\\fn7$F_`o$\"331!zW\"p6W9F\\fn7$Fen$\"3 =#psR2&RD7F\\fn7$F[o$\"3Qdj$p5XR;*Fffn7$F`o$\"3a=`7PfF5oFffn7$Feo$\"3/ jOJ4@OR[Fffn7$Fjo$\"39Xz$>*p>4MFffn7$Fdp$\"3F*[OXYSUP#Fffn7$Fhq$\"3%R' *oN+rvj\"Fffn7$F\\s$\"3Q,pr*z2,7\"Fffn7$Ffs$\"3Oj>7S!=3e(FI7$F`t$\"3nG Wpo,\"pc$FI7$Fjt$\"3@q3gjlCi:FI7$Fdu$\"3@#oz&4Oo.jFT7$F^v$\"3m(QUB$f/* R#FT7$Fhv$\"3y)3/8C1lD)Fin7$Fbw$\"35B\\%Q8%33IFin7$F`y$\"3FZLy9(R:\")) Fidl7$F^[l$\"3UO2,5Gk\"H#Fidl7$Fh[l$\"3y6MR4P'3]&Ffdl7$Fb\\l$\"3b?!**3 0())=8Ffdl7$F\\]l$\"3?94&*zk[r>Fcjn7$Ff]l$\"3ojEaut@uJF\\cv7$F`^l$\"3Q 46s9L/KFFf`n7$Fj^l$\"3-OA*)[gZ&4#Fbg[l7$Fh`l$\"3$e?=3=KZS(F^hu7$Ffbl$ \"3m6(*z'p#et**Fgcv7$F`cl$\"3-k'G\"G/!H:)Fgcv7$Fjcl$\"3#zJv')*H7N]Fgcv Fhcv7$F`el$!31pf>c!p8+&Fgcv7$Fjel$!3+7SS#fgB3)Fgcv7$Fdfl$!3[c\"fbo#)[t *Fgcv7$Fbhl$!33guSc_.koF^hu7$F`jl$!3w%3xI5QBz\"Fbg[l7$Fjjl$!3^%)*G3v!o yDFf`n7$Fd[m$!3sx&*G>X*f6$F\\cv7$F^\\m$!3qVSNp*)*o9#Fcjn7$Fh\\m$!3bnSY XWZ>8Ffdl7$Fb]m$!3W)[OH^M/v&Ffdl7$F\\^m$!3#e>M++XwY#Fidl7$Fj_m$!3?w_-D Xz3')Fidl7$Fham$!3Xsk>ilAgGFin7$Fbbm$!3x\"G@zjrRY)Fin7$F\\cm$!3=,nGlS \")HCFT7$Ffcm$!3E@qQ$\\TbE'FT7$F`dm$!3bF::B1-`:FI7$Fjdm$!3O FI7$F_em$!3-I8Hp$>MD&FI7$Fdem$!3DBvd%za6b(FI7$F^fm$!3mZoj^gqJ6Fffn7$Fb gm$!3;.3IpU#pn\"Fffn7$Ffhm$!3i\\R&3iI?O#Fffn7$F`im$!3?zQ'3Y](*H$Fffn7$ Feim$!3)=bp:V#GsYFffn7$Fjim$!3$Q2x\"pM^flFffn7$F_jm$!3nIbA\"eGq+*Fffn7 $Fdjm$!3F\\fn7$F^[n$!3C,+iE\"*fOBF\\fnF_enF3-F46&FFQ2Th e~13~term~errorF'F>F?7&F[^\\lF_i^lF3-F46&FFQ2The~14~term~errorF'F>F?7& -FL6$7gv7$FP$!3O%[G-++ja\"Fidl7$F^\\y$!3B!3eOOCbC\"Fidl7$Fiiv$!3\"4N:w >Bsu*Ffdl7$Ff\\y$!3EFXQC3@>tFfdl7$Fb`s$!3/C\\?J,V_^Ffdl7$F^_\\l$!31[g_ $y?!HKFfdl7$Fajv$!3*[y2R1/?`\"Ffdl7$Ff_\\l$!3I*[JvS\\\\_%F\\cv7$Fg_o$ \"3U'Gq[bdlC\"Ffdl7$Fijv$\"3$zCL3$ep-LFfdl7$Fj`s$\"3kXDJ&f!yVZFfdl7$Fa [w$\"3%e)HR<5/mcFfdl7$FV$\"3*zVdXNI8'Ffdl7$Fga\\l$\"39I\"*4-0&*[dFfdl7$F_`o$\"3>B(G^wK ))=&Ffdl7$Fjas$\"3q(eU+'4?=PFfdl7$Fen$\"3!eB7kfW\">?Ffdl7$Fg`o$!3!*QYy l$3pr)Fcjn7$F[o$!31Q#>!z!H@7$Ffdl7$F_ao$!3mfBC/;iSWFfdl7$F`o$!3Oz1!=$3 'R![Ffdl7$F]ds$!31zn$\\*4CWUFfdl7$Feo$!3JA$)3Ck(*fHFfdl7$Fjds$!3Uw$Ffdl7$Fhq$\"3mk,+8 *[v9%Ffdl7$F\\s$\"3QJ4J5kLuKFfdl7$Ffs$\"3C!=t)G<:M7Ffdl7$F[t$!3d\"GD$* H\\[F*Fcjn7$F`t$!34dEZ_H[+FFfdl7$Fdfs$!3>*G%*=IB$>LFfdl7$Fet$!3th][>'* p*o$Ffdl7$F\\gs$!3mmB'**=xxx$Ffdl7$Fags$!3al,YZOF,QFfdl7$Ffgs$!3\">h*) )e(3;w$Ffdl7$Fjt$!3rOe?%*z9hOFfdl7$F_u$!3&)Rj-\"p&)=o#Ffdl7$Fdu$!3nQw? WvSs5Ffdl7$Fiu$\"3*zi8QDaCC(Fcjn7$F^v$\"3)p2e'3lS#H#Ffdl7$F\\bw$\"3esp ++0^**GFfdl7$Fcv$\"3xhEY%*HhHLFfdl7$$!3wLe*)4WzvfFT$\"3Jn1Pw)>IZLFfdl7$F]w$\"3UMNupS3$>$Ffdl7$F[fy$\" 3IEIXp?I\"z#Ffdl7$Fbw$\"3uX(ew*Qr\"G#Ffdl7$F`y$!3**pysJ=K3pFcjn7$Fdz$! 37#o;kD*3)3#Ffdl7$F^[l$!35h3u%)*Gi2$Ffdl7$Fi[t$!3)o^m>bOKB$Ffdl7$F^\\t $!3pvd%[](\\bLFfdl7$Fc\\t$!3#f:X\"zM3UMFfdl7$Fc[l$!3SrW.C`W#\\$Ffdl7$$ !3m*****\\7rSI%FT$!3$oQg&o2T1NFfdl7$F[]t$!3nbGq/&yT[$Ffdl7$$!3#****** \\Z<(*>%FT$!31W:V_)3jU$Ffdl7$Fh[l$!3#pd-Z>6PL$Ffdl7$F]\\l$!3Yc!Q2g>!GF Ffdl7$Fb\\l$!3ZsYs:***fx\"Ffdl7$F\\]l$\"3W3:3US0u5Ffdl7$Fa]l$\"3\"*fU+ \">gB:#Ffdl7$Ff]l$\"3$\\F%R(pX]&HFfdl7$F]_]l$\"3!\\^M#3!)*oD$Ffdl7$F[^ l$\"3?+%\\;vQDU$Ffdl7$$!3T+vVtJLeEFT$\"3`iwIsqR_MFfdl7$Fj_]l$\"3KHb/6# omW$Ffdl7$$!3Kn\"z%\\THZDFT$\"3I*y@0'Qf0MFfdl7$F`^l$\"3*)=oa(fp(HLFfdl 7$Fe^l$\"3WQS#*p:**)z#Ffdl7$Fj^l$\"3Vjpc\"ow6#>Ffdl7$Fh`l$!3*zQd*4eY#G 'Fcjn7$F\\bl$!3Eu'pGfL*)z\"Ffdl7$Ffbl$!3+)R[n]Cms#Ffdl7$Fchw$!3;cgo-SW sIFfdl7$F[cl$!3y[(\\b+anI$Ffdl7$Fja]l$!3\"f`:l;M%zLFfdl7$F`iw$!3![WG4P v:U$Ffdl7$Fbb]l$!3cdw$zz]GV$Ffdl7$F`cl$!3#4M%y0$3KT$Ffdl7$$!3uJLe9rc&H (Fin$!370tPYI_(H$Ffdl7$Fecl$!3A$p#*4J;A3$Ffdl7$$!3XLL3xruF`Fin$!3AR$y( p*zRx#Ffdl7$Fjcl$!3vwJuOyC#Q#Ffdl7$Fddl$!3metVco3\")fF\\cv7$F`el$\"3un L%*y\\wnBFfdl7$Feel$\"3BK;'Q%)f+1$Ffdl7$Fjel$\"3M/b_J.y.MFfdl7$F\\d]l$ \"3LH)pqq;0V$Ffdl7$Fgjw$\"313nC8ONGMFfdl7$Fdd]l$\"3Y<5a%flsR$Ffdl7$F_f l$\"3k!ebWznuL$Ffdl7$F_[x$\"3d&z!*G$*)zLJFfdl7$Fdfl$\"3i->%\\&pzBGFfdl 7$F^gl$\"3cbxT\"*)=L*=Ffdl7$Fbhl$\"3!4-E.iZg&oFcjn7$F`jl$!3B$Q1[V$z1=F fdl7$Fejl$!3Mw`:Y4DSFFfdl7$Fjjl$!3Zpu`:(*=8LFfdl7$$\"3S)\\ilPN%QDFT$!3 +X)*fJ\\!eR$Ffdl7$Fdf]l$!3h\\sDL6'GW$Ffdl7$$\"3N)\\(o/q'4l#FT$!3dq2&Q% Ho`MFfdl7$F_[m$!3dhav\"*p'zU$Ffdl7$Fag]l$!3#[ln2^3wE$Ffdl7$Fd[m$!3%fkW $*H$)p'HFfdl7$Fi[m$!3*yN<:[RN6#Ffdl7$F^\\m$!3z\"HC^n\"zj&*Fcjn7$Fh\\m$ \"3R+oYAOkwTv#Ffdl7$Fb]m$\"3.9?UvrveLFfdl7$$\"3 #e;/E&RR8UFT$\"3Nxmpd*y[W$Ffdl7$F`ft$\"3!y(=\"[`PU\\$Ffdl7$$\"3kKe9mqM ?VFT$\"3kqOf1c&f]$Ffdl7$Fg]m$\"3gc!=/MX&zMFfdl7$F]gt$\"3IuUE63T7LFfdl7 $F\\^m$\"3%)Qa&*4G7(*HFfdl7$Ff^m$\"3L-x3Km\\X?Ffdl7$Fj_m$\"341Ln5KveuF cjn7$Fham$!3&eRQQ]YJ<#Ffdl7$F]hz$!3a\"*Qn7]&ft#Ffdl7$F]bm$!3Iiotx*3v<$ Ffdl7$$\"3AmT5lw9ocFT$!3O=,U(>g\\M$Ffdl7$Fjhz$!3-a9)[%QMtMFfdl7$$\"3)= $e9m@*4x&FT$!3Tpc%=U;0c$Ffdl7$Fbbm$!3S0-ZOjx/OFfdl7$$\"3ZKeRZ([i(eFT$! 300-m;l!Qg$Ffdl7$F^ax$!3kU!G)Qml`NFfdl7$$\"3ylT&)3u\"R)fFT$!3#o/QGifTX $Ffdl7$Fgbm$!3D%[#GSh\"eI$Ffdl7$F[bx$!3\"*\\\\D+F.oGFfdl7$F\\cm$!3[/hO q<:eAFfdl7$Facm$!3kBBi&[gI8(Fcjn7$Ffcm$\"3kQb5'yE$\\5Ffdl7$F[dm$\"32C4 UmS#Qm#Ffdl7$F`dm$\"3u[L#)*RAQl$Ffdl7$Fb[u$\"3>)z.ekR%fPFfdl7$Fg[u$\"3 N!pls'4B,QFfdl7$F\\\\u$\"3(fDk**oHmx$Ffdl7$Fedm$\"32'Q^#zR6%o$Ffdl7$Fd \\u$\"3_ZMDN4,&H$Ffdl7$Fjdm$\"3Xv\"o\\#QyXEFfdl7$F_em$\"3y5yqG$4B**)Fc jn7$Fdem$!3Sp5qT*y5@\"Ffdl7$F^fm$!3>j#o9%3p;LFfdl7$Fbgm$!3**HTKX$)zXTF fdl7$Ffhm$!3yroMZV.AKFfdl7$F`im$!3D$>fbJj(ouFcjn7$F^^u$\"3gV[f4(zIa*Fc jn7$Feim$\"3QKJ$e&>'=k#Ffdl7$Fe_u$\"3.6-\"*R%>/-%Ffdl7$Fjim$\"3*=&*Rv? 4$fZFfdl7$$\"3-,vo/*Qj=*FT$\"3nnKAP8:0[Ffdl7$$\"3Z+](oW7;@*FT$\"3cyY!) *Ha@z%Ffdl7$$\"3\"**\\i!*)f)oB*FT$\"3!G#e04_\"pr%Ffdl7$Faip$\"3&**Qt\" >c^wXFfdl7$$\"3Y+]i:mq7$*FT$\"39Gd)Hz-;4%Ffdl7$F_jm$\"3S0D+j]'yK$Ffdl7 $Fiip$\"3>')\\Ai)*425Ffdl7$Fdjm$!3PIwE*4QK1#Ffdl7$F^au$!35KShz#QVv$Ffd l7$Fajp$!3)p5)>\"3YG@&Ffdl7$Fce^l$!3\\=g(>LHcw&Ffdl7$Ffau$!3Q$*QT)e)yT hFfdl7$F`f^l$!3#yQm[#y**)G'Ffdl7$Fijm$!3DZ0)3s=x9'Ffdl7$F`ix$!3(Q$4eWh `]cFfdl7$F^bu$!3eT!3$=\"[9s%Ffdl7$Fhix$!356'fU.3^F$Ffdl7$Fijp$!3w%3%3, l,;7Ffdl7$Fdg^l$\"3%HaJI3aTh(F\\cv7$F`jx$\"3(*3=I2(RBc\"Ffdl7$F\\h^l$ \"3#>G5zYjxD$Ffdl7$Ffbu$\"3=a]y0WSy^Ffdl7$Fba[l$\"3$\\&oPO@4TtFfdl7$Fh jx$\"3:M-Qsr_j(*Ffdl7$Fja[l$\"3q]_k![JkC\"Fidl7$F^[n$\"3O%[G-++ja\"Fid lFb[n-FL6$7[o7$FP$!3W&)**z`c9!3)Fffn7$Fg_o$!3#*)Q&>p..RnFffn7$FV$!3% \\o7r\"fr3cFffn7$F_`o$!3Yq:lDK(zl%Fffn7$Fen$!3mk*)*H14*fQFffn7$F[o$!3 \"oubf)pCoFFffn7$F`o$!3%p,9eaI8(>Fffn7$Feo$!3(=9%=OX'QL\"Fffn7$Fjo$!3! f$[T=QkP*)FI7$Fdp$!3Kgd45mP6fFI7$Fhq$!37\\\"=r7-u'QFI7$F\\s$!3_1%41q!e 1DFI7$Ffs$!3t3k&z`y_g\"FI7$F`t$!3-K(>Hp?;z'FT7$Fjt$!3SKU>)QB$\\EFT7$Fd u$!31UIOKh*zT*Fin7$F^v$!3t)H(R>.9MJFin7$Fhv$!3A&3Y8+!Q2$*Fidl7$Fbw$!3@ tr'))>-4&HFidl7$F`y$!3b$4jv?d[I(Ffdl7$F^[l$!3IihTg(*p!e\"Ffdl7$Fh[l$!3 ]n^*z-V\\7$Fcjn7$Fb\\l$!3(z;R*4yYthF\\cv7$F\\]l$!3G+&eS%)yr8(Ff`n7$Ff] l$!3L;UB>0W\")*)Fbg[l7$F`^l$!32(>@<-(HNbFccv7$Fj^l$!3DR0wu_uqDF^hu7$Fh `l$\"3$)\\,vky5.oFgcv7$Ffbl$\"3x\\.5n,G2!*Fgcv7$F`cl$\"34z8Hc&[5:)Fgcv 7$Fjcl$\"34E5N#)G7N]FgcvFhcv7$F`el$!3M@,Z_*o8+&Fgcv7$Fjel$!3o>\"GSp+43 )Fgcv7$Fdfl$!3[VDa#[`v.*Fgcv7$Fbhl$!3(4x;P'RQ;pFgcv7$F`jl$\"3MqYE;r+l? F^hu7$Fjjl$\"3UZR`+zL#=&Fccv7$Fd[m$\"3sB&RMw/Zz)Fbg[l7$F^\\m$\"3\\&=me ?w@'yFf`n7$Fh\\m$\"3^=$f*\\'*ewhF\\cv7$Fb]m$\"3;:G*p>hkG$Fcjn7$F\\^m$ \"3&e&zF*4^$>!\\m;(Rd*Fidl7$F\\cm$\"3L*f5ivZ*zJFin7$Ffcm$\"3p99%f&>5`$*Fi n7$F`dm$\"3O'R1rd&\\JEFT7$Fjdm$\"3AAPR5@'o!pFT7$Fdem$\"3yyg5J;6)f\"FI7 $F^fm$\"35Pb=_SBODFI7$Fbgm$\"3*y0R3sBP(RFI7$Ffhm$\"3%[$Q**R*fm(eFI7$F` im$\"3wzl&=h20h)FI7$Feim$\"3S]1xc;N\"G\"Fffn7$Fjim$\"35VG)pEZ&))=Fffn7 $F_jm$\"3(=^04#G49FFffn7$Fdjm$\"3nq`?eO>pQFffn7$Fajp$\"3PF?7&-FL6$7gv7$FP$!3li+?++IY:Fi dl7$F^\\y$!3)*)*[^jV_X7Fidl7$Fiiv$!30;[k)>Bsu*Ffdl7$Ff\\y$!3p#[Ya#3@>t Ffdl7$Fb`s$!3hA]HC7%y?!HKFfdl7$Fajv$!3%fOmZ1/?` \"Ffdl7$Ff_\\l$!3Y!3!>`#\\\\_%F\\cv7$Fg_o$\"3)\\]ZMbdlC\"Ffdl7$Fijv$\" 3O3i^Fep-LFfdl7$Fj`s$\"3M&)ev%f!yVZFfdl7$Fa[w$\"3D<0A=5/mcFfdl7$FV$\"3 ^fC23&Ra:'Ffdl7$Fbh_l$\"3#\\&GxLV2iiFfdl7$Fj`\\l$\"3#y-2iu\"\\)G'Ffdl7 $Fjh_l$\"3'Qy_)yi!HC'Ffdl7$Fbas$\"37>i+ca.LhFfdl7$Fga\\l$\"3s'GL<]]*[d Ffdl7$F_`o$\"3m@AwkF$))=&Ffdl7$Fjas$\"3_gtqf4?=PFfdl7$Fen$\"3'=OhdfW\" >?Ffdl7$Fg`o$!3q]B6c$3pr)Fcjn7$F[o$!3zR))Gy!H@7$Ffdl7$F_ao$!3G*HhTg@1W %Ffdl7$F`o$!36ds[J3'R![Ffdl7$F]ds$!3Za4c%*4CWUFfdl7$Feo$!3oH4lBk(*fHFf dl7$Fjds$!3;L3zk4A$H\"Ffdl7$Fjo$\"3wXxgBH[=WFcjn7$Fdp$\"35Ok%HbY\\>$Ff dl7$Fhq$\"34#[VJ\"*[v9%Ffdl7$F\\s$\"3zEEH5kLuKFfdl7$Ffs$\"3?(y)*)G<:M7 Ffdl7$F[t$!30v^`&H\\[F*Fcjn7$F`t$!3a'y*\\_H[+FFfdl7$Fdfs$!3k?wz,LK>LFf dl7$Fet$!3p.b`>'*p*o$Ffdl7$F\\gs$!3JEN&)*=xxx$Ffdl7$Fags$!30uQYZOF,QFf dl7$Ffgs$!3*=f%ze(3;w$Ffdl7$Fjt$!3)yE,U*z9hOFfdl7$F_u$!3]eH/\"p&)=o#Ff dl7$Fdu$!3&Rp+Ua2C2\"Ffdl7$Fiu$\"3\"[\\FMDaCC(Fcjn7$F^v$\"3!QAz'3lS#H# Ffdl7$F\\bw$\"3Qg`++0^**GFfdl7$Fcv$\"3H'e\\W*HhHLFfdl7$F\\_`l$\"3`W@Pw ;\"R(*f$Ffdl7$F\\``l$\"3q0(oL(H?_NFfdl7$F^ey$\"3nd2`\")zenMFfdl7$Fd``l $\"3rO>$>)>IZLFfdl7$F]w$\"3OjavpS3$>$Ffdl7$F[fy$\"3k!He%p?I\"z#Ffdl7$F bw$\"3rnMm(*Qr\"G#Ffdl7$F`y$!3#)fhzJ=K3pFcjn7$Fdz$!3X'*zTc#*3)3#Ffdl7$ F^[l$!3!*Rut%)*Gi2$Ffdl7$Fi[t$!3:\"=l>bOKB$Ffdl7$F^\\t$!3_Yj%[](\\bLFf dl7$Fc\\t$!3/M49zM3UMFfdl7$Fc[l$!3+>C.C`W#\\$Ffdl7$Fgb`l$!3d*3i&o2T1NF fdl7$F[]t$!3()3Lq/&yT[$Ffdl7$F_c`l$!3(p&4V_)3jU$Ffdl7$Fh[l$!3_zMq%>6PL $Ffdl7$F]\\l$!3O3tt+'>!GFFfdl7$Fb\\l$!3n*>Dd\"***fx\"Ffdl7$F\\]l$\"3WU 93US0u5Ffdl7$Fa]l$\"3c;V+\">gB:#Ffdl7$Ff]l$\"307UR(pX]&HFfdl7$F]_]l$\" 3b+YB3!)*oD$Ffdl7$F[^l$\"3_L$\\;vQDU$Ffdl7$F\\e`l$\"30PwIsqR_MFfdl7$Fj _]l$\"3L%[X5@omW$Ffdl7$Fde`l$\"3Wl<_gQf0MFfdl7$F`^l$\"3?()oa(fp(HLFfdl 7$Fe^l$\"3[XS#*p:**)z#Ffdl7$Fj^l$\"3!f(pc\"ow6#>Ffdl7$Fh`l$!3fjt&*4eY# G'Fcjn7$F\\bl$!3+u'pGfL*)z\"FfdlFgf`lFjf`lF]g`lF`g`lFcg`lFfg`lFig`lF \\h`lFah`lFdh`lFih`lF\\i`lF_i`lFbi`lFei`lFhi`lF[j`lF^j`l7$F_fl$\"35!eb WznuL$FfdlFdj`lFgj`l7$F^gl$\"35cxT\"*)=L*=Ffdl7$Fbhl$\"3?ggK?w/coFcjn7 $F`jl$!3N%R1[V$z1=Ffdl7$Fejl$!3:1a:Y4DSFFfdl7$Fjjl$!3j;v`:(*=8LFfdl7$F j[al$!3eE)*fJ\\!eR$Ffdl7$Fdf]l$!3K2sDL6'GW$Ffdl7$Fb\\al$!3?F2&Q%Ho`MFf dl7$F_[m$!3%>Sb<*p'zU$Ffdl7$Fag]l$!3$)4ww5&3wE$Ffdl7$Fd[m$!3=\"oW$*H$) p'HFfdl7$Fi[m$!3czs^\"[RN6#Ffdl7$F^\\m$!3U\"RG^n\"zj&*Fcjn7$Fh\\m$\"37 #onCiVmx\"Ffdl7$F]]m$\"3&>F6W7>Tv#Ffdl7$Fb]m$\"3xVJUvrveLFfdl7$F_^al$ \"3E@#)pd*y[W$Ffdl7$F`ft$\"356;\"[`PU\\$Ffdl7$Fg^al$\"31wOf1c&f]$Ffdl7 $Fg]m$\"3wh'>/MX&zMFfdl7$F]gt$\"3%=zg7\"3T7LFfdl7$F\\^m$\"3%=,c*4G7(*H Ffdl7$Ff^m$\"3Yg%)3Km\\X?Ffdl7$Fj_m$\"3HF1l5KveuFcjn7$Fham$!3/_N#Q]YJ< #Ffdl7$F]hz$!3m2hn7]&ft#Ffdl7$F]bm$!3j#pLx(*3v<$Ffdl7$Fd`al$!3L7^V(>g \\M$Ffdl7$Fjhz$!3A\"=o[%QMtMFfdl7$F\\aal$!3V!pK=U;0c$Ffdl7$Fbbm$!3(*os ZOjx/OFfdl7$Fdaal$!3x3fo;l!Qg$Ffdl7$F^ax$!3zi$*zQml`NFfdl7$F\\bal$!3&4 %3&GifTX$Ffdl7$Fgbm$!3FFfHSh\"eI$Ffdl7$F[bx$!3$zRT-qK!oGFfdl7$F\\cm$!3 LtaNq<:eAFfdl7$Facm$!3E2jg&[gI8(Fcjn7$Ffcm$\"3e'p.hyE$\\5Ffdl7$F[dm$\" 3ieCTmS#Qm#Ffdl7$F`dm$\"3oO*3(*RAQl$Ffdl7$Fb[u$\"3F5rrX'R%fPFfdl7$Fg[u $\"3:&)R?n4B,QFfdl7$F\\\\u$\"3:8P$)*oHmx$Ffdl7$Fedm$\"3LmYDzR6%o$Ffdl7 $Fd\\u$\"3Yd#f^$4,&H$Ffdl7$Fjdm$\"3:kI4DQyXEFfdl7$F_em$\"3w)Ra\"G$4B** )Fcjn7$Fdem$!3N:$\\9%*y5@\"Ffdl7$F^fm$!3i&>'=k#Ffdl7$Fe_u$\"3V*)H\\R%>/-%Ffdl7$Fjim$\"3Xp &)o2#4$fZFfdl7$Fffal$\"3F6&fmL^^![Ffdl7$F[gal$\"3]6(H3Ia@z%Ffdl7$F`gal $\"3`47!y?:pr%Ffdl7$Faip$\"3i@E!*>c^wXFfdl7$Fhgal$\"31*4YLz-;4%Ffdl7$F _jm$\"3q0(oI1lyK$Ffdl7$Fiip$\"3xz+1j)*425Ffdl7$Fdjm$!3V9#\\&)4QK1#Ffdl 7$F^au$!3r11Bz#QVv$Ffdl7$Fajp$!3,GzQ\"3YG@&Ffdl7$Fce^l$!3Ab&y=LHcw&Ffd l7$Ffau$!3QF?7&-F L6$7[o7$FP$\"3d9+?m)\\\"*=#Fffn7$Fg_o$\"3FiB+g\")z%y\"Fffn7$FV$\"3nw&p #fJt^9Fffn7$F_`o$\"3')3k!*4=+y6Fffn7$Fen$\"3s:m8qZ\\N&*FI7$F[o$\"3h?L& o4F7c'FI7$F`o$\"3tgYEe)Q#zWFI7$Feo$\"3.'*)zC-et)GFI7$Fjo$\"3'o=D:[$3T= FI7$Fdp$\"3$>%)[\"*\\Mq:\"FI7$Fhq$\"3nE\\!4UYO=(FT7$F\\s$\"3pq_5:)Fgcv7$Fjcl$\"3;0FU#)G7N]FgcvFhcv7$F`el$!3%y1LD&*o8+&Fgcv7$Fje l$!3GEu;gQ!43)Fgcv7$Fdfl$!3Sqs\"o(e*4/*Fgcv7$Fbhl$!3r1hjxVMquFgcv7$F`j l$!3#HO11Q'[&)[Fgcv7$Fjjl$!3B=+aG9P.6F^hu7$Fd[m$!3l!H*\\\"e]z,#Fccv7$F ^\\m$!3?v_Nx$4yH#Fbg[l7$Fh\\m$!3-6^2UMX'H#Ff`n7$Fb]m$!3)4uhvd`'*[\"F \\cv7$F\\^m$!3%3*G9__q#\\*F\\cv7$Fj_m$!3>!4$['[a^l%Fcjn7$Fham$!3=z?([! e*z9#Ffdl7$Fbbm$!3Em,:g1^i&)Ffdl7$F\\cm$!3sh3 !3@5\"Fin7$F`dm$!37-XrUka;NFin7$Fjdm$!3_?L#[E'QQ5FT7$Fdem$!3ba%pK))GFm #FT7$F^fm$!3n7xLW]bsWFT7$Fbgm$!3ocr-J/&eS(FT7$Ffhm$!3W$Qf52.%\\6FI7$F` im$!3xAKr(oJbw\"FI7$Feim$!3**oan5%3*fFFI7$Fjim$!3s2P`\\2LoUFI7$F_jm$!3 3clrrQ2%F\\cv7$Ff_\\l$!3h%4cF]aSe\"Fcjn7$Fg_o$! 3ISb\"*Q1*)QDFcjn7$Fijv$!3k?\\[tv;**QFcjn7$Fj`s$!3zene(*e3ZYFcjn7$$!3Q LeRA\")*G#)*FT$!3bp!3#3N(=$[Fcjn7$Fa[w$!3IP4>b+&)4\\Fcjn7$$!3Q+voz;l&z *FT$!3g/^K^V<%*[Fcjn7$FV$!3%eh3%)z'HFcjn7$F[o$\"398o\"yQ%ozPFcjn7$$!359zL6/8$Fcjn7$Fh`y$!3gS;b*3]#zJFcjn7$F_p$!3Q cD@azH#>$Fcjn7$$!3=l;z%*3)Ri)FT$!3;9xME')oqJFcjn7$F`ay$!3o%e#e&>`e6$Fc jn7$$!3Om;/waIq&)FT$!3ieH^)H&[HIFcjn7$Fdp$!3Ml6')pp]8HFcjn7$F^q$!3ezp \"=cp!)>#Fcjn7$Fhq$!3(HH#4aQ7+7Fcjn7$F\\s$\"360ofha<%y*F\\cv7$Ffs$\"3] ,v4Z<5KDFcjn7$F^e\\l$\"3ou7Qm`^IFFcjn7$Fce\\l$\"3Z&o*Q?OGgGFcjn7$Fhe \\l$\"3/I\"39;;4#HFcjn7$F[t$\"3')f))f8O^8HFcjn7$F`f\\l$\"3;Xa))*z0hq#F cjn7$F`t$\"3U#z)\\'>xHF#Fcjn7$Fet$\"3I4(*\\A[VZ))F\\cv7$Fjt$!3E)R\"4!H &4]rF\\cv7$F_u$!3\\%\\G&GiJ_?Fcjn7$Fdu$!3eLBNKV(Gp#Fcjn7$Fdg\\l$!3+\"3 rKe#zBFFcjn7$Fig\\l$!3SOG:%\\6Jq#Fcjn7$F^h\\l$!3?V?i\"*QPKEFcjn7$Fiu$! 3j'GJypsR^#Fcjn7$F[i\\l$!3\\\">&QDH^Z@Fcjn7$F^v$!3+/c?4vjO;Fcjn7$Fcv$! 3!R%\\JglALF,#Fcjn7$Fb w$\"3us(\\9fA=a#Fcjn7$Fhfy$\"32T4LzG'4g#Fcjn7$Fgw$\"3sB'>E&pJ>EFcjn7$F `gy$\"3AN(>/(*)=(f#Fcjn7$F\\x$\"3$z5uhF'[NDFcjn7$Ffx$\"33&e?P04)*H#Fcj n7$F`y$\"3C%=Modg-$>Fcjn7$Fdz$\"3%\\#)Gf#3;**))F\\cv7$F^[l$!31$Q$Go@U+ LF\\cv7$Fc[l$!3I=OdWL(yT\"Fcjn7$Fh[l$!3z)\\h4]GE?#Fcjn7$Fcdw$!3x9P_#e7 rT#Fcjn7$F]\\l$!3u6LBZS@JDFcjn7$$!3:nTNrGj5RFT$!3A)[(pP6^\\DFcjn7$F[ew $!3'HFN()y0>a#Fcjn7$$!3aLe*)f(pe\"QFT$!3/y1#*z3n3DFcjn7$Fb\\l$!3*)*>(R w*H.X#Fcjn7$Fg\\l$!3Byn1'3%>f=Fcjn7$F\\]l$!3-)fbw)>[=()F\\cv7$Ff]l$\"3 p?XvcHNn5Fcjn7$F[^l$\"3SZLA5!G&e>Fcjn7$F`^l$\"3Rk)zl(*>$\\CFcjn7$$!3An \"z>YIEW#FT$\"3!Qp,c\"o4$\\#Fcjn7$$!3%RLekG'[$R#FT$\"36*o=*3TA7DFcjn7$ $!3n+v$46UVM#FT$\"3w'\\Y#FT$\"3xW:LFH?WBFcjn7$Fj^l$\"3DRcqJxO@@Fcjn7$Fd_l$\"35Z>/X*\\YO\"Fcj n7$Fh`l$\"37(y;48T%ROF\\cv7$Ffbl$!3B_#)o_t%*\\:Fcjn7$Fchw$!3Cl+J:T(o#> Fcjn7$F[cl$!3-02I\"3Ps@#Fcjn7$Fja]l$!3%G\"zYO(yeK#Fcjn7$F`iw$!3YTvbSC_ 3CFcjn7$Fbb]l$!3eE\\t!*zHkCFcjn7$F`cl$!3'G:u:[KE\\#Fcjn7$$!3JJ$e*)4Avy (Fin$!3c_;B0^C%\\#Fcjn7$F]h`l$!3&zn(4\\>]sCFcjn7$$!3;K$3-87O!oFin$!33c UFm`jFCFcjn7$Fecl$!3Y/y/M24gBFcjn7$Feh`l$!3MrJksXyf@Fcjn7$Fjcl$!3!4Ao6 *p=z=Fcjn7$F_dl$!3()[)[w`wN0\"Fcjn7$Fddl$!3*o;g-HWI$[Ff`n7$F[el$\"3Wa6 9=59-5Fcjn7$F`el$\"3&eY1@vx%o=Fcjn7$$\"38&)**\\ibQq_Fin$\"3U=!o>(4aX@F cjn7$Feel$\"3L^s0c!GnM#Fcjn7$$\"3U()*\\(o>\")4nFin$\"3r>tAa$*[;CFcjn7$ $\"3_@L$3x?'*=(Fin$\"3gk%>'[7ykCFcjn7$$\"3ibm\"HdH%pwFin$\"3IesVc3:\" \\#Fcjn7$Fjel$\"3U!f6/JP`\\#Fcjn7$F\\d]l$\"3u^.3C+([Z#Fcjn7$Fgjw$\"3/m ;LokiGCFcjn7$Fdd]l$\"3f&)*>Fcjn7$Fdfl$\"3iy=L!*zO_;Fcjn7$Fbhl$!3l['zJ,;3<$F\\cv7$Ffil$ !3#f-!eh>5)H\"Fcjn7$F`jl$!3)>!*>LZ\"Rg?Fcjn7$$\"3/)*\\P%G?\"y@FT$!3HoD vA'o$3BFcjn7$Fejl$!3_gOju)f;Y#Fcjn7$$\"3C)\\PM#\\9IBFT$!3E!*e)\\\\:/]# Fcjn7$$\"3I)*\\7.)>3Q#FT$!36@q\\)RVJ^#Fcjn7$$\"3O)\\7Go%\\JCFT$!35X'>U c(f*\\#Fcjn7$Fjjl$!3nJ)*3dA!)fCFcjn7$F_[m$!3Fr$o/Z9'z>Fcjn7$Fd[m$!3tb8 <\">OV3\"Fcjn7$F^\\m$\"3jYe5&*)[Al*F\\cv7$Fc\\m$\"3IoP]Y=l$*=Fcjn7$Fh \\m$\"3-(4cLu(\\]CFcjn7$Fb^x$\"3]^:#po\\-^#Fcjn7$Fg^x$\"3HRY))*=Fcjn7$Fd`m$!3 rcE;L(4%pAFcjn7$F^am$!3Xq)33))>Y^#Fcjn7$Fhfz$!3<@zhn^Z%e#Fcjn7$Fcam$!3 PGp$H$3+:8H[^LFFcjn7$Fdem$!3N$>[x\"*e>a#Fcj n7$F^fm$!3'3Cx,7HMC*F\\cv7$Fbgm$\"3-*HB9FafL\"Fcjn7$F\\hm$\"3'RX(z<\"> 6C#Fcjn7$Ffhm$\"31$4zB4(p)*GFcjn7$Fh\\[l$\"3#e;H)42\"e4$Fcjn7$F[im$\"3 /115\\DM(=$Fcjn7$F`][l$\"39]Y#[zk,>$Fcjn7$Fe][l$\"3$GV%*pC`H;$Fcjn7$Fj ][l$\"3)f@Gq%R$\\5$Fcjn7$F`im$\"3-o.#yb%e:IFcjn7$F^^u$\"3a%z!Hm(eGG#Fc jn7$Feim$\"3S,EyrJ$>.\"Fcjn7$Fe_u$!3o24Z(e[^k&F\\cv7$Fjim$!3#)R))R&z$3 (>#Fcjn7$Faip$!3s[C9qd>'R$Fcjn7$F_jm$!3%oh)\\Q!y8!QFcjn7$Fiip$!33Br/Pz 'G.$Fcjn7$Fdjm$!3j]a)R#>$3[*F\\cv7$F^au$\"3r.%o:oO87'F\\cv7$Fajp$\"3Iz .Uh#[MH#Fcjn7$Ffau$\"3T[a'\\k#[Fcjn7$F^bu$\"3]<&H(3!Rxj%Fcjn7$Fhix$\"32r=U#=1F)QFcj n7$Fijp$\"3)QG_y=]q^#Fcjn7$Fdg^l$\"39/p-t3^g:Fcjn7$F`jx$\"3(fK^I;D.&RF \\cv7$F\\h^l$!3-e:Olj)4+\"Fcjn7$Ffbu$!39V[AUoj]EFcjn7$Fba[l$!3eF?7&F3-F46&FFQ2The~18~term ~errorF'F>F?FhgdlFecel7&F3-F46&FFQ2The~19~term~errorF'F>F?-FL6$7iw7$FP $\"3o8wV&***H*3'Fcjn7$F^\\y$\"3^UK5'pY_n%Fcjn7$Fiiv$\"3NxK^GY'QV$Fcjn7 $Ff\\y$\"3\\V$4'\\i=^BFcjn7$Fb`s$\"3!R5'R?M299Fcjn7$F^_\\l$\"3,mMadHz, hF\\cv7$Fajv$!3s%o6&oZ&)3sFf`n7$Ff_\\l$!3[\"Ra;&3.OkF\\cv7$Fg_o$!3_la# )pCb96Fcjn7$Fj`s$!3@E@Eo*zU<#Fcjn7$FV$!3>E1Si$>JG#Fcjn7$Fbas$!3@F[&HD0 )G=Fcjn7$F_`o$!3MIWb@f'Q4\"Fcjn7$Fjas$!3k#4]\"G0noFF\\cv7$Fen$\"3!H^Qy *\\*p*[F\\cv7$Fg`o$\"3T??n%QLJ[\"Fcjn7$F[o$\"3t(\\iCX$*4#>Fcjn7$F\\hel $\"371p\"*[+cW>Fcjn7$Fahel$\"3/rcbLSyO>Fcjn7$Ffhel$\"3]/!fBD#z**=Fcjn7 $F_ao$\"3!p8nE*p%f$=Fcjn7$F^iel$\"3$R$*R&4d&zj\"Fcjn7$F`o$\"3V,z/%GAUO \"Fcjn7$F]ds$\"3g5XLp=fQeF\\cv7$Feo$!3WX*)oQ$GBU#F\\cv7$Fjds$!3vUNZ434 %Q*F\\cv7$Fjo$!3Ub.A-ho&R\"Fcjn7$Fc`y$!3_*=%3]pB>:Fcjn7$F_p$!3[!R(R.I4 q:Fcjn7$F^[fl$!3ptzwj]0p:Fcjn7$F`ay$!3'[b&[Bu7^:Fcjn7$Ff[fl$!3cP_S(e`q ^\"Fcjn7$Fdp$!3\\s$QKs*pn9Fcjn7$F^q$!3+tho(*p?Q6Fcjn7$Fhq$!38X-)>;Tza' F\\cv7$F\\s$\"3#zhUo`DDW%F\\cv7$Ffs$\"3?.9`Ahoq7Fcjn7$F^e\\l$\"3N\\#3w ;iQQ\"Fcjn7$Fce\\l$\"3_A4qLwui9Fcjn7$Fhe\\l$\"3U^_qV\"*o1:Fcjn7$F[t$\" 3]cq_kl&e^\"Fcjn7$$!3zM$3-.I&ywFT$\"3)Hhn%\\Km2:Fcjn7$$!3inm\"zHmPl(FT $\"3sO4S_9=\"\\\"Fcjn7$$!3X+]ilD+HwFT$\"3aB@')*zMmY\"Fcjn7$F`f\\l$\"3' )fL]6^GM9Fcjn7$$!3/++vo8ravFT$\"3k0gt1')RZ8Fcjn7$F`t$\"3CED..rFL7Fcjn7 $Fet$\"3)e\"Gv$)e%RX&F\\cv7$Fjt$!3'Hr2!R<0EGF\\cv7$F_u$!3$)f!p7X9U+\"F cjn7$Fdu$!375Hm>0*=Q\"Fcjn7$Fdg\\l$!3KDM2y+&*49Fcjn7$Fig\\l$!3yyu<)y\" )4T\"Fcjn7$F^h\\l$!3l'*=pu]`&Q\"Fcjn7$Fiu$!3y4r(y-\"Fcjn7$Fbw$\"3Kr=eB3]J8Fcjn7$Fhfy $\"3kl#oC3R=P\"Fcjn7$Fgw$\"39O92&\\v0R\"Fcjn7$F`gy$\"3lL4*3<+xQ\"Fcjn7 $F\\x$\"3@3K-R8`j8Fcjn7$Ffx$\"38>(4e\")RTD\"Fcjn7$F`y$\"3;;\"\\x$p)32 \"Fcjn7$Fdz$\"3+L5X]]EK`F\\cv7$F^[l$!3Nu5+PE'o;\"F\\cv7$Fc[l$!3[sGP)ft ]5(F\\cv7$Fh[l$!3Nwzj&GEA:\"Fcjn7$Fcdw$!3PegnQYhy7Fcjn7$F]\\l$!38Z0M0q ]^8Fcjn7$Fjbfl$!3lW[y*f5rO\"Fcjn7$F[ew$!3!3**yaW)oo8Fcjn7$Fbcfl$!3*y^G U()>jN\"Fcjn7$Fb\\l$!3))*f)p&*GAI8Fcjn7$Fg\\l$!3!)*)\\)[!*f:-_!\\8Fcjn7$F[ffl$\"3 yhf()H0J$G\"Fcjn7$Fj^l$\"3enN+QbQn6Fcjn7$Fd_l$\"3IB))\\5^_LwF\\cv7$Fh` l$\"3CK!))HM\"*e?#F\\cv7$Ffbl$!3R$*fM'fP^I)F\\cv7$Fchw$!32x%)e$*y`R5Fc jn7$F[cl$!3E.3>#3[9?\"Fcjn7$Fja]l$!3wS3)4uECE\"Fcjn7$F`iw$!3MM)fYO`\"4 8Fcjn7$Fbb]l$!3cXB;Df7T8Fcjn7$F`cl$!3cq'>0r2!e8Fcjn7$F^hfl$!35P!pTIr+O \"Fcjn7$F]h`l$!3[:)*Rc4E\\8Fcjn7$Ffhfl$!3uF0)3)HpD8Fcjn7$Fecl$!3^)fdY! Gg*G\"Fcjn7$Feh`l$!3,#zT5Ox8=\"Fcjn7$Fjcl$!3K+:kd6tG5Fcjn7$F_dl$!3/1CW 0V7udF\\cv7$Fddl$!31F%=1?b(\\EFf`n7$F[el$\"3o2qWUaW#\\&F\\cv7$F`el$\"3 9D'>G(H*G-\"Fcjn7$F`jfl$\"3Y'[hxz[O<\"Fcjn7$Feel$\"3u0vOUQU#G\"Fcjn7$F hjfl$\"3D%p[Fso(>8Fcjn7$F][gl$\"3i='[4/d_M\"Fcjn7$Fb[gl$\"3mXj;x\"['e8 Fcjn7$Fjel$\"3!y'Gb,*3)f8Fcjn7$F\\d]l$\"3o.*)fk\"*HZ8Fcjn7$Fgjw$\"3u5% >)>*31K\"Fcjn7$Fdd]l$\"3y^koD')**z7Fcjn7$F_fl$\"3'*G*\\n^ueA\"Fcjn7$F_ [x$\"3nVb6>]Tz5Fcjn7$Fdfl$\"31jgws))F\\cv7$Fbhl$!3R$[.Xwa.&>F\\cv7$ Ffil$!3iGu#\\u,ZF(F\\cv7$F`jl$!3Z2^!4y[_8\"Fcjn7$Fe]gl$!3+Ju^jf!\\E\"F cjn7$Fejl$!3(G>-T/]?M\"Fcjn7$F]^gl$!3N*))o:3:(f8Fcjn7$Fb^gl$!3S*H%3Xl2 j8Fcjn7$Fg^gl$!3A#y4GGW?N\"Fcjn7$Fjjl$!3/\\QN4$ymK\"Fcjn7$F_[m$!3\"GR$ )[37'\\5Fcjn7$Fd[m$!3yGNtjZ?7bF\\cv7$F^\\m$\"3)=A6b:'36cF\\cv7$Fc\\m$ \"3r\"**)G'z?00\"Fcjn7$Fh\\m$\"37%o'*y*3II8Fcjn7$Fb^x$\"3N45(3?\")pN\" Fcjn7$Fg^x$\"3qktL[1-p8Fcjn7$F\\_x$\"3n6z9$3(=m8Fcjn7$F]]m$\"3!)o.e8hS [8Fcjn7$Fd_x$\"3L:*y[;6&o7Fcjn7$Fb]m$\"3%)p%f0\")y>8\"Fcjn7$Fg]m$\"3!z $pBp40\\mF\\cv7$F\\^m$\"3&e5P?ux(\\ZFf`n7$Ff^m$!3SBVQ=H*[a&F\\cv7$Fj_m $!3!os$QMp,b5Fcjn7$Fd`m$!3mMu%e\"fRR7Fcjn7$F^am$!3-uy&o\"\\[a8Fcjn7$Fh fz$!3`G;(G%zG$Q\"Fcjn7$Fcam$!3/-<3$*H'>R\"Fcjn7$F`gz$!3js,=#QJ,Q\"Fcjn 7$Fham$!3Q(yb!G#3xM\"Fcjn7$F]bm$!3;yrCr/dJ5Fcjn7$Fbbm$!3)4\"RZ`9LtYF \\cv7$Fgbm$\"3uM&\\$3v`_DF\\cv7$F\\cm$\"3TNKOw0v[#*F\\cv7$Fc]^l$\"3#pI \\DX(Hp6Fcjn7$Facm$\"3Ch1&e_&>O8Fcjn7$F`^^l$\"3f>V'='pv&Q\"Fcjn7$Fe^^l $\"3i,Rr'H83T\"Fcjn7$Fj^^l$\"3Wf[^[(4/T\"Fcjn7$Ffcm$\"3\"HNj#)[C1SQV\\\"Fcjn7$F_em$! 3#\\m^vtUi^\"Fcjn7$Ff`^l$!3xDq=On'e]\"Fcjn7$F[a^l$!3vzm_RQPi9Fcjn7$F`a ^l$!3a!R->Y>cQ\"Fcjn7$Fdem$!3%=Ls;?0iF\"Fcjn7$F^fm$!35q+A,:=jTF\\cv7$F bgm$\"3Hew^/^8:sF\\cv7$F\\hm$\"3'=-W'>2ie6Fcjn7$Ffhm$\"3/ZavVFBh9Fcjn7 $$\"30P)3:Fcjn7$Fh\\[l$\"3vp^N\\*yNa\"Fcjn7$$\"3 W*\\Pf.YZh)FT$\"3BNF`%HdZc\"Fcjn7$F[im$\"3lCTlsnvr:Fcjn7$F`][l$\"3al6w Y(eSc\"Fcjn7$Fe][l$\"3.,mj;BDT:Fcjn7$Fj][l$\"3W;Iv9_/.:Fcjn7$F`im$\"3U >uK$zu#\\9Fcjn7$F^^u$\"3\"odinghw/\"Fcjn7$Feim$\"3PH&RkYX^)RF\\cv7$Fe_ u$!3pE(*pxmx_SF\\cv7$Fjim$!3=ZMQU;*f?\"Fcjn7$Faip$!3^0f#*\\5TtFcjn7$Fiip$!3o,7,?(4'>:Fcjn7$Fdjm$!3'p)*=i9Jfr%F\\cv7$F ^au$\"3)\\boh%yw[HF\\cv7$Fajp$\"3W0nl1.;46Fcjn7$Ffau$\"39_$yDrV)Q=Fcjn 7$Fijm$\"3lwlA!)*pfG#Fcjn7$F^bu$\"3'Gj%G,(y$p@Fcjn7$Fijp$\"3EVfM^#[P5 \"Fcjn7$Fdg^l$\"3TY;%>N?-K'F\\cv7$F`jx$\"3AB!