{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 52 "Integration Using the G auss Quadrature Rule - Method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 78 "2004 Autar Kaw, Loubna Guennoun, Univer sity of South Florida, kaw@eng.usf.edu," }}{PARA 257 "" 0 "" {TEXT -1 35 "http://numericalmethods.eng.usf.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "NOTE: This worksheet demonstrates t he use of Maple to illustrate the Gauss Quadrature rule of integration ." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Section I: Introduction" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 312 "Gauss Qu adrature rule is another method of estimating an integral. The two poi nt Gauss Quadrature Rule is an extension of the Trapezoidal Rule appro ximation where the integral estimate was based on taking the area unde r the straight line connecting the function values at the limits of th e integration interval, " }{TEXT 264 1 "a" }{TEXT -1 5 " and " }{TEXT 265 1 "b" }{TEXT -1 209 ". However, unlike the Trapezoidal Rule approx imation, the two point Gauss Quadrature rule is based on evaluating th e area under a straight line connecting two points on the curve that a re not predetermined as " }{TEXT 266 1 "a" }{TEXT -1 5 " and " }{TEXT 267 1 "b" }{TEXT -1 18 ", but as unknowns " }{OLE 1 3605 1 "[xm]Br=Wfo RrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::NDYmq^H;C:ELq^H_mvJ::::::::gjl?Xxoq_r:fB]mtFFcmnvGWMJ nC==nHE=;:::::JJNZ:vyyuy:>:<::::::=J:>A>:F:AlqfG[maNFO=;::::::::_J;Zy= J:B::::::N:;B:G=;:wAwAA:j:>b=HCsf[M<;j;@j:HZDFZLVjrs:YR:hj:DJ>A=]Z:>`;B:<:F:wyyAbr:B:bKi:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCF;B=BKaDBETV:;r:vn;B:id:r:__:N:e:;j<>:Mb:B:C: ?R:=j ryyYGwyyyy;T:nr;j?<:G;Sj`@Pt\\Pd`QrP@[lDB:qi:;sy>Z:JBA:;B:Cb:;N`D>v;>v:FZ:B:]C:e:qQ:uY;vCS=[LsfFaMR>`:J:<::: 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyy ya:nYf::G:I:K:wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::J:ElxQYQDv]Aj;JZ]Z:B:F:YLpfF>:::::::::J?NZ;vyyyyyY:vYxY:B::: ::::c:;:ElrfH=MtFGYMq>>Wlj:gmlJ::::::>>?B:yyyxI:;Z::::::j:>:;Y:B:F:Alq fG[maNFO=;::::::::_J;Zy=J:B::::::N:;B:G=;:wAK:AJ:nYf:n:v::M:OJ:V;^ ;;j@>:WJ:nY><;jBJC>:a:c:e:gJ:v<>=F=N=V=^=f=n=v=>>b=HCsf[M<;j;@j:HZDFZ=>=@JCHRmNZ<@?_rZkM;DrNN\\=`F_rZkm:nyyY ZDjysy?bm?:;JZC:bKi:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCF;B=BKaDBETV:;r:yayQZ:J:j^mm?j:F;HJ`E:?jDB:;j<> :Mb:>Z:^:NZ;F:E:=b:yyyyI:C:WS:kjryyYAwyyyy;ED^;UTRcETcTX[USR:[B:B=B:?ja^G>D_mlVH[KRJ:<::: ::::>=?R:>Z:N:;jysy;J:<::::::C:_X;>Z:J;vCJbNHVH>@>Z::::::::kJ:vYxI:;Z: :::::JywYB:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<::::::::?b[uK CHrkVZIF:QR:?B\\][Kbb[QKCHB:\\rR:_rZkm;x::J;Z==b:KfF>ZDN:rZ:B\\a[KJC::];V::H=;ZN:K;==:[[ KZ@\\=:DrN>ZEFZLV j[C:=b>D@Ub:GvD?[:>::::::b:;b::Cb::CB:>X=J>JS>[;VD=Z:NFK:_KjLjdF: FFe:qQ:K:<:[>;JX;J>JS>RebC=J:;N@Fs^F:FFK:_;EWB=J:>IJ Sj_IJSjxlk:jj:_;QX@=:==N@FyQF:;jkJSjAMk:Jv:_;At>=:== N@FcJF:>IJSjquj:jj:_;qE==:k=N@VcEF:FFJSjF]j:jj:_;]i;=J:Vf::_;AB;?J:^fr ?;N@frJSjLTk:jN:_;YB<=:K;N@Fi=F:F?JSjNDj:>:KkDjw?>B:>LB:Cb:^D::: f?=J;bYkhB=J:B:ZD>>_dB=: <:b<>\\[F:;Z::d:odA=:<:b;N@^kTF:>@K:_;GG@=:<:b>uH>=:]C:JSJTyj:Z::d:YR==:];N@vxEF:F@JSJg `j:JR:_;Ev;=:]K?N;yayI:>:[Z:VY[j=B:;JXE:: C:[q:VZ:B:;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::2:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Equating th e last two equations gives" }}{PARA 0 "" 0 "" {OLE 1 7189 1 "[xm]Br=Wf oRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:K:M:O:Q :S:wAyA::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::_lqvGcMJ:::::::JEf:yyyxI^:NZHQ:R<:T><::[E 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::::::::F:D:<::::::::::s:?jVIXDD:<:CDZMLp:;j;@j:HZD^Z;FZ;N\\=PF?b[ _bB_rZ=M;TrXN\\=XFAr==j?@j:nyyYZDjysy?bm?:;JZC:bKi:UTTAeVYuVYeScEBETVe URcUTYeU;sFWCF;B=BKaDBETV:;rZ:::::::yayY:^:;j:jysy?B: >:nqHJ[r:Oi:VZ:f<DZ:>Z:^:NZ;F:E:=b:yyyyI:C:WS:kjryyYewyyyy;S=fhHB:QB:n>^;UT RcETcTX[US>j=F;<<:N>C:US:f:Nl>f:^fZ:JDNwFN:YLpJbNHEms>@[C:>Z::::::::kJ;@:;B:?B:;B:yayA:;B::::::^:n[@ J:<:?ja:[@>Z::::::::kJ::<::::::wqy[:::::::::::::vYxI:;Z:::: ::::::::::::::::yay=J:B::::::::NZBH@@[DNZ@X?_rZ:QJ;\\RSR>Z=v;:J:dj:DJ>[AFXHjA::DZej;;J>E=]ZL :::;b:ZmZDN:_:::B:OR:?:N::F:jRR>d:::j;x:ZlJ:dJ;V?HB:\\B@;_:::V::J :kljBb>::Z=:jSJ:::: JZf:f^yM:<:[V:>Z:^Z<>ZJ^dcgg_WhZnc_whZNdigg[oG@R<>Z:f?=J >>:_c;N@OW=QB==:ZDj^ej:Z::dJ;?aBF:V@JSNOOx;=J:F`:>;N@ Np@F:F@JS^oEFw=F:;JT>;N@V^;FZ:>:c;N@IDJSJEG:F@JSFhA^t=F:;jS:_;sR:=B:V@JSNa?VlEF:B:ZDjbd j:Z::d:Ox;=:]K>JSJSUj:jR:_KXHjnIj:>:<:bx;F:K:_KutjADj:>:c;N@qc<=b:=:a;N@Gb;Yb:=:< jDjw;;B:>L=J:^:^D::RZjPF:C:[Q;JSdJ_pJihj:Jv>;N@^]DF:VFK:_;gs;=:k=N@ 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"" 20 "6\"" }0 }{CELL 92 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 92 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 93 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 93 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 94 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 94 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 95 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 95 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 96 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 96 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 97 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 97 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 98 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 98 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 99 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 99 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 100 2 {CELLOPTS 0 -1 7 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }{CELL 100 3 {CELLOPTS 0 -1 9 0 0 0 0 0 } {R5MATHOBJ "" 20 "6\"" }0 }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C:=array(1..4,1..4):\n" }{TEXT -1 64 "The weighting factor for Gauss Quadrature Rule with one point is" }{MPLTEXT 1 0 12 "\nC[1,1]:=2;\n" }{TEXT -1 67 "The weighting factors \+ for Gauss Quadrature Rule with two points are" }{MPLTEXT 1 0 1 " " } {TEXT -1 0 "" }{MPLTEXT 1 0 22 "\nC[1,2]:=1;C[2,2]:=1;\n" }{TEXT -1 69 "The weighting factors for Gauss Quadrature Rule with three points \+ are" }{MPLTEXT 1 0 62 "\nC[1,3]:=0.555555556;C[2,3]:=0.888888889;C[3,3 ]:=0.555555556;\n" }{TEXT -1 67 "The weighting factors for Gauss Quadr ature Rule with four point are" }{MPLTEXT 1 0 81 "\nC[1,4]:=0.34785484 5;C[2,4]:=0.652145155;C[3,4]:=0.652145155;C[4,4]:=0.347854845;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6$\"\"\"F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6$\"\"\"\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6$\"\"#F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6$\"\"\"\"\"$$\"*cbbb&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"CG6$\"\"#\"\"$$\"**))))))))!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"CG6$\"\"$F'$\"*cbbb&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"CG6$\"\"\"\"\"%$\"*X[&yM!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"CG6$\"\"#\"\"%$\"*b^9_'!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"CG6$\"\"$\"\"%$\"*b^9_'!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"CG6$\"\"%F'$\"*X[&yM!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "X:=array(1..4,1..4):\n" }{TEXT -1 65 "The function argument for Ga uss Quadrature Rule with one point is" }{MPLTEXT 1 0 12 "\nX[1,1]:=0; \n" }{TEXT -1 68 "The function arguments for Gauss Quadrature Rule wit h two points are" }{MPLTEXT 1 0 43 "\nX[1,2]:=-0.577350269;X[2,2]:=0.5 77350269;\n" }{TEXT -1 71 "The function arguments for Gauss Quadrature Rule with three points are\n" }{MPLTEXT 1 0 52 "X[1,3]:=-0.774596669; X[2,3]:=0;X[3,3]:=0.774596669;\n" }{TEXT -1 70 "The function arguments for Gauss Quadrature Rule with four points are\n" }{MPLTEXT 1 0 82 "X [1,4]:=-0.861136312;X[2,4]:=-0.339981044;X[3,4]:=0.339981044;X[4,4]:=0 .861136312;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"\"F'\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"\"\"\"#$!*p-Nx&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"#F'$\"*p-Nx&!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"\"\"\"$$!*pmfu(!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"#\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"$F'$\"*pmfu(!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"XG6$\"\"\"\"\"%$!*7j8h)!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"#\"\"%$!*W5)*R$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"$\"\"%$\"*W5)*R$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6$\"\"%F'$\"*7j8h)!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Section II: Data " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "The \+ following simulation will illustrate the Gauss Quadrature Rule of inte gration. This section is the only section where the user may interacts with the program." }}{PARA 0 "" 0 "" {TEXT -1 32 "The user may enter \+ any function " }{TEXT 257 4 "f(x)" }{TEXT -1 366 ", and the lower and \+ upper limit for the function. By entering these data, the program will calculate the exact value of the integral, followed by the results us ing the Gauss Quadrature Rule with 2, 3, and 4 points. The program wil l also display the true error, the absolute relative true % error, the approximate error, and the absolute relative approximate % error." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Fu nction in " }{TEXT 258 7 "f(x)=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x->x*exp(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%$expG6#,$*&\"\"$F .F-F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The lower limi t of the integral (" }{TEXT 259 1 "a" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:=0.2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The upper limit of the integral (" }{TEXT 260 1 "b" }{TEXT -1 2 ") :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b:=0.9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"\"*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "This is the end of the user's section. All information m ust be entered before proceeding to the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Section III: The exact value of the integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "In this section, the program will evaluat e the exact value for the integral of the function " }{TEXT 261 4 "f(x )" }{TEXT -1 25 " evaluated at the limits " }{TEXT 262 1 "a" }{TEXT -1 5 " and " }{TEXT 263 1 "b" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s_exact:=int(f(x),x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(s_exactG$\"+]!*f\"*G!\"*" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 69 "Section IV: The value of the integral using the Gauss Q uadrature Rule" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "Conversion of t he limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "The integral given above has the limits of [a,b]. It nee ds to be converted into an integral with limits [-1,1]" }}{PARA 0 "" 0 "" {TEXT -1 106 "f_new(x) is the new function that will be used for \+ evaluating the integral using the Gauss Quadrature rule" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f_new :=x->f((b-a)/2*x+(b+a)/2)*(b-a)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&f_newGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"#F/*&-%\"fG 6#,(*&F.F/*&,&%\"bGF/%\"aG!\"\"F/9$F/F/F/*&F0F;F9F/F/*&F0F;F:F/F/F/F8F /F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "One Point" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " s_1:=f_new(X[1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_1G$\"+ " 0 "" {MPLTEXT 1 0 18 "AV[1]:=C[1,1]*s_1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#AVG6#\"\"\"$\"+Mso/?!\"*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 28 "The approximate error (E_a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E_a[1]:=undefined;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$E_aG6#\"\"\"%*undefinedG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 60 "The absolute approximate percentage relative error (E_a rel):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "E_arel[1]:=undefin ed;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6#\"\"\"%*undefinedG " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Two Points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s_1:=f_new(X[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_1G$\"++HKeM!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s_2:=f_new(X[2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$s_2G$\"+'))*)H^#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The \+ approximate value of the integral using two-points Gauss quadrature ru le is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "AV[2]:=C[1,2]*s_1+ C[2,2]*s_2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#AVG6#\"\"#$\"+w@#)e G!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The approximate error (E _a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "E_a[2]:=AV[2]-AV[1] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$E_aG6#\"\"#$\"*U\\8a)!\"*" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The absolute approximate percenta ge relative error (E_arel):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "E_arel[2]:=abs(E_a[2]/AV[2]*100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6#\"\"#$\"+Kir()H!\")" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "Three Points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s_1:=f_new(X[1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_1G$\"+56a `A!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s_2:=f_new(X[2,3]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_2G$\"+ " 0 "" {MPLTEXT 1 0 19 "s_3:=f_new(X[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_3G$\"+c=.vL!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "The approximate value of the integral using three-points \+ Gauss quadrature rule is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "AV[3]:=C[1,3]*s_1+C[2,3]*s_2+C[3,3]*s_3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#AVG6#\"\"$$\"+Jl=\"*G!\"*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 28 "The approximate error (E_a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "E_a[3]:=AV[3]-AV[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$E_aG6#\"\"$$\")bVOK!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The absolute approximate percentage relative error (E_arel):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "E_arel[3]:=abs(E_a[3]/AV[3]* 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6#\"\"$$\"+%G9%>6! \"*" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Four Points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s_1:=f_new(X[1,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_1G$\"+S5JM=!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s_2:=f_new(X[2,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$s_2G$\"++ur'\\&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " s_3:=f_new(X[3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_3G$\"+\\! \\Au\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s_4:=f_new(X[ 4,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s_4G$\"+ZZTKQ!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The approximate value of the integ ral using four-points Gauss quadrature rule is" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "AV[4]:=C[1,4]*s_1+C[2,4]*s_2+C[3,4]*s_3+C[4,4] *s_4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#AVG6#\"\"%$\"+ukf\"*G!\"* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The approximate error (E_a): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "E_a[4]:=AV[4]-AV[3];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$E_aG6#\"\"%$\"'V*4%!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The absolute approximate percentag e relative error (E_arel):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "E_arel[4]:=abs(E_a[4]/AV[4]*100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'E_arelG6#\"\"%$\"+JZq<9!#6" }}}{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 }