{VERSION 6 1 "Windows XP" "6.1" } {USTYLETAB {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 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 0 0 0 1 }1 1 0 -1 3 3 1 0 1 0 2 2 -1 3 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "MS Serif" 1 12 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }1 1 0 -1 0 0 1 0 1 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 " MS Serif" 1 14 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }1 1 0 -1 0 0 1 0 1 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "A uthor" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 1 0 -1 8 8 1 0 1 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "MS Serif" 1 16 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }1 1 0 -1 8 2 1 0 1 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "MS Serif" 1 18 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }1 1 0 -1 8 4 1 0 1 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 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 0 0 1 1 2 2 2 0 0 0 1 }3 1 0 -1 12 12 1 0 1 0 2 2 -1 1 }{PSTYLE "Map le Output" -1 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 -1 3 3 1 0 1 0 2 2 -1 5 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 -1 3 3 1 0 1 0 2 2 -1 2 } {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 0 0 0 2 1 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "MS Serif" 1 12 0 128 128 1 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "Dictionary Hyperlink" -1 45 "MS Serif" 1 12 147 0 15 1 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 201 "Courier" 1 12 200 0 200 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times " 1 10 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle1" -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 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle1" -1 202 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle2" -1 201 1 {CSTYLE "" -1 -1 "Couri er" 1 12 255 0 0 1 0 1 0 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "_cstyle2" -1 203 "Courier" 1 12 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{PSTYLE "_pstyle3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }3 3 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{CSTYLE "_csty le3" -1 204 "Times" 1 12 0 0 255 1 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_psty le4" -1 203 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle5" -1 204 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{CSTYLE "_cstyle4" -1 205 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 202 37 "Andrew J. Sommese, Sept ember 29, 2004" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 149 "Let's now d o the whole procedure for finding the points and weights of for Gaussi an integration with an arbitrarily prescribed weight, h(t) on [a,b]." }{TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 13 "Digit s := 17;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I'D igitsG6\"\"#<" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 22 "a := -1;b:=1; #b:= Pi;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"!\"\"" }{TEXT 204 0 "" }}{PARA 202 "" 1 " " {XPPMATH 20 "6#>I\"bG6\"\"\"\"" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 20 "f := x -> exp(-x*x);" }{MPLTEXT 1 203 0 " " }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I\"xGF%F%6$I)operato rGF%I&arrowGF%F%-I$expG6$I*protectedGF/I(_syslibGF%6#,$*&9$\"\"\"F4F5! \"\"F%F%F%" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 32 "h := t -> 1;#sqrt(t);#sin(t)^2; " }{MPLTEXT 1 203 0 "" }} {PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"hG6\"f*6#I\"tGF%F%6$I)operatorGF% I&arrowGF%F%\"\"\"F%F%F%" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 29 "Here is the new inner product" }{TEXT 202 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 48 "IP := proc(f,g) evalf(In t(f*g*h(t),t=a..b)) end;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I#IPG6\"f*6$I\"fGF%I\"gGF%F%F%F%-I&evalfGI*protectedGF ,6#-I$IntGI(_syslibGF%6$*(9$\"\"\"9%F4-I\"hGF%6#I\"tGF%F4/F9;I\"aGF%I \"bGF%F%F%F%" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 5 "n:=4;" } {MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"nG6\"\"\"%" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 17 "p := array(0..n);" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 10 "p[0] := 1:" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 14 "j:='j':k:='k':" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 25 "for j from 0 to n-1 do; " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 13 "vv := t*p[j]:" } {MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 22 "for k fro m 0 to j do; " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 45 "vv := vv-IP(t*p[j],p[k])/IP(p[k],p[k])*p[k]: " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 22 "od:p[j+1]:=expand(vv ):" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 3 "od:" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"pG6\"-I&ar rayGI*protectedGF(6$;\"\"!\"\"%7\"" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 30 "for j from 0 to n do p[j]; od;" } {MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#\"\"\"" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#I\"tG6\"" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#,&*$I\"tG6\"\"\"#\"\"\"$!2MLLLLLLL$ !# " 0 "" {MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 46 "tempX:= vector(n):x := vector(n);y:=vector(n);" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 23 "tempX:= \+ fsolve(p[n],t):" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 40 "for j from 1 to n do x[j] :=tempX[j];od;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"xG6\"-I&arrayGI*protectedGF( 6$;\"\"\"\"\"%7\"" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6# >I\"yG6\"-I&arrayGI*protectedGF(6$;\"\"\"\"\"%7\"" }{TEXT 204 0 "" }} {PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"\"$!2d_Sf6j8h)!#<" } {TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"#$!2 Fc[eV5)*R$!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I \"xG6\"6#\"\"$$\"2Fc[eV5)*R$!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"%$\"2d_Sf6j8h)!#<" }{TEXT 204 0 "" }}} {EXCHG {PARA 200 "" 0 "" {TEXT 202 24 "And the weights on [a,b]" } {TEXT 202 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 28 "w:= vector(n); j:='j';k:='k';" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 21 "for j from 1 to n do " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 20 "for k from 1 to n do" }{MPLTEXT 1 203 0 "" }} {PARA 201 "> " 0 "" {MPLTEXT 1 203 40 "if (k=j) then y[k]:=1; else y[k ]:=0; fi;" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 3 "od;" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 28 " w[j] := IP(interp(x,y,t),1);" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 3 "od;" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"wG6\"-I&arr ayGI*protectedGF(6$;\"\"\"\"\"%7\"" }{TEXT 204 0 "" }}{PARA 202 "" 1 " " {XPPMATH 20 "6#>I\"jG6\"F$" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"kG6\"F$" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"\"$\"2!QXP^%[&yM!