*\\I?4:gFf`n7$F\\h^l$!3(y v*HN!)p>iF\\cv7$Ffbu$!3OM\\xRA7D9Fcjn7$Fba[l$!3L\\:F')))zgBFcjn7$Fhjx$ !3\"eR[ItP7W$Fcjn7$Fja[l$!336el0&e%zYFcjn7$F^[n$!3o8wV&***H*3'FcjnFb[n -FL6$7_o7$FP$!38`)**z:w0y%FI7$Fb`s$!3CO>,lBbpUFI7$Fg_o$!37Xg)oVC2\"QFI 7$Fj`s$!31/>`IG,*R$FI7$FV$!3V'e:3Z2)HIFI7$F_`o$!3k$3:KdzDS#FI7$Fen$!3- tf#G&=2+>FI7$F[o$!3o-HHI.wa7FI7$F`o$!3cT'\\=s^W@)FT7$Feo$!3c6ve3CDY]FT 7$Fjo$!3T@5e)=NI1$FT7$Fdp$!3As<@m>qH=FT7$Fhq$!3_OBUoWVy5FT7$F\\s$!3n5h 5j$[QG'Fin7$Ffs$!3Y,6;UH43OFin7$F`t$!3[40&=:ArB\"Fin7$Fjt$!3)f\"[L5]:P QFidl7$Fdu$!3zY%3Am`71\"Fidl7$F^v$!3aq`sE:W1FFfdl7$Fhv$!3qkAEmGl(*fFcj n7$Fbw$!3M,)y2Y9FW\"Fcjn7$F`y$!3*eg!oo()4bDF\\cv7$F^[l$!3o\"4>hlQ+$QFf `n7$Fh[l$!3CQ^X%Fgcv7$Fh`l$\"3?0#GhNO\"4uFgcv7$Ffbl$\"3wqc*4*>C7!*Fgcv7$F`cl$\" 3M&>OXp_5:)Fgcv7$Fjcl$\"3FqEU#)G7N]FgcvFhcv7$F`el$!3ZPI`_*o8+&Fgcv7$Fj el$!3fDjqaQ!43)Fgcv7$Fdfl$!3sSC]hB)4/*Fgcv7$Fbhl$!3)zoJ))G!RmuFgcv7$F` jl$!3ylkaV'Q%*f%Fgcv7$Fjjl$!3^(=u1d#GkDFgcv7$Fd[m$!3u?\"QQ&)fH$))Fgjr7 $F^\\m$\"3/34;#[z.A%F^hu7$Fh\\m$\"3i15h!\\=Jm'Fccv7$Fb]m$\"3cjxE[TPTaF bg[l7$F\\^m$\"3Dg:?ol-^UFf`n7$Fj_m$\"3qVt*oPREZ#F\\cv7$Fham$\"3W4*4Si6 PM\"Fcjn7$Fbbm$\"3;'>k;nF:@'Fcjn7$F\\cm$\"3aGRk%)ojbFFfdl7$Ffcm$\"3qG< [Tq<_5Fidl7$F`dm$\"3zI\\KCG30QFidl7$Fjdm$\"3q1=2Z))Gj7Fin7$Fdem$\"3c;O L&>_!)e$Fin7$F^fm$\"3ub]d-EbwjFin7$Fbgm$\"3q\\8Ir7\\:6FT7$Ffhm$\"3Ec2T K4K;=FT7$F`im$\"3KNDrm&4R#HFT7$Feim$\"3Yr'=<@;(*z%FT7$Fjim$\"3F97^Z7O' y(FT7$F_jm$\"3fF]9)3,UA\"FI7$Fdjm$\"3b+3\"RYzd!>FI7$Fajp$\"3Ofb\"4\"z# zS#FI7$Fijm$\"3![%G(***3DMIFI7$F^bu$\"3q7ks(fHFS$FI7$Fijp$\"3YF?Fd^hlF^fjl-%%VIEWG6$;$!#5F9Fg[n;$F9F9$F-!\"#-%*GRIDS TYLEG6#%,RECTANGULARG" 1 2 2 1 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" " Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curv e 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27 " "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "C urve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Cu rve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve \+ 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Cur ve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 7 8" "Curve 79" "Curve 80" }}{TEXT 213 0 "" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 210 10 "Exercises:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 209 208 "1) \+ For our example function, what degree Taylor approximation do we need to be within 0.1 on the domain [-0.5, 0.5]? What degree Legandre app roximation do we need to be within 0.1 on the domain [-0.5, 0.5]?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 101 "2) With the same function what is the maximum error of \+ a 20th degree Taylor approximation on [-1, 1]?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 219 "3) W ith the same function, what degree approximation do we need to be with in 0.1 on the domain [-1, 1]? What degree for an accuracy of 0.01 on \+ the same domain? What degree for an accuracy of 0.001 on the same dom ain?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 209 212 "4) Repeat the process above for the function s in(2\271x) + cos(\271x). What degree Legandre approximation gives an \+ accuracy of 0.01 on the whole domain? (Be warned that you need to rec ompute the coefficients.)" }}}{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 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }