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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 55 "Interpolation with unevenly space
d interpolation points" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone,
 Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  24.3.
2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "load root-finding procedure
s including: " }{TEXT 0 9 "allbisect" }}{PARA 0 "" 0 "" {TEXT -1 17 "T
he Maple m-file " }{TEXT 276 7 "roots.m" }{TEXT -1 37 " contains the c
ode for the procedure " }{TEXT 0 6 "bisect" }{TEXT -1 25 " used in thi
s worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a M
aple session by a command similar to the one that follows, where the f
ile path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Chebyshev polynomials" }}
{PARA 0 "" 0 "" {TEXT -1 105 "\nThe Chebyshev polynomials of the 1st k
ind are based on the following family of trigonometric expansions." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
99 "u := 'u':\nexpand(cos(2*u));\nexpand(cos(3*u));\nexpand(cos(4*u));
\nexpand(cos(5*u));\nexpand(cos(6*u));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*$)-%$cosG6#%\"uG\"\"#\"\"\"F*F+!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*$)-%$cosG6#%\"uG\"\"$\"\"\"\"\"%*&F*F+F&F+!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)-%$cosG6#%\"uG\"\"%\"\"\"\"\")*&F
,F+)F&\"\"#F+!\"\"F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)-%$cosG
6#%\"uG\"\"&\"\"\"\"#;*&\"#?F+)F&\"\"$F+!\"\"*&F*F+F&F+F+" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,**$)-%$cosG6#%\"uG\"\"'\"\"\"\"#K*&\"#[F+)F&\"
\"%F+!\"\"*&\"#=F+)F&\"\"#F+F+F+F1" }}}{PARA 0 "" 0 "" {TEXT -1 150 "
\nIf we replace each occurrence of cos(u) by x, we get the family of C
hebyshev polynomials of the 1st kind.\nThis can be done automatically \+
as follows.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "simplify(e
xpand(cos(2*arccos(x))));\nsimplify(expand(cos(3*arccos(x))));\nsimpli
fy(expand(cos(4*arccos(x))));\nsimplify(expand(cos(5*arccos(x))));\nsi
mplify(expand(cos(6*arccos(x))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
&*$)%\"xG\"\"#\"\"\"F'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)
%\"xG\"\"$\"\"\"\"\"%*&F'F(F&F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,(*$)%\"xG\"\"%\"\"\"\"\")*&F)F()F&\"\"#F(!\"\"F(F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"&\"\"\"\"#;*&\"#?F()F&\"\"$F(!\"\"*&
F'F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"'\"\"\"\"#
K*&\"#[F()F&\"\"%F(!\"\"*&\"#=F()F&\"\"#F(F(F(F." }}}{PARA 0 "" 0 "" 
{TEXT -1 10 "\nThus the " }{TEXT 266 1 "n" }{TEXT -1 57 " th Chebyshev
 polynomial of the first kind is defined by " }{XPPEDIT 18 0 "T(n,x) =
 cos(n*arccos(x))" "6#/-%\"TG6$%\"nG%\"xG-%$cosG6#*&F'\"\"\"-%'arccosG
6#F(F-" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 35 "By using the \+
trigonometric formula:" }}{PARA 256 "" 0 "" {TEXT -1 1 "\n" }{TEXT 23 
8 "        " }{XPPEDIT 18 0 "cos(n*theta) + cos((n-2)*theta) = 2 cos(t
heta) * cos ((n-1)*theta)" "6#/,&-%$cosG6#*&%\"nG\"\"\"%&thetaGF*F*-F&
6#*&,&F)F*\"\"#!\"\"F*F+F*F**(F0F*-F&6#F+F*-F&6#*&,&F)F*F*F1F*F+F*F*" 
}{TEXT -1 3 ", \n" }}{PARA 0 "" 0 "" {TEXT -1 68 "one can check that t
hey satisfy the following recurrence relations: " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "PIEC
EWISE([ T(0,x)=1 , `` ],[ T(1,x)=x ,``  ],[T(n,x)=2*x*T(n,x)-T(n-2,x) \+
,n>1 ])" "6#-%*PIECEWISEG6%7$/-%\"TG6$\"\"!%\"xG\"\"\"%!G7$/-F)6$F-F,F
,F.7$/-F)6$%\"nGF,,&*(\"\"#F-F,F--F)6$F7F,F-F--F)6$,&F7F-F:!\"\"F,F@2F
-F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 38 "This suggests the following procedure." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "ChebT \+
:= proc(n::nonnegint,x)\n   local i,t,prevt,newt;\n   prevt := 1;\n   \+
if n=0 then return prevt end if;\n   t := x;\n   if n=1 then return t \+
end if;\n   for i from 1 to n-1 do\n      newt := 2*x*t - prevt;\n    \+
  prevt := t;\n      t := newt;\n   end do;\n   t;\nend proc:" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "If the 2
nd argument is a variable, the output is given in an expanded form. To
 see the polynomials as above we need to use the procedure " }{TEXT 0 
7 "collect" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "collect(ChebT(6,x),x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"'\"\"\"\"#K*&\"#[F()F&\"\"%F(!\"
\"*&\"#=F()F&\"\"#F(F(F(F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 55 "These polynomials are available as the Maple pr
ocedure " }{TEXT 0 10 "ChebyshevT" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 64 "To see the polynomials in the same form as above we need \+
to use " }{TEXT 0 6 "expand" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "expand(ChebyshevT(
6,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"'\"\"\"\"#K*&
\"#[F()F&\"\"%F(!\"\"*&\"#=F()F&\"\"#F(F(F(F." }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "They are generally used w
ith the domain restricted to the interval " }{XPPEDIT 18 0 "[-1,1]" "6
#7$,$\"\"\"!\"\"F%" }{TEXT -1 26 " in line with the formula " }
{XPPEDIT 18 0 "T(n,x) = cos(n*arccos(x));" "6#/-%\"TG6$%\"nG%\"xG-%$co
sG6#*&F'\"\"\"-%'arccosG6#F(F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 46 "The values will then also lie in the interval " }
{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pl
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_l7$Fggl$\"+TkU*y*F^_l7$Fjgl$\"+Ms&*H**F^_l7$F]hl$\"+zTv!***F^_l7$$!)]
u*G&!#7$\"+w(*******F^_l7$F`hl$\"+G_$4***F^_l7$$\"++`bR?Fc]o$\"+cbtm**
F^_l7$Fdhl$\"+>XOF**F^_l7$Fghl$\"+-rd)y*F^_l7$Fjhl$\"+o2C6'*F^_l7$F]il
$\"+aEFX$*F^_l7$F`il$\"+\\*H\\-*F^_l7$Fcil$\"+m&[Tk)F^_l7$Ffil$\"+r))Q
1#)F^_l7$Fiil$\"+)GDIt(F^_l7$F\\jl$\"+XislrF^_l7$F_jl$\"+R/;ulF^_l7$Fb
jl$\"+--O,fF^_l7$Fejl$\"+(>`6D&F^_l7$Fhjl$\"+;'G5\\%F^_l7$F[[m$\"+(eki
s$F^_l7$F^[m$\"+0I0KHF^_l7$Fa[m$\"+iE@u?F^_l7$Fd[m$\"+M5Sj7F^_l7$Fg[m$
\"*Os`<%F^_l7$Fj[m$!)Z\\\"G&Fbhl7$F]\\m$!*UD))Q\"Fbhl7$F`\\m$!*Q;pE#Fb
hl7$Fc\\m$!*2;3:$Fbhl7$Ff\\m$!*)G')[RFbhl7$Fi\\m$!*/ftx%Fbhl7$F\\]m$!*
:`&pbFbhl7$F_]m$!**)y]P'Fbhl7$Fb]m$!**\\FUqFbhl7$Fe]m$!*,];v(Fbhl7$Fh]
m$!*qPtK)Fbhl7$F[^m$!*Wx!G))Fbhl7$F^^m$!*wn-G*Fbhl7$Fa^m$!*!o_L'*Fbhl7
$Fd^m$!*?]v&)*Fbhl7$$\"+l5anoF^_l$!*\"phN**Fbhl7$Fg^m$!*(>c$)**Fbhl7$$
\"+&*p&z1(F^_l$!*_%)*****Fbhl7$Fj^m$!*o#4&)**Fbhl7$$\"+XXCusF^_l$!*GR?
$**Fbhl7$F]_m$!*\\=*Q)*Fbhl7$$\"+b?RuuF^_l$!*)*pYs*Fbhl7$F`_m$!*K!)zd*
Fbhl7$Fc_m$!*E\")=6*Fbhl7$Ff_m$!*Fy#*\\)Fbhl7$Fi_m$!*3]Pr(Fbhl7$F\\`m$
!*(*4st'Fbhl7$F_`m$!*gt?`&Fbhl7$Fb`m$!*3s!4SFbhl7$Fe`m$!*FumP#Fbhl7$$
\"+g$e**3*F^_l$!*#>l$[\"Fbhl7$Fh`m$!)(pxI&Fbhl7$$\"+?h5\"H*F^_l$\")HVe
bFbhl7$F[am$\"*c#p9<Fbhl7$$\"+bUt)\\*F^_l$\"*+=]%HFbhl7$F^am$\"*\"fG^U
Fbhl7$$\"+gH^&p*F^_l$\"*0#4!\\&Fbhl7$Faam$\"*Y>Kz'Fbhl7$$\"+g`H%*)*F^_
l$\"*pUIN)FbhlF_^l-Fa^l6&Fc^lFd^lFf^lFd^l-F$6$7asFj^l7$$!+9%er%**F^_l$
!*xwmq)Fbhl7$F`go$!*WnzY(Fbhl7$$!+U_ZT)*F^_l$!*6bEG'Fbhl7$F\\_l$!*eP&
\\^Fbhl7$Fhgo$!*x?')H$Fbhl7$F`_l$!*OAgf\"Fbhl7$F`ho$\")L?!\\\"Fbhl7$Fc
_l$\"*-]Cs\"Fbhl7$Fhho$\"*8R79$Fbhl7$Ff_l$\"*k*\\-WFbhl7$F`io$\"*!f=4b
Fbhl7$Fi_l$\"*jQ_Z'Fbhl7$$!+gaW')))F^_l$\"*h$p^sFbhl7$F\\`l$\"*cP$>zFb
hl7$F_`l$\"*s$))z*)Fbhl7$Fb`l$\"*_/:m*Fbhl7$$!+J$HKG)F^_l$\"*Hi*e)*Fbh
l7$Fe`l$\"*r9(p**Fbhl7$$!+<!oz7)F^_l$\"*3&y%***Fbhl7$$!+<gBv!)F^_l$\"*
*e>****Fbhl7$$!+;S]A!)F^_l$\"*N&p$)**Fbhl7$Fh`l$\"*C>!\\**Fbhl7$$!+m#z
o(yF^_l$\"*b#\\V)*Fbhl7$F[al$\"*cpXo*Fbhl7$F^al$\"*q-1:*Fbhl7$Faal$\"*
\\*=-%)Fbhl7$Fdal$\"*nyc^(Fbhl7$Fgal$\"*#oU(f'Fbhl7$Fjal$\"*'\\8(R&Fbh
l7$F]bl$\"*lhDJ%Fbhl7$F`bl$\"*B]++$Fbhl7$Fcbl$\"*WWS\"=Fbhl7$Ffbl$\")y
z'3&Fbhl7$Fibl$!)\"pW>(Fbhl7$F\\cl$!*2lr'>Fbhl7$F_cl$!*AL%oIFbhl7$Fbcl
$!*>][>%Fbhl7$Fecl$!*-;TG&Fbhl7$Fhcl$!*\"\\!f:'Fbhl7$F[dl$!*W#H4qFbhl7
$F^dl$!*N!R(y(Fbhl7$Fadl$!*n/3W)Fbhl7$Fddl$!*s\"Gn*)Fbhl7$Fgdl$!+%QilU
*F^_l7$Fjdl$!+@D,D(*F^_l7$$!+CZ+HMF^_l$!+&=M)R)*F^_l7$F]el$!+<ZqB**F^_
l7$$!+Z,XHKF^_l$!+,J2t**F^_l7$F`el$!+(pps***F^_l7$$!+'Rt3.$F^_l$!+W5:&
***F^_l7$Fcel$!+=/]j**F^_l7$$!+tidHGF^_l$!+)e*42**F^_l7$Ffel$!+\\=FD)*
F^_l7$Fiel$!+r9*Rd*F^_l7$F\\fl$!+O'*3F#*F^_l7$F_fl$!+fu=i()F^_l7$Fbfl$
!+fC7A#)F^_l7$Fefl$!+zF3$e(F^_l7$Fhfl$!+v=@qoF^_l7$F[gl$!+'[8E:'F^_l7$
F^gl$!+22Hl_F^_l7$Fagl$!+Gz9@WF^_l7$Fdgl$!+hIqxMF^_l7$Fggl$!+GocTDF^_l
7$Fjgl$!+Ne$\\Z\"F^_l7$F]hl$!+CV1t`Fc]o7$F`hl$\"+9&R-K&Fc]o7$Fdhl$\"+n
\\\"=]\"F^_l7$Fghl$\"+I:fYDF^_l7$Fjhl$\"+\\-1EMF^_l7$F]il$\"+1)RJR%F^_
l7$F`il$\"+pL&HG&F^_l7$Fcil$\"+**)>%>hF^_l7$Ffil$\"+mRk\"*oF^_l7$Fiil$
\"+NrKovF^_l7$F\\jl$\"+m6#*>#)F^_l7$F_jl$\"+@jUc()F^_l7$Fbjl$\"+y*>[A*
F^_l7$Fejl$\"+si=e&*F^_l7$Fhjl$\"+b8)4#)*F^_l7$$\"++9?GGF^_l$\"+xB71**
F^_l7$F[[m$\"+#=PU'**F^_l7$$\"+&=%>HIF^_l$\"+IG([***F^_l7$F^[m$\"+h?&y
***F^_l7$$\"+!44WB$F^_l$\"+4u6r**F^_l7$Fa[m$\"+p'GO\"**F^_l7$$\"+?C\"e
V$F^_l$\"+5%4L$)*F^_l7$Fd[m$\"+OIlE(*F^_l7$Fg[m$\"+!f0fU*F^_l7$Fj[m$\"
**R2m*)Fbhl7$F]\\m$\"*]%)eV)Fbhl7$F`\\m$\"*(e%[y(Fbhl7$Fc\\m$\"*s@W,(F
bhl7$Ff\\m$\"*u(3:iFbhl7$Fi\\m$\"*I/PF&Fbhl7$F\\]m$\"*f2nD%Fbhl7$F_]m$
\"*Pv))3$Fbhl7$Fb]m$\"*t(p+?Fbhl7$Fe]m$\");8KpFbhl7$Fh]m$!)\"=8>&Fbhl7
$F[^m$!*p:Gt\"Fbhl7$F^^m$!*BYR.$Fbhl7$Fa^m$!*[&48VFbhl7$Fd^m$!**\\HJaF
bhl7$Fg^m$!*avka'Fbhl7$Fj^m$!*\"*4p`(Fbhl7$F]_m$!*]Y'o%)Fbhl7$F`_m$!*0
%*p7*Fbhl7$Fc_m$!*,Qwn*Fbhl7$$\"+:cv!)yF^_l$!*>!**[)*Fbhl7$Ff_m$!**H(z
&**Fbhl7$$\"+oxjI!)F^_l$!*Pbt)**Fbhl7$$\"+NdF!3)F^_l$!*hY'****Fbhl7$$
\"+.P\"*H\")F^_l$!*wJU***Fbhl7$Fi_m$!*4'[q**Fbhl7$$\"+!e1*y#)F^_l$!*yT
`')*Fbhl7$F\\`m$!*a3!z'*Fbhl7$F_`m$!*^iY-*Fbhl7$Fb`m$!*a$z%)yFbhl7$Fe`
m$!*d;eQ'Fbhl7$F[^q$!*+c0Z&Fbhl7$Fh`m$!*:77V%Fbhl7$Fc^q$!*Iie<$Fbhl7$F
[am$!*N1Pw\"Fbhl7$F[_q$!)K-.>Fbhl7$F^am$\"*!pYb:Fbhl7$Fc_q$\"*S4hF$Fbh
l7$Faam$\"*qP'[^Fbhl7$$\"+SIWT)*F^_l$\"*!*[>G'Fbhl7$F[`q$\"*?vuY(Fbhl7
$$\"+!oZr%**F^_l$\"*g>kq)FbhlF_^l-Fa^l6&Fc^lFf^lFd^lFd^l-%+AXESLABELSG
6$Q\"x6\"Q!6\"-%%VIEWG6$;F(F+%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 38 "The zeros of the degree 10 polynomial " }{XPPEDIT 
18 0 "T(10,x)" "6#-%\"TG6$\"#5%\"xG" }{TEXT -1 25 " can be found as fo
llows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "expand(ChebyshevT(10,x));\nfsolve(%,x=-1..1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"#5\"\"\"\"$7&*&\"%!G\"F()F
&\"\")F(!\"\"*&\"%?6F()F&\"\"'F(F(*&\"$+%F()F&\"\"%F(F.*&\"#]F()F&\"\"
#F(F(F(F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$!+1M)o()*!#5$!+U_15*)F%$
!+7y1rqF%$!+(*\\!*RXF%$!+]YMk:F%$\"+]YMk:F%$\"+(*\\!*RXF%$\"+7y1rqF%$
\"+U_15*)F%$\"+1M)o()*F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "T(n,x) = cos(n*arccos(x));
" "6#/-%\"TG6$%\"nG%\"xG-%$cosG6#*&F'\"\"\"-%'arccosG6#F(F-" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "T(n,x)=0" "6#/-%\"TG6$%\"nG%\"xG\"\"!" }
{TEXT -1 6 " when " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "n*arccos(x)=(2*k-1)*Pi/2" "6#/*&%\"nG\"\"\"-%'arccosG6#%\"xGF&*(,&*
&\"\"#F&%\"kGF&F&F&!\"\"F&%#PiGF&F.F0" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{TEXT 267 1 "k" }{TEXT -1 25 " is an integer
, that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccco
s(x)=(2*k-1)*Pi/(2*n)" "6#/-%(arcccosG6#%\"xG*(,&*&\"\"#\"\"\"%\"kGF,F
,F,!\"\"F,%#PiGF,*&F+F,%\"nGF,F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x \+
= cos((2*k-1)*Pi/(2*n))" "6#/%\"xG-%$cosG6#*(,&*&\"\"#\"\"\"%\"kGF,F,F
,!\"\"F,%#PiGF,*&F+F,%\"nGF,F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Hence the zeros of " }
{XPPEDIT 18 0 "T(10,x)" "6#-%\"TG6$\"#5%\"xG" }{TEXT -1 19 " are obtai
ned from:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "n := 10;\n[seq(cos((2*k-1)*Pi/(2*n)),k=1..n)];\nsort(
evalf(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7,-%$cosG6#,$%#PiG#\"\"\"\"#?-F%6#,$F(#\"\"$F+,$
*$-%%sqrtG6#\"\"#F*#F*F6-F%6#,$F(#\"\"(F+-F%6#,$F(#\"\"*F+,$F=!\"\",$F
8FC,$F2#FCF6,$F,FC,$F$FC" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,$!+1M)o(
)*!#5$!+U_15*)F&$!+5y1rqF&$!+&*\\!*RXF&$!+_YMk:F&$\"+_YMk:F&$\"+&*\\!*
RXF&$\"+5y1rqF&$\"+U_15*)F&$\"+1M)o()*F&" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 54 "Interpolation using the zeros of Chebyshev poly
nomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 
0 "" {TEXT -1 71 "In this section we find an interpolating polynomial \+
appproximation for " }{XPPEDIT 18 0 "f(x)=sin(Pi*x/2)" "6#/-%\"fG6#%\"
xG-%$sinG6#*(%#PiG\"\"\"F'F-\"\"#!\"\"" }{TEXT -1 17 " on the interval
 " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 70 ", which
 has good error characteristics with respect to absolute error." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 335 "The idea
 is to use non-uniformly spaced interpolation points. You may have not
iced that, when using equally spaced interpolation points, the absolut
e error tends to be worse towards the edges of the interval. Perhaps w
e can improve the situation if we take more interpolation points near \+
the edges than near the centre of the interval." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 269 "We shall use the zeros o
f a Chebyshev polynomial to obtain a suitable distribution of interpol
ation points. We won't go into the theory behind this approach, but yo
u can see that the zeros Chebyshev polynomials tend to be crowded more
 towards the edges of the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"
\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 136 "We shall attempt to find an interpolating poly
nomial which will serve to approximate sin(x) to the default Maple pre
cision of 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 163 "In order to ensu
re that rounding errors do not effect the computed of the polynomial c
oefficients we shall perform the calculations using a precision of 20 \+
digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
52 "The zeros of the degree 11 Chebyshev polynomial are:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "n := \+
11:\nxvals := sort([seq(evalf[20](cos((2*k-1)*Pi/(2*n))),k=1..n)]);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7-$!5QKF$4)=W@)*)*!#?$!5Sr$=
XN&*>j4*F($!5w$GeUNu&\\dvF($!55#e(fbu\"3kS&F($!5q(pH9%obK<GF($\"\"!F2$
\"5q(pH9%obK<GF($\"55#e(fbu\"3kS&F($\"5w$GeUNu&\\dvF($\"5Sr$=XN&*>j4*F
($\"5QKF$4)=W@)*)*F(" }}}{PARA 0 "" 0 "" {TEXT -1 19 "\nThe correspond
ing " }{TEXT 270 1 "y" }{TEXT -1 18 " values are . . .\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f := x -> sin(Pi/2*x);\nyvals := ev
alf[20](map(f,xvals));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%
\"xG6\"6$%)operatorG%&arrowGF(-%$sinG6#,$*&#\"\"\"\"\"#F2*&%#PiGF29$F2
F2F2F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7-$!5xvkn)e(=s)
***!#?$!5femB'*G/U**)*F($!5#eHVg6yzHF*F($!5<Ah;h=Ow2vF($!5T3$=r2$>S#G%
F($\"\"!F2$\"5T3$=r2$>S#G%F($\"5<Ah;h=Ow2vF($\"5#eHVg6yzHF*F($\"5femB'
*G/U**)*F($\"5xvkn)e(=s)***F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 59 "We are now ready to construct the interpo
lating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 
5 " for " }{XPPEDIT 18 0 "f(x) = sin(Pi*x/2)" "6#/-%\"fG6#%\"xG-%$sinG
6#*(%#PiG\"\"\"F'F-\"\"#!\"\"" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "px := evalf[20](evalf[30](expand(interp(xvals,
yvals,x))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#pxG,8*&$\"5x$)R%=R%
RId9!#Y\"\"\")%\"xG\"#5F*F**&$\"5*Q!z8;kv83:!#BF*)F,\"\"*F*F*$\"\"(!#I
!\"\"*&$\"(ouS$!#LF*)F,\"\")F*F7*&$\"5_ZUGy*G'zq:!#>F*F,F*F**&$\"5W9q/
K.E@sY!#AF*)F,F5F*F7*&$\"#!*F6F*)F,\"\"#F*F**&$\"'H<F!#KF*)F,\"\"'F*F*
*&$\"5K(=YJncL'fk!#?F*)F,\"\"$F*F7*&$\"54zw^([!p%)oz!#@F*)F,\"\"&F*F**
&$\"%![)!#JF*)F,\"\"%F*F7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 61 "Since the coefficients of the terms involving eve
n powers of " }{TEXT 268 1 "x" }{TEXT -1 87 " are \"small\", it looks \+
as though this polynomial is attempting to be an odd polynomial." }}
{PARA 0 "" 0 "" {TEXT -1 65 "Let's help it out by removing the terms i
nvolving even powers of " }{TEXT 269 1 "x" }{TEXT -1 40 ", and constru
ct our polynomial function " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 177 "chp := proc(u) if irem(degree(u),2)=0 then 0 el
se u end if end proc;\npx := map(chp,px):\npx := expand(convert(px,hor
ner)): # to get the terms in desired order\np := unapply(px,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$chpGf*6#%\"uG6\"F(F(@%/-%%iremG6$-%
'degreeG6#9$\"\"#\"\"!F3F1F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,,*&$\"5_ZUGy*G'zq:!#>\"\"\"9$F
1F1*&$\"5K(=YJncL'fk!#?F1)F2\"\"$F1!\"\"*&$\"54zw^([!p%)oz!#@F1)F2\"\"
&F1F1*&$\"5W9q/K.E@sY!#AF1)F2\"\"(F1F9*&$\"+kv83:!#8F1)F2\"\"*F1F1F(F(
F(" }}}{PARA 0 "" 0 "" {TEXT -1 126 "\nThe coefficients bear more than
 a passing resemblance to the coefficients of the first few terms of t
he Maclaurin series for " }{XPPEDIT 18 0 "f(x) = sin(Pi*x/2)" "6#/-%\"
fG6#%\"xG-%$sinG6#*(%#PiG\"\"\"F'F-\"\"#!\"\"" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
77 "f := x -> sin(Pi/2*x);\nseries(f(x),x=0,10);\nconvert(%,polynom);
\nevalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6
$%)operatorG%&arrowGF(-%$sinG6#,$*&#\"\"\"\"\"#F2*&%#PiGF29$F2F2F2F(F(
F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG,$*&\"\"#!\"\"%#PiG\"\"\"
F*F*,$*&\"#[F(F)\"\"$F(F.,$*&\"%SQF(F)\"\"&F*F2,$*&\"'?^kF(F)\"\"(F(F6
,$*&\"*gXz&=F(F)\"\"*F*F:-%\"OG6#F*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,,*&#\"\"\"\"\"#F&*&%#PiGF&%\"xGF&F&F&*&#F&\"#[F&*&)F)\"\"$F&)F*
F0F&F&!\"\"*&#F&\"%SQF&*&)F)\"\"&F&)F*F8F&F&F&*&#F&\"'?^kF&*&)F)\"\"(F
&)F*F?F&F&F2*&#F&\"*gXz&=F&*&)F)\"\"*F&)F*FFF&F&F&" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#,,*&$\"5#>m*[zEjzq:!#>\"\"\"%\"xGF(F(*&$\"5o`iC1v4kfk
!#?F()F)\"\"$F(!\"\"*&$\"5E^/nhCEEpz!#@F()F)\"\"&F(F(*&$\"565)o=`8a<o%
!#AF()F)\"\"(F(F0*&$\"5*=#)fty%=T/;!#BF()F)\"\"*F(F(" }}}{PARA 0 "" 0 
"" {TEXT -1 92 "\nThe following graph shows the interpolation points a
long with the interpolating polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "points := evalf(zip((x
,y)->[x,y],xvals,yvals)):\np1 := plot(p(x),x=-1..1):\np2 := plot([poin
ts$4],x=-1..1,symbol=[circle$2,diamond,cross],\n       symbolsize=[15,
10$3],color=[black,green$3,red],style=point):\nplots[display]([p1,p2])
;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6
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$\"3egww.jew7F17$$\"3GKLLe\"*[H7F1$\"3+4'p%GQH>>F17$$\"3I*******pvxl\"
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$\"3#emmmT9C#eF1$\"3)[dkTh!3BzF17$$\"33****\\i!*3`iF1$\"34:pQ+4R<$)F17
$$\"3%QLLL$*zym'F1$\"3iyw2Xn?h')F17$$\"3wKLL3N1#4(F1$\"3#QU$)o?%yu*)F1
7$$\"3Nmm;HYt7vF1$\"31]*>[%=VY#*F17$$\"3Y*******p(G**yF1$\"3e!GCN7\"\\
g%*F17$$\"3]mmmT6KU$)F1$\"3qhZyzE!Hm*F17$$\"3fKLLLbdQ()F1$\"3NHjKVgL/)
*F17$$\"3[++]i`1h\"*F1$\"3HZB![L'H8**F17$$\"3W++]P?Wl&*F1$\"358%zOt6n(
**F17$$\"\"\"F*$\"3]F:J.+++5F--%'COLOURG6&%$RGBG$\"#5F)$F*F*Fb[l-F$6&7
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***G/U**)*F17$$!37+++Wd\\dvF1$!3I+++\"yzHF*F17$$!3P+++v\"3kS&F1$!3y+++
>Ow2vF17$$!3y*****zcDt\"GF1$!3++++J>S#G%F17$Fb[lFb[l7$$\"3y*****zcDt\"
GF1$\"3++++J>S#G%F17$$\"3P+++v\"3kS&F1$\"3y+++>Ow2vF17$$\"37+++Wd\\dvF
1$\"3I+++\"yzHF*F17$$\"3[*****R&*>j4*F1$\"3')******G/U**)*F17$$\"3))**
****=W@)*)*F1$\"3i*****f(=s)***F1-F][l6&F_[lF*F*F*-%'SYMBOLG6$%'CIRCLE
G\"#:-%&STYLEG6#%&POINTG-F$6&Fe[l-F][l6&F_[lFb[l$\"*++++\"!\")Fb[l-F\\
_l6$F^_lFa[lF`_l-F$6&Fe[lFf_l-F\\_l6$%(DIAMONDGFa[lF`_l-F$6&Fe[lFf_l-F
\\_l6$%&CROSSGFa[lF`_l-%+AXESLABELSG6%Q\"x6\"Q!F[al-%%FONTG6#%(DEFAULT
G-%%VIEWG6$;F(FhzF`al" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The absol
ute error curve of the interpolating polynomial is . . ." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "e := \+
p-f;\nplot1 := plot(e(x),x=-1..1,color=blue):\nplot1;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"eG,&%\"pG\"\"\"%\"fG!\"\"" }}{PARA 13 "" 1 "" 
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" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 242 "This error graph looks very similar to the graph of a \+
Chebyshev polynomial, except that it is scaled vertically so that the \+
magnitude of maximum and minimum values are much reduced. It certainly
 has the same zeros as the Chebyshev polynomial " }{XPPEDIT 18 0 "T(11
,x)" "6#-%\"TG6$\"#6%\"xG" }{TEXT -1 68 " by the very method of constr
uction of the interpolating polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 232 "The fact that the magnitude of the \+
maximum and minimum values at the turning points are all close togethe
r means that \"no effort is wasted\" in trying to reduce the absolute \+
error by choosing the location of the interpolation points." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "We shall make a
 more careful comparison of the error curve with the Chebyshev polynom
ial " }{XPPEDIT 18 0 "T(11,x)" "6#-%\"TG6$\"#6%\"xG" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 20 "Using the procedure " }{TEXT 0 9 "allbis
ect" }{TEXT -1 64 " from the appropriate worksheet on root-finding we \+
can find the " }{TEXT 271 1 "x" }{TEXT -1 64 " coordinates of the turn
ing points on this absolute error graph." }}{PARA 0 "" 0 "" {TEXT -1 
65 "The magnitudes of the associated errors are quite close together.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 78 "[allbisect(D(e)(x),x=-1..1)];\nymax :=  evalf[20](map(abs@e,%)
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The maxim
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!\"\"" }{TEXT -1 1 " " }{TEXT 277 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "10^(-9)" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 68 ", and the average of t
he magnitudes of the maximum errors is . . .  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
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" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Now we use this average
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0 "T(11,x)" "6#-%\"TG6$\"#6%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y=c*T(11,x)" "6#/%\"yG*&%
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e absolute error graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 65 "The error curve is characteristic of a  minimax approx
imation to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 18 "minimax polynomial" 
}{TEXT -1 25 " approximation of degree " }{TEXT 272 1 "n" }{TEXT -1 
16 " for a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 16 " on an interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }
{TEXT -1 19 " is the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"
xG" }{TEXT -1 60 " such that the maximum absolute error is a minimum, \+
that is," }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 262 53 "           \+
                          max             " }{XPPEDIT 18 0 "abs( f(x) \+
- p(x) )" "6#-%$absG6#,&-%\"fG6#%\"xG\"\"\"-%\"pG6#F*!\"\"" }{TEXT 
263 6 "      " }}{PARA 0 "" 0 "" {TEXT 264 26 "                x in [a
,b]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "is
 minimized over all polynomials " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"
xG" }{TEXT -1 10 " of degree" }{XPPEDIT 18 0 "``<=n" "6#1%!G%\"nG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 64 "In fact we can construct
 the minimax polynomial of degree 9 for " }{XPPEDIT 18 0 "f(x) = sin(P
i/2*x)" "6#/-%\"fG6#%\"xG-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"F'F-" }{TEXT 
-1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%
" }{TEXT -1 27 " using the Maple procedure " }{TEXT 0 7 "minimax" }
{TEXT -1 16 " in the package " }{TEXT 0 9 "numapprox" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 84 "f := x -> sin(Pi/2*x);\nqx := expand(evalf[20](numapprox[minimax
](f(x),x=-1..1,10)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"
xG6\"6$%)operatorG%&arrowGF(-%$sinG6#,$*&#\"\"\"\"\"#F2*&%#PiGF29$F2F2
F2F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#qxG,8$\"5lTl\"\\#y%p*el
!#N\"\"\"*&$\"5]zyB-!H'zq:!#>F)%\"xGF)F)*&$\"5:#HlCd.&eSR!#LF))F.\"\"#
F)!\"\"*&$\"5ZFl-m)fL'fk!#?F))F.\"\"$F)F5*&$\"5T$fG9[C^*eO!#KF))F.\"\"
%F)F)*&$\"5q1?\"*3a![)oz!#@F))F.\"\"&F)F)*&$\"5D8oY^kjOc6!#JF))F.\"\"'
F)F5*&$\"5$HHSfS#zAsY!#AF))F.\"\"(F)F5*&$\"5lMB5[iUhb9FKF))F.\"\")F)F)
*&$\"5c^*ft#\\c?3:!#BF))F.\"\"*F)F)*&$\"5z%>i$Q!pIRE'F?F))F.\"#5F)F5" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Again \+
it looks as though we can safely ignore the terms of even degree." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
177 "chp := proc(u) if irem(degree(u),2)=0 then 0 else u end if end pr
oc:\nqx := map(chp,qx):\nqx := expand(convert(qx,horner)): # to get th
e terms in desired order\nq := unapply(qx,x);" }}{PARA 12 "" 1 "" 
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he magnitudes of the maximum and minimum values at the turning points \+
are even closer together than for the polynomial " }{XPPEDIT 18 0 "p(x
)" "6#-%\"pG6#%\"xG" }{TEXT -1 67 ", constructed by interpolation, but
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "As far as the eye can see
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\\3H$F-7$Fd]o$\"3?pZ+l[3:KF-7$Fi]o$\"3g\\>#3Z]I&HF-7$F^^o$\"32<rlYJ;UD
F-7$Fc^o$\"3G[stU'eME#F-7$Fh^o$\"3Ele\\%QP^%>F-7$F]_o$\"3su/QT6d*e\"F-
7$Fb_o$\"3Ku^$o)y)**>\"F-7$Fg_o$\"3%=%=+<Rj0yFH7$F\\`o$\"37WQq:f<lLFH7
$Fa`o$!3gy&)Gr\\!yD\"FH7$Ff`o$!3o**3>9k/()fFH7$F[ao$!3#3X>D,5I0\"F-7$F
`ao$!3aQP\\N'Q'*\\\"F-7$Feao$!3f]!\\(\\S0G>F-7$Fjao$!3k[z6;%[iK#F-7$F_
bo$!3E\"o]1)Q(e(HF-7$Fdbo$!36b#Ge'yU=LF-7$Fibo$!3;Hj1Vs4QLF-7$F^co$!3=
ZN&>/p0K$F-7$Fcco$!3e]#Hl/&3iKF-7$Fhco$!3%*)HU-)\\oeJF-7$F]do$!3$*>ufA
QB+GF-7$Fbdo$!3^Z%y,R&=4AF-7$Fgdo$!3)RFB()fhT\"=F-7$F\\eo$!3Y\\!3(H)od
M\"F-7$Faeo$!3CV)\\P1!3%)zFH7$Ffeo$!3[Sx&RwL@m\"FH7$F[fo$\"3^?8@@C0O=F
H7$F`fo$\"3=DiN,[dpbFH7$Fefo$\"3%*yKYHCcY&*FH7$Fjfo$\"3EqzR)3KvP\"F-7$
F_go$\"3&yQ/rLMk#=F-7$Fdgo$\"3')\\BeQ<B-BF-7$Figo$\"3MhhH$p7e!GF-7$F^h
o$\"37OBTJS3QLF--%&COLORG6&FehoF^hoFfho$\"\"(F)-%*THICKNESSG6#\"\"#-%+
AXESLABELSG6%Q\"x6\"Q!Fghr-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F^hoF\\ir" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "A polynomi
al approximation for the sine function " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 75 "In this section we \+
find an interpolating polynomial appproximation for the " }{XPPEDIT 
18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " on the interval " }
{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F
(F)" }{TEXT -1 205 " in which we pay attention to relative error inste
ad of absolute error. As in the last section, we shall use the zeros o
f a Chebyshev polynomial to obtain a suitable distribution of interpol
ation points. ." }}{PARA 0 "" 0 "" {TEXT -1 7 "\nSince " }{XPPEDIT 18 
0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 60 " is an odd function, we ca
n find a polynomial approximation " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%
\"xG" }{TEXT -1 19 " for the function  " }{XPPEDIT 18 0 "sin(x)/x;" "6
#*&-%$sinG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 32 " , and then take the pol
ynomial " }{XPPEDIT 18 0 "p(x) = x*q(x);" "6#/-%\"pG6#%\"xG*&F'\"\"\"-
%\"qG6#F'F)" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "sin(x)" "
6#-%$sinG6#%\"xG" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 10 "Not
e that " }{XPPEDIT 18 0 "Limit(sin(x)/x,x = 0) = 1;" "6#/-%&LimitG6$*&
-%$sinG6#%\"xG\"\"\"F+!\"\"/F+\"\"!F," }{TEXT -1 38 ", so the function
 we approximate is   " }{XPPEDIT 18 0 "g(x) = PIECEWISE([sin(x)/x, x <
> 0],[1, x = 0]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$*&-%$sinG6#F'\"\"
\"F'!\"\"0F'\"\"!7$F0/F'F3" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "g := x -> if x<>0 \+
then sin(x)/x else 1 end if;\nplot(g(x),x=-Pi/2..Pi/2,y=0..1,color=mag
enta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operat
orG%&arrowGF(@%09$\"\"!*&-%$sinG6#F.\"\"\"F.!\"\"F4F(F(F(" }}{PARA 13 
"" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$!3+++lBj
zq:!#<$\"3O*f@kt(>mj!#=7$$!3WNzQW&=B]\"F*$\"36?V&)fuxSmF-7$$!3.42![ROF
W\"F*$\"3;\\bFzl^uoF-7$$!3?xt3O+tv8F*$\"3,mFK-E,JrF-7$$!3xxIt:&z#38F*$
\"3/o)\\22b<Q(F-7$$!3\")zz#fd\\6C\"F*$\"3J)=(Qn!HKi(F-7$$!3W:#)\\G:\"*
y6F*$\"3R\"***=RqNRyF-7$$!3,%4``jnW6\"F*$\"3MP#e'\\Xva!)F-7$$!3)yYu)o'
>y/\"F*$\"3?%H_vQ!*zE)F-7$$!3EWfpKW&Q\")*F-$\"3e5X[k[Nq%)F-7$$!3(fe1Vw
'\\I\"*F-$\"3%[(\\m&\\Ytm)F-7$$!333*H!e\\fG&)F-$\"3jrh'Gw]5$))F-7$$!35
`u@%3'*4&yF-$\"3?*Gw2\\(*Q+*F-7$$!3%ewEV#\\hqrF-$\"3qq'Q>u,[;*F-7$$!3S
I*3_gT\\^'F-$\"3fY.I5UX2$*F-7$$!3B!RBoTF&>fF-$\"32>B(GyLhU*F-7$$!3!=qR
rNA:@&F-$\"3]P$*Q2:W`&*F-7$$!3Z[5T-#\\<h%F-$\"3%HJ%oO*z#\\'*F-7$$!3#4J
\\*R/29RF-$\"3M?Y6kihY(*F-7$$!3wFQ0=l]'H$F-$\"3u5:aEc')>)*F-7$$!3)eE=r
,T*=EF-$\"3YY1k\">xg))*F-7$$!3eSV%*H%QP(>F-$\"3rf)3*f*)>N**F-7$$!3)*RW
U;s`+8F-$\"3yviuzQ$=(**F-7$$!3.gSQ<NGBo!#>$\"35^s6tAC#***F-7$$!3y]Py0u
l]:!#?$\"2y\\OC*f******F*7$$\"3O=#e5YQ8x'Fir$\"3=JMJ9*fB***F-7$$\"3I^(
**zOz+G\"F-$\"3A()GI8Brs**F-7$$\"3dNv()\\qFJ>F-$\"3.Xyqw>&z$**F-7$$\"3
y0j*\\(z-/EF-$\"3#o9rB`mt))*F-7$$\"3_\"oMd]$=iKF-$\"3@KNwJtdB)*F-7$$\"
3SH4XBG)*)*QF-$\"3+/Br;6b[(*F-7$$\"3=DLY%*)Rgg%F-$\"3=%p5dcQ,l*F-7$$\"
3U6;S?@OT_F-$\"3i7Z$\\g$Q[&*F-7$$\"3KMYS.Uq>fF-$\"3!)*QQk5+hU*F-7$$\"3
g!oi?&HQMlF-$\"39\"*H3CKS.$*F-7$$\"3Jr$zA=*Q1sF-$\"33)H_l\\km:*F-7$$\"
3gHJCuYpQyF-$\"3ywpa$e?p+*F-7$$\"3)fG\"*Q5O'*\\)F-$\"3ch\"4U%))oQ))F-7
$$\"3#=wm/;Fe9*F-$\"3!H%fqs_0j')F-7$$\"37E5$3JHB#)*F-$\"3GFO))4'RyY)F-
7$$\"31h>eG\")QZ5F*$\"3YO)43IQ$p#)F-7$$\"3PK#)fG(=S6\"F*$\"3iayLAQAc!)
F-7$$\"330`g$f(4!=\"F*$\"3o*4#Ga6JNyF-7$$\"3&oOhy?<3C\"F*$\"3\">15x_.W
i(F-7$$\"3XnP+Q(3/J\"F*$\"3/5>@!ygRP(F-7$$\"3/%yp@B_EP\"F*$\"3BTc(e)>i
UrF-7$$\"3cNK@zn,R9F*$\"3)*[#R*zs$*))oF-7$$\"3*p@f'=h`-:F*$\"3%p5d&oZ
\"*RmF-7$$\"3+++lBjzq:F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`[lF
\\[l-%+AXESLABELSG6$Q\"x6\"Q\"yFe[l-%%VIEWG6$;$!+Fjzq:!\"*$\"+Fjzq:F]
\\l;F_[l$\"\"\"F`[l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 70 "We shall attempt to find a polynomial which will s
erve to approximate " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }
{TEXT -1 45 " to the default Maple precision of 10 digits." }}{PARA 0 
"" 0 "" {TEXT -1 163 "In order to ensure that rounding errors do not e
ffect the computed of the polynomial coefficients we shall perform the
 calculations using a precision of 20 digits." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The zeros of the degree 1
1 Chebyshev polynomial are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "n := 11:\nT11zeros := sort([seq(eva
lf[20](cos((2*k-1)*Pi/(2*n))),k=1..n)]);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%)T11zerosG7-$!5QKF$4)=W@)*)*!#?$!5Sr$=XN&*>j4*F($!5w$GeUNu&\\d
vF($!55#e(fbu\"3kS&F($!5q(pH9%obK<GF($\"\"!F2$\"5q(pH9%obK<GF($\"55#e(
fbu\"3kS&F($\"5w$GeUNu&\\dvF($\"5Sr$=XN&*>j4*F($\"5QKF$4)=W@)*)*F(" }}
}{PARA 0 "" 0 "" {TEXT -1 30 "\nSince we want to approximate " }
{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 18 " on the interval \+
[" }{XPPEDIT 18 0 "-Pi/2;" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 
3 " , " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 
31 "], we multiply these zeros by  " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG
\"\"\"\"\"#!\"\"" }{TEXT -1 16 "  to obtain our " }{TEXT 274 1 "x" }
{TEXT -1 9 " values.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "xv
als := evalf[20](Pi/2*T11zeros);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
&xvalsG7-$!5LQ(R*3&)y![b\"!#>$!5K#**)z.(fY)G9F($!5i^vCPb'Gr=\"F($!553H
ku,hO#\\)!#?$!5q!>_]UlWaU%F/$\"\"!F3$\"5q!>_]UlWaU%F/$\"553Hku,hO#\\)F
/$\"5i^vCPb'Gr=\"F($\"5K#**)z.(fY)G9F($\"5LQ(R*3&)y![b\"F(" }}}{PARA 
0 "" 0 "" {TEXT -1 19 "\nThe corresponding " }{TEXT 273 1 "y" }{TEXT 
-1 18 " values are . . .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
103 "g := x -> if x<>0 then sin(x)/x else 1 end if:\nyvals := evalf[20
]([seq(g(xvals[i]),i=1..nops(xvals))]);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%&yvalsG7-$\"5N;ZfsI2%3V'!#?$\"5/&fQPKzf#GpF($\"56@3OH]xE6yF($
\"5*zv)*\\j(GgS))F($\"5$f(el5)3snn*F($\"\"\"\"\"!F/F-F+F)F&" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "We are now read
y to construct the interpolating polynomial " }{XPPEDIT 18 0 "q(x)" "6
#-%\"qG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sin(x)/x;" "6#*&-%
$sinG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 36 " , and the approximating poly
nomial " }{XPPEDIT 18 0 "p(x) = x*q(x)" "6#/-%\"pG6#%\"xG*&F'\"\"\"-%
\"qG6#F'F)" }{TEXT -1 6 "  for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%
\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "qx := evalf[20](expand(interp(xvals,yvals
,x)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#qxG,8*&$\"58@tQM;jX)R#!#F
\"\"\")%\"xG\"#5F*!\"\"*&$\")`sV;F)F*)F,\"\"*F*F.$\"5+++++++++5!#>F**&
$\"5)yKN7IH:Jv#!#DF*)F,\"\")F*F**&$F*!#?F*F,F*F.*&$\"'j\"f)F:F*)F,\"\"
(F*F**&$\"5sl*=R^mmmm\"F?F*)F,\"\"#F*F.*&$\"5Aq?*fU#))4%)>!#BF*)F,\"\"
'F*F.*&$\"##*!#@F*)F,\"\"$F*F**&$\"&(p9FMF*)F,\"\"&F*F.*&$\"56/ZVZ%4KL
L)!#AF*)F,\"\"%F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 78 "Since the magnitudes of the coefficients of the terms i
nvolving odd powers of " }{TEXT 275 1 "x" }{TEXT -1 33 " are \"small\"
, we can remove them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 135 "Digits := 20:\nchp := proc(u) if irem(de
gree(u),2)=1 then 0 else u end if end proc:\nqx := map(chp,qx):\nq := \+
unapply(qx,x);\nDigits := 10:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"q
Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,.*&$\"58@tQM;jX)R#!#F\"\"\")9$\"#
5F1!\"\"$\"5+++++++++5!#>F1*&$\"5)yKN7IH:Jv#!#DF1)F3\"\")F1F1*&$\"5sl*
=R^mmmm\"!#?F1)F3\"\"#F1F5*&$\"5Aq?*fU#))4%)>!#BF1)F3\"\"'F1F5*&$\"56/
ZVZ%4KLL)!#AF1)F3\"\"%F1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 91 "The following graph shows the interpolati
on points along with the interpolating polynomial." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "pts := eval
f(zip((x,y)->[x,y],xvals,yvals));\np1 := plot(q(x),x=-Pi/2..Pi/2,y=0..
1,color=magenta):\np2 := plot([pts$4],x=-Pi/2..Pi/2,y=0..1,color=[blac
k,red$3],style=[point$4,line],\n   symbol=[circle$2,diamond,cross],sym
bolsize=[15,10$3]):\nplots[display]([p1,p2]);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$ptsG7-7$$!+&)y![b\"!\"*$\"+J2%3V'!#57$$!+(fY)G9F)$\"
+$zf#GpF,7$$!+b'Gr=\"F)$\"+]xE6yF,7$$!+-hO#\\)F,$\"+wGgS))F,7$$!+aYWDW
F,$\"+)3snn*F,7$$\"\"!FC$\"\"\"FC7$$\"+aYWDWF,F?7$$\"+-hO#\\)F,F:7$$\"
+b'Gr=\"F)F57$$\"+(fY)G9F)F07$$\"+&)y![b\"F)F*" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$!3+++lBjzq:!#<$
\"3W#*H3Ox>mj!#=7$$!3WNzQW&=B]\"F*$\"3sBi<guxSmF-7$$!3.42![ROFW\"F*$\"
3sx&\\$zl^uoF-7$$!3?xt3O+tv8F*$\"36IP5-E,JrF-7$$!3xxIt:&z#38F*$\"3oN3Z
q]v\"Q(F-7$$!3\")zz#fd\\6C\"F*$\"3k%>Os1HKi(F-7$$!3W:#)\\G:\"*y6F*$\"3
w0B@RqNRyF-7$$!3,%4``jnW6\"F*$\"3]UE#)\\Xva!)F-7$$!3)yYu)o'>y/\"F*$\"3
q>gx(Q!*zE)F-7$$!3EWfpKW&Q\")*F-$\"3E@\"yY'[Nq%)F-7$$!3(fe1Vw'\\I\"*F-
$\"3gYpw&\\Ytm)F-7$$!333*H!e\\fG&)F-$\"3;n<(Gw]5$))F-7$$!35`u@%3'*4&yF
-$\"36h0p!\\(*Q+*F-7$$!3%ewEV#\\hqrF-$\"3?6>!=u,[;*F-7$$!3SI*3_gT\\^'F
-$\"3OL&e,@auI*F-7$$!3B!RBoTF&>fF-$\"3\"e$pv#yLhU*F-7$$!3!=qRrNA:@&F-$
\"3QpsK2:W`&*F-7$$!3Z[5T-#\\<h%F-$\"3)\\mqm$*z#\\'*F-7$$!3#4J\\*R/29RF
-$\"3o'eXTE;mu*F-7$$!3wFQ0=l]'H$F-$\"3()[QfEc')>)*F-7$$!3)eE=r,T*=EF-$
\"3-ecp\">xg))*F-7$$!3eSV%*H%QP(>F-$\"3?p6&*f*)>N**F-7$$!3)*RWU;s`+8F-
$\"3G-(o(zQ$=(**F-7$$!3.gSQ<NGBo!#>$\"3i&4CJFUA***F-7$$!3y]Py0ul]:!#?$
\"2A]OC*f******F*7$$\"3O=#e5YQ8x'Fir$\"3!p<?V\"*fB***F-7$$\"3I^(**zOz+
G\"F-$\"338ZK8Brs**F-7$$\"3dNv()\\qFJ>F-$\"3A$**[n(>&z$**F-7$$\"3y0j*
\\(z-/EF-$\"31/gUKlO())*F-7$$\"3_\"oMd]$=iKF-$\"3$)4l\"=LxN#)*F-7$$\"3
SH4XBG)*)*QF-$\"3)>,Wn6^&[(*F-7$$\"3=DLY%*)Rgg%F-$\"3=zupl&Q,l*F-7$$\"
3U6;S?@OT_F-$\"3c(=q[g$Q[&*F-7$$\"3KMYS.Uq>fF-$\"3b'*HK1,5E%*F-7$$\"3g
!oi?&HQMlF-$\"39d2%RA.MI*F-7$$\"3Jr$zA=*Q1sF-$\"3%=/<k\\km:*F-7$$\"3gH
JCuYpQyF-$\"3k=*fMe?p+*F-7$$\"3)fG\"*Q5O'*\\)F-$\"3n!G5U%))oQ))F-7$$\"
3#=wm/;Fe9*F-$\"3=F.\"GFbIm)F-7$$\"37E5$3JHB#)*F-$\"3=w!y+hRyY)F-7$$\"
31h>eG\")QZ5F*$\"3ANO.,$Q$p#)F-7$$\"3PK#)fG(=S6\"F*$\"3I!)H]AQAc!)F-7$
$\"330`g$f(4!=\"F*$\"3:K7Ia6JNyF-7$$\"3&oOhy?<3C\"F*$\"35`*fv_.Wi(F-7$
$\"3XnP+Q(3/J\"F*$\"3K_7$*z2'RP(F-7$$\"3/%yp@B_EP\"F*$\"3Ow$[c)>iUrF-7
$$\"3cNK@zn,R9F*$\"3;*>$**zs$*))oF-7$$\"3*p@f'=h`-:F*$\"3e_#z)oZ\"*RmF
-7$$\"3+++lBjzq:F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`[lF\\[l-F$
6&7-7$$!3++++&)y![b\"F*$\"3U+++J2%3V'F-7$$!3%******pfY)G9F*$\"31+++$zf
#GpF-7$$!3)******\\lGr=\"F*$\"3S******\\xE6yF-7$$!3k*****>5mB\\)F-$\"3
%******f(GgS))F-7$$!3w*****RlWaU%F-$\"3m*****z3snn*F-7$F_[l$\"\"\"F`[l
7$$\"3w*****RlWaU%F-F[]l7$$\"3k*****>5mB\\)F-Ff\\l7$$\"3)******\\lGr=
\"F*Fa\\l7$$\"3%******pfY)G9F*F\\\\l7$$\"3++++&)y![b\"F*Fg[l-Fiz6&F[[l
F`[lF`[lF`[l-%&STYLEG6#%&POINTG-%'SYMBOLG6$%'CIRCLEG\"#:-F$6&Fc[l-Fiz6
&F[[lF\\[lF_[lF_[lFa^l-Ff^l6$Fh^l\"#5-F$6&Fc[lF\\_lFa^l-Ff^l6$%(DIAMON
DGF`_l-F$6&Fc[lF\\_lFa^l-Ff^l6$%&CROSSGF`_l-%+AXESLABELSG6%Q\"x6\"Q\"y
F_`l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+Fjzq:!\"*$\"+Fjzq:F[al;F_[lF^]l
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C
urve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 33 "The polynomial approximation for " }
{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 10 " is . . . " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "px := expand(x*qx):\np := unapply(px,x);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,.*&$\"58@tQM;
jX)R#!#F\"\"\")9$\"#6F1!\"\"*&$\"5+++++++++5!#>F1F3F1F1*&$\"5)yKN7IH:J
v#!#DF1)F3\"\"*F1F1*&$\"5sl*=R^mmmm\"!#?F1)F3\"\"$F1F5*&$\"5Aq?*fU#))4
%)>!#BF1)F3\"\"(F1F5*&$\"56/ZVZ%4KLL)!#AF1)F3\"\"&F1F1F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "\nThe co
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:!\"*$\"+Fjzq:F[jn%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 69 "The maximum relative error appears to be
 at the ends of the interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalf[20](subs(x=Pi/2,(sin(x
)-p(x))/sin(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5++++!*[r%)=`!
#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "T
hus we see that the maximum relative truncation error is about half a \+
unit in the last decimal place if we were working with 10 digit precis
ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "W
e now round the coefficients of the polynomial to 10 digits and again \+
check the relative error." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "r := unapply(evalf(p(x)),x);\nplot(
(sin(x)-r(x))/sin(x),x=-Pi/2..Pi/2,color=blue);" }}{PARA 11 "" 1 "" 
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442 "The errors obtained by the degree 11 approximation found in the l
ast section were of the order of rounding errors for the precision use
d in the computations. If the same thing happens for a lower degree ap
proximation, then we have wasted some effort in using the higher degre
e approximation. Test this by reworking the previous section with 9 in
terpolation points obtained using the zeros of the degree 9 Chebyshev \+
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{TEXT -1 79 "Use the method of the last section to construct a polynom
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G" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"
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