#<" }{TEXT 204 0 "" }} {PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"#$\"2+YD'[:X@l!#<" } {TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"$$\" 2BYD'[:X@l!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I \"wG6\"6#\"\"%$\"2(QXP^%[&yM!#<" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 " " 0 "" {TEXT 202 65 "And we compute an integral and compare it to the \+ actual integral." }{TEXT 202 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 8 "j:= 'j':" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 33 "GInt := sum(w[j]*f(x[j]),j=1..n);" }{MPLTEXT 1 203 0 "" }} {PARA 201 "> " 0 "" {MPLTEXT 1 203 38 "trueInt:=evalf(Int(f(t)*h(t),t= a..b));" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 13 "GInt-trueInt;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 " 6#>I%GIntG6\"$\"2(Q&\\CiML\\\"!#;" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I(trueIntG6\"$\"2T&[il#[O\\\"!#;" }{TEXT 204 0 "" }} {PARA 202 "" 1 "" {XPPMATH 20 "6#$!.aJvJk8$!#;" }{TEXT 204 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 6 "n:=10;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"nG6\"\"#5" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 17 "p := array(0..n);" } {MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 10 "p[0] := 1 :" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 14 "j:='j ':k:='k':" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 25 "for j from 0 to n-1 do; " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 13 "vv := t*p[j]:" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 22 "for k from 0 to j do; " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 45 "vv := vv-IP(t*p[j],p [k])/IP(p[k],p[k])*p[k]: " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 22 "od:p[j+1]:=expand(vv):" }{MPLTEXT 1 203 0 "" }} {PARA 201 "> " 0 "" {MPLTEXT 1 203 3 "od:" }{MPLTEXT 1 203 0 "" }} {PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"pG6\"-I&arrayGI*protectedGF(6$;\" \"!\"#57\"" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 30 "for j from 0 to n do p[j]; od:" }{MPLTEXT 1 203 0 "" }}} {EXCHG {PARA 200 "" 0 "" {TEXT 202 193 "Here we compute the roots of t he nth polynomial. There are fast solvers adapted to orthogonal polyn omials --- if there is time we will discuss in class when we get to nu merical linear algebra." }{TEXT 202 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 46 "tempX:= vector(n):x := vector(n):y:=vector(n):" } {MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 23 "tempX:= f solve(p[n],t):" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 40 "for j from 1 to n do x[j] :=tempX[j];od;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"\"$!2wrr^Gl!R(*!#< " }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"# $!2\\%)*)omL1l)!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6 #>&I\"xG6\"6#\"\"$$!2KC!*Ho&4%z'!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"%$!2?Z#HTR&RL%!#<" }{TEXT 204 0 "" } }{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"&$!2?J;)*QV()[\"!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"'$ \"2?J;)*QV()[\"!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6 #>&I\"xG6\"6#\"\"($\"2?Z#HTR&RL%!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\")$\"2KC!*Ho&4%z'!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"\"*$\"2\\%)*)omL1l)! #<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"xG6\"6#\"# 5$\"2wrr^Gl!R(*!#<" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 24 "And the weights on [a,b]" }{TEXT 202 0 "" }}{PARA 201 "> " 0 " " {MPLTEXT 1 203 28 "w:= vector(n):j:='j':k:='k':" }{MPLTEXT 1 203 0 " " }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 21 "for j from 1 to n do " } {MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 20 "for k fro m 1 to n do" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 40 "if (k=j) then y[k]:=1; else y[k]:=0; fi;" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 3 "od;" }{MPLTEXT 1 203 0 "" }} {PARA 201 "> " 0 "" {MPLTEXT 1 203 28 "w[j] := IP(interp(x,y,t),1);" } {MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 3 "od;" } {MPLTEXT 1 203 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"\"$\"2<3p3VMrm'!#=" }{TEXT 204 0 "" }} {PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"#$\"2'oa]\"\\8X\\\"!# <" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\" $$\"2Agf^ij3>#!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6# >&I\"wG6\"6#\"\"%$\"2W\"**4$>nEp#!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"&$\"2PdZrCU_&H!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"'$\"2U_ZrCU_&H!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"($ \"2x'**4$>nEp#!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6# >&I\"wG6\"6#\"\")$\"2myf^ij3>#!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"\"*$\"2\">e]\"\\8X\\\"!#<" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>&I\"wG6\"6#\"#5$\"2Fuo3VMrm'!#= " }{TEXT 204 0 "" }}}{PARA 200 "" 0 "" {TEXT 202 65 "And we compute an integral and compare it to the actual integral." }{TEXT 202 0 "" }} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 8 "j:= 'j':" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 33 "GInt := sum(w[j]*f(x [j]),j=1..n);" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 38 "trueInt:=evalf(Int(f(t)*h(t),t=a..b));" }{MPLTEXT 1 203 0 "" } }{PARA 201 "> " 0 "" {MPLTEXT 1 203 13 "GInt-trueInt;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I%GIntG6\"$\"2/KCcE[O\\\" !#;" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I(trueIntG6\" $\"2T&[il#[O\\\"!#;" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 " 6#$!%P`!#;" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 15 " with(student):" }{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 43 " evalf(simpson(f(x), x=-1..1, 10))-trueInt;" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#$\"-!Qe>We#!#; " }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 1 " " } {MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 44 " \+ evalf(simpson(f(x), x=-1..1, 100))-trueInt;" }{MPLTEXT 1 203 0 "" }} {PARA 202 "" 1 "" {XPPMATH 20 "6#$\")=y:E!#;" }{TEXT 204 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 0 "" }}}{PARA 204 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }