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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 61 "Conversions between Taylor Serie
s \nand Orthogonal Polynomials" }}{PARA 19 "" 0 "" {TEXT -1 15 "Marc A
. Murison" }}{PARA 256 "" 0 "" {TEXT -1 74 "Astronomical Applications \+
Department\nU.S. Naval Observatory\nWashington, DC" }}{PARA 258 "" 0 "
" {TEXT -1 57 "murison@aa.usno.navy.mil\nhttp://aa.usno.navy.mil/muris
on/" }}{PARA 257 "" 0 "" {TEXT -1 12 "May 14, 1999" }}{SECT 0 {PARA 3 
"" 0 "Table of Contents" {TEXT -1 17 "Table of Contents" }}{EXCHG 
{PARA 268 "" 0 "" {HYPERLNK 17 "Introduction" 1 "" "Introduction" }}
{PARA 269 "" 0 "" {HYPERLNK 17 "Usage Examples" 1 "" "Usage Examples" 
}}{PARA 260 "" 0 "" {HYPERLNK 17 "Setup" 1 "" "Setup" }}{PARA 261 "" 
0 "" {HYPERLNK 17 "A convenient plotting procedure" 1 "" "A convenient
 plotting procedure" }}{PARA 262 "" 0 "" {HYPERLNK 17 "Legendre" 1 "" 
"Legendre" }}{PARA 263 "" 0 "" {HYPERLNK 17 "Chebyshev type 1" 1 "" "C
hebyshev type 1" }}{PARA 264 "" 0 "" {HYPERLNK 17 "Chebyshev type 2" 
1 "" "Chebyshev type 2" }}{PARA 265 "" 0 "" {HYPERLNK 17 "Hermite" 1 "
" "Hermite" }}{PARA 266 "" 0 "" {HYPERLNK 17 "Laguerre" 1 "" "Laguerre
" }}{PARA 267 "" 0 "" {HYPERLNK 17 "Gegenbauer (ultraspherical)" 1 "" 
"Gegenbauer (ultraspherical)" }}{PARA 270 "" 0 "" {HYPERLNK 17 "A Nume
rical Illustration" 1 "" "A Numerical Illustration" }}{PARA 271 "" 0 "
" {HYPERLNK 17 "The Polynomials Package" 1 "" "The Polynomials Package
" }}}}{SECT 1 {PARA 3 "" 0 "Introduction" {TEXT -1 12 "Introduction" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "This worksheet illustrates my " }
{TEXT 263 11 "polynomials" }{TEXT -1 68 " Maple package.  Its purpose \+
is to allow one to convert a truncated " }{HYPERLNK 17 "Taylor" 2 "Tay
lor" "" }{TEXT -1 36 " series polynomial in some variable " }{XPPEDIT 
18 0 "x;" "6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "   \+
 " }{XPPEDIT 18 0 "S[N](x) = sum(a[k]*x^k,k = 0 .. N);" "6#/-&%\"SG6#%
\"NG6#%\"xG-%$sumG6$*&&%\"aG6#%\"kG\"\"\")F*F2F3/F2;\"\"!F(" }{TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 17 "to an equivalent " }
{HYPERLNK 17 "orthogonal polynomial" 2 "orthopoly" "" }{TEXT -1 3 ",  \+
" }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "S[N](x) = sum(c[
k]*Q[k](x),k = 0 .. N);" "6#/-&%\"SG6#%\"NG6#%\"xG-%$sumG6$*&&%\"cG6#%
\"kG\"\"\"-&%\"QG6#F26#F*F3/F2;\"\"!F(" }{TEXT -1 2 "  " }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Q;" "6#%\"QG" }{TEXT -1 86 " \+
is one of P, T, U, H, L, or G.  These stand for the following orthogon
al polynomials:" }}{PARA 17 "" 0 "" {TEXT -1 4 "    " }{TEXT 257 1 "P
" }{TEXT -1 4 "    " }{HYPERLNK 17 "Legendre" 2 "orthopoly,P" "" }
{TEXT -1 5 "\n    " }{TEXT 258 1 "T" }{TEXT -1 4 "    " }{HYPERLNK 17 
"Chebyshev type 1" 2 "orthopoly,T" "" }{TEXT -1 6 " \n    " }{TEXT 
259 1 "U" }{TEXT -1 4 "    " }{HYPERLNK 17 "Chebyshev type 2" 2 "ortho
poly,U" "" }{TEXT -1 6 " \n    " }{TEXT 260 1 "H" }{TEXT -1 4 "    " }
{HYPERLNK 17 "Hermite" 2 "orthopoly,H" "" }{TEXT -1 6 " \n    " }
{TEXT 261 1 "L" }{TEXT -1 4 "    " }{HYPERLNK 17 "Laguerre" 2 "orthopo
ly,L" "" }{TEXT -1 6 " \n    " }{TEXT 262 1 "G" }{TEXT -1 4 "    " }
{HYPERLNK 17 "Gegenbauer" 2 "orthopoly,G" "" }{TEXT -1 18 " (ultrasphe
rical) " }}{PARA 0 "" 0 "" {TEXT -1 285 "There is also a conversion pr
ocedure to transform back to a truncated Taylor series.  Use of the co
nversion procedures is documented by way of examples in the next secti
on.  Following that section is a numerical illustration.  Finally, the
 last section contains the source code of the " }{TEXT 264 11 "polynom
ials" }{TEXT -1 9 " package." }}}}{SECT 1 {PARA 3 "" 0 "Usage Examples
" {TEXT -1 14 "Usage Examples" }}{SECT 1 {PARA 4 "" 0 "Setup" {TEXT 
-1 5 "Setup" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;" "6#%(re
startG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "with(polynomials);" "
6#-%%withG6#%,polynomialsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,%*conv
ert/GG%*convert/HG%*convert/LG%*convert/PG%*convert/TG%*convert/UG%/co
nvert/taylorG%%initG%*orthosubsG%)plotpolyG" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "s := series(cos(x+phi)*sinh(x)*exp(-x*y)+x*tan(x),x,4
);" "6#>%\"sG-%'seriesG6%,&*(-%$cosG6#,&%\"xG\"\"\"%$phiGF/F/-%%sinhG6
#F.F/-%$expG6#,$*&F.F/%\"yGF/!\"\"F/F/*&F.F/-%$tanG6#F.F/F/F.\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG++%\"xG-%$cosG6#%$phiG\"\"\",(*&
F'\"\"\"%\"yGF.!\"\"-%$sinGF)F0F.F.\"\"#,(*&F'\"\"\")F/\"\"#F6#F.F8F'#
F0\"\"$*&F1F.F/F6F.\"\"$-%\"OG6#F.\"\"%" }}}}{SECT 1 {PARA 4 "" 0 "A c
onvenient plotting procedure" {TEXT -1 31 "A convenient plotting proce
dure" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "plotpoly := proc( intlis
t::list(posint), ptype::name )\n\n  local p, k, xmin, xmax, ymin, ymax
, ytext, ttext, pargs;\n\n  if ptype='G' then\n    pargs := args[3], x
;\n  else\n    pargs := x;\n  fi;\n\n  xmin := -1;\n  xmax :=  1;\n  y
min := -1;\n  ymax :=  1;\n  if ptype='L' then\n    xmin :=  0;\n    x
max :=  5;\n    ymin := -5;\n    ymax :=  5;\n  elif ptype='H' then\n \+
   xmin :=  0;\n    xmax :=  3;\n    ymin := -2;\n    ymax :=  8;\n  e
lif ptype='U' then\n    ymin := -5;\n    ymax :=  5;\n  elif ptype='G'
 and args[3] > 1/2 then\n    ymin := -5;\n    ymax :=  5;\n  fi;\n\n  \+
p := [];\n  for k from 1 to nops(intlist) do\n    if ptype='H' then\n \+
     p := [ op(p),\n             plot( orthopoly[ptype](intlist[k],par
gs)/intlist[k]^3,\n                   x=xmin..xmax, \n                \+
   color=blue ) ];\n    else\n      p := [ op(p),\n             plot( \+
orthopoly[ptype](intlist[k],pargs),\n                   x=xmin..xmax, \+
\n                   color=blue ) ];\n    fi;\n  od;\n\n  ytext := cat
(ptype,\"[n](x)\");\n  if ptype='P' then\n    ttext := \"Legendre\";\n
  elif ptype='T' then\n    ttext := \"Chebyshev type 1\";\n  elif ptyp
e='U' then\n    ttext := \"Chebyshev type 2\";\n  elif ptype='H' then
\n    ttext := \"Hermite\";\n    ytext := cat(ptype,\"[n](x)/n^3\");\n
  elif ptype='L' then\n    ttext := \"Laguerre\";\n  elif ptype='G' th
en\n    ttext := cat(\"Gegenbauer (a=\",convert(args[3],string),\")\")
;\n  fi;\n\n  plots[display]( [op(p),plot(0,x=xmin..xmax,color=black)]
, \n                  view=[xmin..xmax,ymin..ymax],\n                 \+
 title=ttext, labels=[\"x\",ytext],\n                  axes=box );\n\n
end:" "6#>%)plotpolyGR6$'%(intlistG-%%listG6#%'posintG'%&ptypeG%%nameG
7+%\"pG%\"kG%%xminG%%xmaxG%%yminG%%ymaxG%&ytextG%&ttextG%&pargsG6\"F:C
-@%/F..%\"GG>F96$&%%argsG6#\"\"$%\"xG>F9FF>F3,$\"\"\"!\"\">F4\"\"\">F5
,$\"\"\"FK>F6\"\"\"@*/F..%\"LGC&>F3\"\"!>F4\"\"&>F5,$\"\"&FK>F6\"\"&/F
..%\"HGC&>F3FY>F4\"\"$>F5,$\"\"#FK>F6\"\")/F..%\"UGC$>F5,$\"\"&FK>F6\"
\"&3/F..F?2*&\"\"\"\"\"\"\"\"#FK&FC6#\"\"$C$>F5,$\"\"&FK>F6\"\"&>F17\"
?(F2\"\"\"Ffp-%%nopsG6#F(%%trueG@%/F..F]o>F17$-%#opG6#F1-%%plotG6%*&-&
%*orthopolyG6#F.6$&F(6#F2F9Ffp*$&F(6#F2\"\"$FK/FF;F3F4/%&colorG%%blueG
>F17$-F_r6#F1-Fbr6%-&Fgr6#F.6$&F(6#F2F9/FF;F3F4/FcsFds>F7-%$catG6$F.Q'
[n](x)6\"@./F..%\"PG>F8Q)LegendreFit/F..%\"TG>F8Q1Chebyshev~type~1Fit/
F..Fio>F8Q1Chebyshev~type~2Fit/F..F]oC$>F8Q(HermiteFit>F7-Fft6$F.Q+[n]
(x)/n^3Fit/F..FV>F8Q)LaguerreFit/F..F?>F8-Fft6%Q/Gegenbauer~(a=Fit-%(c
onvertG6$&FC6#\"\"$%'stringGQ\")Fit-&%&plotsG6#%(displayG6'7$-F_r6#F1-
Fbr6%FY/FF;F3F4/Fcs%&blackG/%%viewG7$;F3F4;F5F6/%&titleGF8/%'labelsG7$
Q\"xFitF7/%%axesG%$boxGF:F:F:" }}}}{SECT 1 {PARA 4 "" 0 "Legendre" 
{TEXT -1 8 "Legendre" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "convert(
s,P,x);" "6#-%(convertG6%%\"sG%\"PG%\"xG" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#,.*&,(*&-%$cosG6#%$phiG\"\"\")%\"yG\"\"#\"\"\"#\"\"$\"#5F'#\"\"%
\"\"&*&-%$sinGF)F+F-F+#F1F5F+-&%\"PG6#F+6#%\"xGF+F+*&,(*&F'F/F-F/#!\"#
F1F7FC#F.F1F+F+-&F<6#F.F>F+F+*&,(F&#F+F5F'#FD\"#:F6#F.F5F+-&F<6#F1F>F+
F+FB#!\"\"F1F7FR#F+F1F+" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "conv
ert(%, taylor, x);" "6#-%(convertG6%%\"%G%'taylorG%\"xG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*&-%$cosG6#%$phiG\"\"\"%\"xGF)F)*&,(*&F%\"\"\"
%\"yGF)!\"\"-%$sinGF'F0F)F)F))F*\"\"#F.F)*&,(*&F%F.)F/F4F.#F)F4F%#F0\"
\"$*&F1F)F/F.F)F))F*F;F.F)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "s
implify(convert(s,polynom)-%);" "6#-%)simplifyG6#,&-%(convertG6$%\"sG%
(polynomG\"\"\"%\"%G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "plotpoly([1, 2, 3, 4, 5],P);" "
6#-%)plotpolyG6$7'\"\"\"\"\"#\"\"$\"\"%\"\"&%\"PG" }}{PARA 13 "" 1 "" 
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{PARA 3 "" 0 "A Numerical Illustration" {TEXT -1 24 "A Numerical Illus
tration" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Start with some functio
n of " }{TEXT 265 1 "t" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "f := (f1,f2,f3,A1,A2,phi1,phi2,t) ->\n     (1+sin(f1*t)
)*(A1*cos(f2*t+phi1) + A2*cos(f3*t+phi2)):" "6#>%\"fGR6*%#f1G%#f2G%#f3
G%#A1G%#A2G%%phi1G%%phi2G%\"tG7\"6$%)operatorG%&arrowG6\"*&,&\"\"\"\"
\"\"-%$sinG6#*&F'F7F.F7F7F7,&*&F*F7-%$cosG6#,&*&F(F7F.F7F7F,F7F7F7*&F+
F7-F?6#,&*&F)F7F.F7F7F-F7F7F7F7F3F3F3" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "F(t) = f(nu[1],nu[2],nu[3],A[1],A[2],phi[1],phi[2],t+b)
;" "6#/-%\"FG6#%\"tG-%\"fG6*&%#nuG6#\"\"\"&F,6#\"\"#&F,6#\"\"$&%\"AG6#
\"\"\"&F66#\"\"#&%$phiG6#\"\"\"&F=6#\"\"#,&F'\"\"\"%\"bGFD" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"tG*&,&\"\"\"F*-%$sinG6#*&&%#nuG6
#F*F*,&F'F*%\"bGF*F*F*F*,&*&&%\"AGF1F*-%$cosG6#,&*&&F06#\"\"#F*F2\"\"
\"F*&%$phiGF1F*F*F**&&F7F>F*-F96#,&*&&F06#\"\"$F*F2F@F*&FBF>F*F*F*F*" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "It looks like this:" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "plot([f(5,2,8,.4,.5,.5,.3,t), 0],t = 0 \+
.. 2*Pi,color = [blue, black],labels = [\"t\", \"F(t)\"]);" "6#-%%plot
G6&7$-%\"fG6*\"\"&\"\"#\"\")$\"\"%!\"\"$\"\"&!\"\"$\"\"&!\"\"$\"\"$!\"
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1#=_!=?*zi'F+7$$\"1d5t08z._F\\w$!1\"\\%zANfIZF+7$$\"1!f![/*ojB&F\\w$!1
X$4aN!o7AF+7$$\"16Z[al-`_F\\w$!1a5*>K()o](F/7$$\"1J))[/Uop_F\\w$\"1:mF
#GOa%yF/7$$\"1_H\\a=M'G&F\\w$\"1?'fZ@z\\N#F+7$$\"1tq\\/&**HI&F\\w$\"1n
*\\DN\"*3#RF+7$$\"1%>,X:d'>`F\\w$\"1<Jw^$pAW&F+7$$\"1:`]/[JO`F\\w$\"1;
DT'=V-)oF+7$$\"1O%4XXsHN&F\\w$\"1,7yi%o%)>)F+7$$\"1dN^/,jp`F\\w$\"1!pX
^09WO*F+7$$\"1)><q`pES&F\\w$\"1[^$Q5xB6\"F\\w7$$\"1S3_p*3dV&F\\w$\"1!H
[^\"Q$Q?\"F\\w7$$\"1\\nkF)oRW&F\\w$\"1na*o7JG@\"F\\w7$$\"1gEx&oGAX&F\\
w$\"107&)[-K;7F\\w7$$\"1q&)*Qa)[gaF\\w$\"1]xFq[R97F\\w7$$\"1\"[C?S[(oa
F\\w$\"1px]@$*=27F\\w7$$\"1-jF=\"o_[&F\\w$\"11(R$4tmx6F\\w7$$\"1A\"GX$
yy,bF\\w$\"1%)*4+(ecH6F\\w7$$\"1]ilTw9KbF\\w$\"1*4;z8=2+\"F\\w7$$\"1yV
y[u]ibF\\w$\"1j'*)o:<nM)F+7$$\"10D\"fDnGf&F\\w$\"1z'*\\:=*)4lF+7$$\"1L
1/jqABcF\\w$\"1%*\\nl3k(o%F+7$$\"1W>6GG-ecF\\w$\"1m#o'Q08DGF+7$$\"1aK=
$f=Gp&F\\w$\"1l(yJtkTO\"F+7$$\"1mXDeVhFdF\\w$\"1BHI\"[n:y$F/7$$\"1xeKB
,TidF\\w$!1$)z3wFEU:F/7$$\"1`m4g0>qdF\\w$!1P[HSu[)>#F/7$$\"1Iu'o*4(zx&
F\\w$!1Qs'R\\a')o#F/7$$\"12#QOV^dy&F\\w$!1an;=t5GIF/7$$\"1%)*3/(=`$z&F
\\w$!1&y%HKGyKKF/7$$\"1Q0&Ru#44eF\\w$!1ex+5_9.LF/7$$\"1#4#\\<OlCeF\\w$
!1S9qM![/.$F/7$$\"1+_dk`xbeF\\w$!10]M+8zZ>F/7$$\"12$e;6(*o)eF\\w$!1)>Z
K^Pl?)Fbu7$$\"1q>qe;E`fF\\w$\"1*>(G>Kyp9Fhu7$$\"1Lcu0ii>gF\\w$\"1gclv*
Q9)*)Fbu7$$\"1:=>Xb9$3'F\\w$\"1Ed`*z3<U)F/7$$\"1**zj%)[mYhF\\w$\"1j-KA
3GNFF+7$$\"1*\\Lr)\\z!='F\\w$\"1yM_u(zn;%F+7$$\"1***G'*3D\\@'F\\w$\"1K
A%*\\M]8dF+7$$\"1+X7#>b!\\iF\\w$\"1$H4v;p`;(F+7$$\"1++i%H&=$G'F\\w$\"1
)y<%pB,(G)F+-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-F$6$7S7$F(F(7$FQF(7$F
_oF(7$FgqF(7$F[sF(7$F_tF(7$FitF(7$F^uF(7$F`wF(7$FjwF(7$FhyF(7$F\\[lF(7
$Fd]lF(7$Fh^lF(7$F`alF(7$FjalF(7$FdblF(7$FbdlF(7$F\\elF(7$FjflF(7$Fdgl
F(7$F\\jlF(7$F`[mF(7$Fd\\mF(7$F\\_mF(7$F``mF(7$F^bmF(7$F\\dmF(7$FdfmF(
7$F^gmF(7$FfimF(7$Fd[nF(7$Fb]nF(7$Fj_nF(7$F^anF(7$FbbnF(7$F`dnF(7$F^fn
F(7$FhfnF(7$F\\hnF(7$F`inF(7$Fh[oF(7$F`^oF(7$Fd_oF(7$Fh`oF(7$F`coF(7$F
jcoF(7$FddoF(7$FheoF(-F]fo6&F_foF(F(F(F\\fo-%&TITLEG6$%!G-%%FONTG6$%*H
ELVETICAG\"#=-%(SCALINGG6#%.UNCONSTRAINEDG-%+AXESLABELSG6%Q\"t6\"Q%F(t
)FjjoF]jo-%*AXESTICKSG6%%(DEFAULTGF_[p-F^jo6$F`jo\"#7-%*AXESSTYLEG6#%&
FRAMEG-%%VIEWG6$;F($\"+3`=$G'!\"*F_[p" 1 2 0 1 0 2 6 1 3 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "To fourt
h order in t, we have, analytically, " }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "N := 5:" "6#>%\"NG\"\"&" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "F(t) = series(f(nu[1],nu[2],nu[3],A[1],A[2],phi[1],phi[
2],t+b),t,N);" "6#/-%\"FG6#%\"tG-%'seriesG6%-%\"fG6*&%#nuG6#\"\"\"&F/6
#\"\"#&F/6#\"\"$&%\"AG6#\"\"\"&F96#\"\"#&%$phiG6#\"\"\"&F@6#\"\"#,&F'
\"\"\"%\"bGFGF'%\"NG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"tG
+/F'*&,&\"\"\"F+-%$sinG6#*&&%#nuG6#F+F+%\"bGF+F+F+,&*&&%\"AGF2F+-%$cos
G6#,&*&&F16#\"\"#F+F3\"\"\"F+&%$phiGF2F+F+F+*&&F7F>F+-F96#,&*&&F16#\"
\"$F+F3F@F+&FBF>F+F+F+F+\"\"!,&*&F*F@,&*(F6F@-F-F:F+F=F@!\"\"*(FDF@-F-
FFF+FIF@FSF+F+*(-F9F.F+F0F@F4F@F+\"\"\",(*(FWF@F0F@FPF@F+*&F*F@,&*(FDF
@FEF@)FIF?F@#FSF?*(F6F@F8F@)F=F?F@FinF+F+*(F,F+)F0F?F@F4F@Fin\"\"#,**(
FWF@F0F@FfnF@F+*(F,F@F]oF@FPF@Fin*&F*F@,&*(FDF@FUF@)FIFKF@#F+\"\"'*(F6
F@FRF@)F=FKF@FfoF+F+*(FWF@)F0FKF@F4F@#FSFgo\"\"$,,*(F,F@)F0\"\"%F@F4F@
#F+\"#C*(FWF@F0F@FcoF@F+*&F*F@,&*(F6F@F8F@)F=FapF@Fbp*(FDF@FEF@)FIFapF
@FbpF+F+*(F,F@F]oF@FfnF@Fin*(FWF@F[pF@FPF@F\\p\"\"%-%\"OGF2\"\"&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "We'll actually use 20th order for \+
numerical calculations." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "S :=
 series(f(nu[1],nu[2],nu[3],A[1],A[2],phi[1],phi[2],t+b),t,21):" "6#>%
\"SG-%'seriesG6%-%\"fG6*&%#nuG6#\"\"\"&F,6#\"\"#&F,6#\"\"$&%\"AG6#\"\"
\"&F66#\"\"#&%$phiG6#\"\"\"&F=6#\"\"#,&%\"tG\"\"\"%\"bGFEFD\"#@" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Create a function that numerically
 evaluates the Taylor series representation." }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "polyS := fn(convert(S,polynom),nu[1],nu[2],nu[3],A[1]
,A[2],phi[1],phi[2],b,t):" "6#>%&polySG-%#fnG6,-%(convertG6$%\"SG%(pol
ynomG&%#nuG6#\"\"\"&F.6#\"\"#&F.6#\"\"$&%\"AG6#\"\"\"&F86#\"\"#&%$phiG
6#\"\"\"&F?6#\"\"#%\"bG%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "F
or example, " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "t0 := .8:" "6#>
%#t0G$\"\")!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "polyS(5,2,8
,.4,.5,.5,.3,t0,t);" "6#-%&polySG6+\"\"&\"\"#\"\")$\"\"%!\"\"$\"\"&!\"
\"$\"\"&!\"\"$\"\"$!\"\"%#t0G%\"tG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
,L$\"+%[#p2i!#6\"\"\"%\"tG$!+h6+'R\"!\"**$)F(\"\"#\"\"\"$\"+'y*Q/lF+*$
)F(\"\"$F/$\"+$\\#4VK!\")*$)F(\"\"%F/$!+Mw#35#!\"(*$)F(\"\"&F/$!+9*z\"
GDF=*$)F(\"\"'F/$\"+efsU9!\"'*$)F(\"\"(F/$\"+5(*z'e*F=*$)F(\"\")F/$!+@
\"zMl%FH*$)F(\"\"*F/$!+z<4\"=#FH*$)F(\"#5F/$\"+l.A`*)FH*$)F(\"#6F/$\"+
]9N.LFH*$)F(\"#7F/$!+*z'yc6!\"&*$)F(\"#8F/$!+p\\!ob$FH*$)F(\"#9F/$\"+M
Q(y2\"Fao*$)F(\"#:F/$\"+[S>bGFH*$)F(\"#;F/$!+$Qx**f(FH*$)F(\"#<F/$!+J<
@s<FH*$)F(\"#=F/$\"+T4Y*>%FH*$)F(\"#>F/$\"+at$Rv)F=*$)F(\"#?F/$!+Cv+o=
FH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Compare the Taylor series w
ith the original function:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "p
lot([polyS(5,2,8,.4,.5,.5,.3,t0,t), f(5,2,8,.4,.5,.5,.3,t+t0), 0],t = \+
0 .. .8,color = [red, blue, black],labels = [\"t\", \"F(t)\"]);" "6#-%
%plotG6&7%-%&polySG6+\"\"&\"\"#\"\")$\"\"%!\"\"$\"\"&!\"\"$\"\"&!\"\"$
\"\"$!\"\"%#t0G%\"tG-%\"fG6*\"\"&\"\"#\"\")$\"\"%!\"\"$\"\"&!\"\"$\"\"
&!\"\"$\"\"$!\"\",&F:\"\"\"F9FN\"\"!/F:;FO$\"\")!\"\"/%&colorG7%%$redG
%%blueG%&blackG/%'labelsG7$Q\"t6\"Q%F(t)6\"" }}{PARA 13 "" 1 "" 
{GLPLOT2D 562 299 299 {PLOTDATA 2 "6,-%'CURVESG6$7co7$\"\"!$\"1+++%[#p
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F(7$FiqF(7$F^rF(7$FcrF(7$FhrF(7$F]sF(7$FbsF(7$FgsF(7$FatF(7$F[uF(7$Feu
F(7$F_vF(7$FdvF(7$FivF(7$F^wF(7$FcwF(7$FhwF(7$F]xF(7$FbxF(7$FgxF(7$F\\
yF(7$FayF(7$F[zF(7$FezF(7$Fi[lF(7$Fc\\lF(7$F]]lF(7$Fg]lF(7$F[_lF(7$Fda
lF(-Fial6&F[blF(F(F(Fa^m-%&TITLEG6$%!G-%%FONTG6$%*HELVETICAG\"#=-%(SCA
LINGG6#%.UNCONSTRAINEDG-%+AXESLABELSG6%Q\"t6\"Q%F(t)FjbmF]bm-%*AXESTIC
KSG6%%(DEFAULTGF_cm-F^bm6$F`bm\"#7-%*AXESSTYLEG6#%&FRAMEG-%%VIEWG6$;F(
$\"\")!\"\"F_cm" 1 2 0 1 0 2 6 1 3 2 1.000000 45.000000 45.000000 0 }}
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The Taylor series is good to abo
ut " }{XPPEDIT 18 0 "t = .6;" "6#/%\"tG$\"\"'!\"\"" }{TEXT -1 2 ". " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Check the conversion routines:" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "convert(S,P,t):" "6#-%(conver
tG6%%\"SG%\"PG%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "simplify
(convert(S,polynom)-convert(%,taylor,t));" "6#-%)simplifyG6#,&-%(conve
rtG6$%\"SG%(polynomG\"\"\"-F(6%%\"%G%'taylorG%\"tG!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Ta
ke a look at the first few terms of the Legendre polynomial representa
tion:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F(t) = collect(convert
(series(f(nu[1],nu[2],nu[3],A[1],A[2],phi[1],phi[2],t+b),t,N),P,t),[se
q(P[k](t),k = 1 .. N-1), A[1], A[2], sin, cos],factor);" "6#/-%\"FG6#%
\"tG-%(collectG6%-%(convertG6%-%'seriesG6%-%\"fG6*&%#nuG6#\"\"\"&F56#
\"\"#&F56#\"\"$&%\"AG6#\"\"\"&F?6#\"\"#&%$phiG6#\"\"\"&FF6#\"\"#,&F'\"
\"\"%\"bGFMF'%\"NG%\"PGF'7'-%$seqG6$-&FP6#%\"kG6#F'/FX;\"\"\",&FOFM\"
\"\"!\"\"&F?6#\"\"\"&F?6#\"\"#%$sinG%$cosG%'factorG" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#/-%\"FG6#%\"tG,.*&,&*&,(**&%#nuG6#\"\"#\"\"\",(*$)&F/
6#\"\"\"F1F2\"\"$!#5F8*$)F.F1F2F8F8-%$sinG6#,&*&F.F8%\"bGF8F8&%$phiGF7
F8F8-F>6#*&F6F8FBF2F8#F8\"#5*(F.F2,&F:F8F;F8F8F=F2FH**F6F2,(F:F8F4F8F;
F9F8-%$cosGF?F8-FOFFF8#!\"\"FIF8&%\"AGF7F8F8*&,(**&F/6#F9F2,(*$)FXF1F2
F8F4F9F:F8F8-F>6#,&*&FXF8FBF2F8&FDF0F8F8FEF2FH*(FXF2,&FenF8F:F8F8FgnF2
FH**F6F2,(F:F8FenF9F4F8F8-FOFhnF8FPF2FQF8&FTF0F8F8F8-&%\"PGF7F&F8F8*&,
&*&,(*(,,*$)F.\"\"%F2#F8\"#UF;#FRF9*$)F6F]pF2F^pF4F`p*&F5F2F<F2#F8\"\"
(F8FNF2FEF2F8*,F6F2F.F2,(F;F8!\"(F8F4F8F8FPF2F=F2#F1\"#@*(F<F2,&F;F8!#
9F8F8FNF2F^pF8FSF2F8*&,(*(,,F4F`pFenF`p*$)FXF]pF2F^pFapF^p*&F5F2FfnF2F
dpF8F`oF2FEF2F8*,F6F2FXF2,(F4F8FenF8FhpF8F8FPF2FgnF2Fip*(FfnF2,&F]qF8F
enF8F8F`oF2F^pF8FaoF2F8F8-&FdoF0F&F8F8*&,&*&,(**F.F2,&F4F9F;F8F8F=F2FE
F2#F8\"#:*&F=F2)F.F9F2Far**F6F2,&F;F9F4F8F8FNF2FPF2#FRFbrF8FSF2F8*&,(*
*FXF2,&FenF8F4F9F8FgnF2FEF2Far*&FgnF2)FXF9F2Far**F6F2,&FenF9F4F8F8F`oF
2FPF2FgrF8FaoF2F8F8-&FdoFYF&F8F8*&,&*&,(*(,(F[p#F8\"$0\"FapFhsFcp#F1\"
#NF8FNF2FEF2F8*,F6F2F.F2,&F4F8F;F8F8FPF2F=F2#F]pFis*&FNF2F\\pF2FhsF8FS
F2F8*&,(*(,(FbqFhsFdqFjsFapFhsF8F`oF2FEF2F8*,F6F2FXF2,&FenF8F4F8F8FPF2
FgnF2F^t*&F`oF2FcqF2FhsF8FaoF2F8F8-&Fdo6#F]pF&F8F8*&,(*(,.Fcp#F8\"#?F4
#FR\"\"'F[p#F8\"$?\"F;F`uFapFbuF8F8F8FNF2FEF2F8*,F6F2F.F2,(F:F8F;F8F4F
8F8FPF2F=F2#F8\"#I*&,(F;F`uF8F8F[pFbuF8FNF2F8F8FSF2F8*&,(*(,.FdqF^uFap
FbuFbqFbuF4F`uFenF`uF8F8F8F`oF2FEF2F8*,F6F2FXF2,(FenF8F:F8F4F8F8FPF2Fg
nF2Ffu*&,(FenF`uF8F8FbqFbuF8F`oF2F8F8FaoF2F8" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 91 "Convert the Taylor series to different representations,
 for example Legendre and Chebyshev:" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "polyP := fn(convert(S,P,t),nu[1],nu[2],nu[3],A[1],A[2],
phi[1],phi[2],b,t):" "6#>%&polyPG-%#fnG6,-%(convertG6%%\"SG%\"PG%\"tG&
%#nuG6#\"\"\"&F/6#\"\"#&F/6#\"\"$&%\"AG6#\"\"\"&F96#\"\"#&%$phiG6#\"\"
\"&F@6#\"\"#%\"bGF-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "polyT :=
 fn(convert(S,T,t),nu[1],nu[2],nu[3],A[1],A[2],phi[1],phi[2],b,t):" "6
#>%&polyTG-%#fnG6,-%(convertG6%%\"SG%\"TG%\"tG&%#nuG6#\"\"\"&F/6#\"\"#
&F/6#\"\"$&%\"AG6#\"\"\"&F96#\"\"#&%$phiG6#\"\"\"&F@6#\"\"#%\"bGF-" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Numerically, " }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 19 1 "polyP(5,2,8,.4,.5,.5,.3,t0,t);" "6#-%&polyPG6+
\"\"&\"\"#\"\")$\"\"%!\"\"$\"\"&!\"\"$\"\"&!\"\"$\"\"$!\"\"%#t0G%\"tG
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L-&%\"PG6#\"#96#%\"tG$!+9$p/h%!\"
*-&F&6#\"#7F)$!+N&==l\"!\")-&F&6#\"\"'F)$!+'3!4*>\"!\"(-&F&6#\"#?F)$!+
tF'4U\"!#6-&F&6#\"\")F)$!+hiyUyF4-&F&6#\"#<F)$\"+^^(pF\"!#5-&F&6#\"#>F
)$\"+30])H\"FB-&F&6#\"\"&F)$\"+,6bDnF4-&F&6#\"#5F)$!+(>u`-%F4-&F&6#\"#
8F)$\"+sf\\8PF--&F&6#\"#:F)$\"+T4.?#*FO-&F&6#\"#=F)$!++ls:9FO-&F&6#\"
\"*F)$\"+7A\"Rx#F4$!+%H;YI#F4\"\"\"-&F&6#\"\"(F)$\"+(=nY)\\F4-&F&6#\"#
6F)$\"+HFU57F4-&F&6#\"#;F)$!+]=186F--&F&6#\"\"#F)$!+6#=s+\"F;-&F&6#\"
\"%F)$!+$4m%R8F;-&F&6#FfpF)$\"+')e2ZMF4-&F&6#\"\"$F)$\"+V#=NX'F4" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "polyT(5,2,8,.4,.5,.5,.3,t0,t);" 
"6#-%&polyTG6+\"\"&\"\"#\"\")$\"\"%!\"\"$\"\"&!\"\"$\"\"&!\"\"$\"\"$!
\"\"%#t0G%\"tG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L-&%\"TG6#\"#96#%\"
tG$!+Oi*[b\"!\"*-&F&6#\"\")F)$!+nem/T!\")-&F&6#\"\"*F)$\"+F%R&*H\"F4-&
F&6#\"\"&F)$\"+O+yl]F4-&F&6#\"#;F)$!+%4O=K$!#5-&F&6#\"#8F)$\"+>S!QI\"F
--&F&6#\"#7F)$!+7J(z<'F--&F&6#\"\"(F)$\"+&G(*=&GF4-&F&6#\"#<F)$\"+mugS
O!#6-&F&6#\"#5F)$!+%p2Tw\"F4-&F&6#\"#>F)$\"+'3i$RL!#7-&F&6#\"\"$F)$\"+
SH)oO(F4-&F&6#\"#?F)$!++;%Hc$Fgo-&F&6#\"\"'F)$!*iB>*y!\"($!+,6c;))F4\"
\"\"-&F&6#\"\"#F)$!+<'zAh\"Fjp-&F&6#\"\"%F)$!+LK%fB\"Fjp-&F&6#\"#=F)$!
+l([>#RFjn-&F&6#\"#:F)$\"+mW(e&GFG-&F&6#\"#6F)$\"+C\"R-z%F--&F&6#F]qF)
$\"+*RGn%))F4" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "polyS(5,2,8,.4
,.5,.5,.3,t0,t);" "6#-%&polySG6+\"\"&\"\"#\"\")$\"\"%!\"\"$\"\"&!\"\"$
\"\"&!\"\"$\"\"$!\"\"%#t0G%\"tG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L$
\"+%[#p2i!#6\"\"\"%\"tG$!+h6+'R\"!\"**$)F(\"\"#\"\"\"$\"+'y*Q/lF+*$)F(
\"\"$F/$\"+$\\#4VK!\")*$)F(\"\"%F/$!+Mw#35#!\"(*$)F(\"\"&F/$!+9*z\"GDF
=*$)F(\"\"'F/$\"+efsU9!\"'*$)F(\"\"(F/$\"+5(*z'e*F=*$)F(\"\")F/$!+@\"z
Ml%FH*$)F(\"\"*F/$!+z<4\"=#FH*$)F(\"#5F/$\"+l.A`*)FH*$)F(\"#6F/$\"+]9N
.LFH*$)F(\"#7F/$!+*z'yc6!\"&*$)F(\"#8F/$!+p\\!ob$FH*$)F(\"#9F/$\"+MQ(y
2\"Fao*$)F(\"#:F/$\"+[S>bGFH*$)F(\"#;F/$!+$Qx**f(FH*$)F(\"#<F/$!+J<@s<
FH*$)F(\"#=F/$\"+T4Y*>%FH*$)F(\"#>F/$\"+at$Rv)F=*$)F(\"#?F/$!+Cv+o=FH
" }}}}{SECT 1 {PARA 3 "" 0 "The Polynomials Package" {TEXT -1 23 "The \+
Polynomials Package" }}{PARA 5 "" 0 "" {TEXT -1 55 "Contents of the fi
le c:\\Maple\\polynomials\\polynomials.p" }}{PARA 17 "" 0 "" {TEXT -1 
11890 "#==============================================================
===============\n# Polynomial conversion functions\n#=================
============================================================\n# Marc A
. Murison\n# Astronomical Applications Dept.\n# U.S. Naval Observatory
\n# 3450 Massachusetts Ave., NW\n# Washignton, DC 20392\n# murison@aa.
usno.navy.mil\n# http://aa.usno.navy.mil/murison/\n#==================
===========================================================\n\n\n#----
-----------------------------------------------------------\n# Convert
 a Taylor series in x to orthogonal polynomials in x.\n#\n# Usage:  co
nvert( p, ptype, x ), where \n#    p     = a polynomial\n#    ptype = \+
one of the polynomial types\n#    x     = the independent variable (i.
e., p = p(x))\n#\n# Polynomial types are:\n#    T    Chebyshev type 1
\n#    U    Chebyshev type 2\n#    P    Legendre\n#    H    Hermite\n#
    L    Laguerre\n#--------------------------------------------------
-------------\npolynomials[`convert/T`] := proc( poly::\{`+`,series\},
 x::name )\n    global T, __T;\n    T := 'T';\n    `polynomials/conver
t_terms`(poly,x,T,args[3..nargs]);\n    subs( __T=T, % );\nend:\n\npol
ynomials[`convert/U`] := proc( poly::\{`+`,series\}, x::name )\n    gl
obal U, __T;\n    U := 'U';\n    `polynomials/convert_terms`(poly,x,U,
args[3..nargs]);\n    subs( __T=U, % );\nend:\n\npolynomials[`convert/
P`] := proc( poly::\{`+`,series\}, x::name )\n    global P, __T;\n    \+
P := 'P';\n    `polynomials/convert_terms`(poly,x,P,args[3..nargs]);\n
    subs( __T=P, % );\nend:\n\npolynomials[`convert/H`] := proc( poly:
:\{`+`,series\}, x::name )\n    global H, __T;\n    H := 'H';\n    `po
lynomials/convert_terms`(poly,x,H,args[3..nargs]);\n    subs( __T=H, %
 );\nend:\n\npolynomials[`convert/L`] := proc( poly::\{`+`,series\}, x
::name )\n    global L, __T;\n    L := 'L';\n    `polynomials/convert_
terms`(poly,x,L,args[3..nargs]);\n    subs( __T=L, % );\nend:\n\npolyn
omials[`convert/G`] := proc( poly::\{`+`,series\}, x::name, a::algebra
ic )\n    global G, __T;\n    G := 'G';\n    `polynomials/convert_term
s`(poly,x,G,a,args[4..nargs]);\n    subs( __T=G, % );\nend:\n\n\n#----
-----------------------------------------------------------\n# Convert
 an orthogonal polynomial in x to a Taylor series in x.\n#\n# Usage:  \+
convert( p, taylor, x ), where \n#    p = a polynomial\n#    x = the i
ndependent variable (i.e., p = p(x))\n#\n# Polynomial types are:\n#   \+
 T    Chebyshev type 1\n#    U    Chebyshev type 2\n#    P    Legendre
\n#    H    Hermite\n#    L    Laguerre\n#----------------------------
-----------------------------------\npolynomials[`convert/taylor`] := \+
proc( poly::`+`, x::name )\n    orthosubs( args );\n    collect( %, x \+
);\nend:\n\n\n\n#-----------------------------------------------------
----------\n# Substitute for T, U, P, H, L, or G, in the polynomial po
ly,\n# the corresponding orthogonal polynomial terms.  For example,\n#
 \n# > randpoly(x);\n# \n#                  5       4       3       2
\n#                 x  - 47 x  - 91 x  - 47 x  - 61 x + 41\n# \n# > co
nvert(%,P,x);\n# \n#     1222           239   1618           4031     \+
      376\n#   - ---- P[2](x) + --- - ---- P[3](x) - ---- P[1](x) - --
- P[4](x)\n#      21            15     45             35            35
\n# \n#          + 8/63 P[5](x)\n# \n# > orthosubs(%,x);\n# \n#       \+
           5       4       3       2\n#                 x  - 47 x  - 9
1 x  - 47 x  - 61 x + 41\n# \n#---------------------------------------
------------------------\npolynomials[orthosubs] := proc( poly::`+`, x
::name )\n\n    local N, k, p, pargs, ptype, xpart;\n\n    #detect the
 orthogonal polynomial type Q\n    xpart := select( has, poly, x );\n \+
   if type(xpart,`+`) then                #xpart = A*Pn(x) + B*Pm(x) +
 ...\n        op(1,select(has,poly,x));        #grab a term\n        o
p(select(has,[op(%)],x));        #isolate Q[n](x)\n    else           \+
                     #xpart = A*Pn(x)\n        select( has, xpart, x )
;        #isolate Q[n](x)\n    fi;\n    ptype := op([0,0],%);         \+
       #select Q\n    debug_print(procname,cat(\"polynomial type = \",
ptype),5);\n    \n    if ptype='G' then\n        if nargs=2 then\n    \+
        pargs := 0,x;\n        else\n            pargs := args[3],x;\n
        fi;\n    else\n        pargs := x;\n    fi;\n\n    #determine \+
the polynomial order\n    N := 0;\n    for p in poly do\n        if ha
s(p,ptype) then\n            N := max( N, op([0,1],op(select(has,[op(p
)],x))) );\n        fi;\n    od;\n\n    #substitute for Q[k](x) the or
thogonal polynomial term in x\n    subs( seq( ptype[k](x)=orthopoly[pt
ype](k,pargs), k=0..N ), poly );\n\nend:\n\n\n#-----------------------
----------------------------------------\n# A plotting procedure for d
emonstration purposes.\n#---------------------------------------------
------------------\npolynomials[plotpoly] := proc( intlist::list(posin
t), ptype::name )\n\n    local p, k, xmin, xmax, ymin, ymax, ytext, tt
ext, pargs;\n\n    if ptype='G' then\n        pargs := args[3], x;\n  \+
  else\n        pargs := x;\n    fi;\n\n    xmin := -1;\n    xmax :=  \+
  1;\n    ymin := -1;\n    ymax :=    1;\n    if ptype='L' then\n     \+
   xmin :=    0;\n        xmax :=    5;\n        ymin := -5;\n        \+
ymax :=    5;\n    elif ptype='H' then\n        xmin :=    0;\n       \+
 xmax :=    3;\n        ymin := -2;\n        ymax :=    8;\n    elif p
type='U' then\n        ymin := -5;\n        ymax :=    5;\n    elif pt
ype='G' and args[3] > 1/2 then\n        ymin := -5;\n        ymax :=  \+
  5;\n    fi;\n\n    p := [];\n    for k from 1 to nops(intlist) do\n \+
       if ptype='H' then\n            p := [ op(p),\n                 \+
  plot( orthopoly[ptype](intlist[k],pargs)/intlist[k]^3,\n            \+
             x=xmin..xmax, color=blue ) ];\n        else\n            \+
p := [ op(p),\n                   plot( orthopoly[ptype](intlist[k],pa
rgs),\n                         x=xmin..xmax, color=blue ) ];\n       \+
 fi;\n    od;\n\n    ytext := cat(ptype,\"[n](x)\");\n    if ptype='P'
 then\n        ttext := \"Legendre\";\n    elif ptype='T' then\n      \+
  ttext := \"Chebyshev type 1\";\n    elif ptype='U' then\n        tte
xt := \"Chebyshev type 2\";\n    elif ptype='H' then\n        ttext :=
 \"Hermite\";\n        ytext := cat(ptype,\"[n](x)/n^3\");\n    elif p
type='L' then\n        ttext := \"Laguerre\";\n    elif ptype='G' then
\n        ttext := cat(\"Gegenbauer (a=\",convert(args[3],string),\")
\");\n    fi;\n\n    plots[display]( [op(p),plot(0,x=xmin..xmax,color=
black)], \n                    view=[xmin..xmax,ymin..ymax],\n        \+
            title=ttext, labels=[\"x\",ytext] );\n\nend:\n\n\n\n#-----
----------------------------------------------------------\n# Initiali
zation function, where we define the workhorse \n# routines used by th
e public polynomials package routines.\n#-----------------------------
----------------------------------\npolynomials[init] := proc()\n\n   \+
 global `polynomials/convert_terms`, `polynomials/get_horner_coeffs`;
\n\n#---------------------------------------------------------------\n
# The polynomial conversion workhorse routine.  \n# Not intended for p
ublic use.\n#---------------------------------------------------------
------\n`polynomials/convert_terms` := proc( poly::\{`+`,series\}, x::
name, ptype::name )\n\n    global __T, `polynomials/get_horner_coeffs`
;\n    local n,        #order of the polynomial \n          Q,        \+
#local storage of the polynomial\n          r,        #structure to ho
ld the recursion formulae for x*T[n]\n         r1,        #structure t
o hold the recursion formulae for x (usually T[1])\n         k, c, p, \+
j, tmp, m, S, a, pargs, astart, i, nterms;\n\n    if not type(poly,\{p
olynom(anything,x),series\}) then \n        ERROR(cat(\"not a polynomi
al or series in \",x)); \n    fi;\n\n    if type(poly,series) then\n  \+
      S := convert(poly,polynom);\n    else\n        S := poly;\n    f
i;\n    \n    __T := '__T';\n    if ptype='G' then\n        if nargs=3
 then\n            a := 0;\n            astart := 4;\n        else\n  \+
          a := args[4];\n            astart := 5;\n        fi;\n      \+
  if a=0 then\n            ERROR(\"Computation of G[n](x,0) is erroneo
us in the polynomials package.\");\n        fi;\n        pargs := __T,
 a;\n    else\n        astart := 4;\n        pargs  := __T;\n    fi;\n
\n    #assign the recursion formulae for r=x*T[n](x) and r1=x for each
 polynomial type\n    r[T]  := (P,n) -> (P[n+1] + P[n-1])/2;\n    r1[T
] := (P)   -> P[1];\n    r[U]  := (P,n) -> (P[n+1] + P[n-1])/2;\n    r
1[U] := (P)   -> P[1]/2;\n    r[P]  := (P,n) -> ((n+1)*P[n+1] + n*P[n-
1])/(2*n+1);\n    r1[P] := (P)   -> P[1];\n    r[H]  := (P,n) -> P[n+1
]/2 + n*P[n-1];\n    r1[H] := (P)   -> P[1]/2;\n    r[L]  := (P,n) -> \+
(n+1)*(P[n] - P[n+1]) + n*(P[n] - P[n-1]);\n    r1[L] := (P)   -> 1-P[
1];\n    r[G]  := proc(P,a,n)\n        ((n+1)*P[n+1] + (n+2*a-1)*P[n-1
])/(2*(n+a));\n    end;\n    r1[G] := proc(P,a)\n        if a=0 then\n
            P[1]/2;\n        else\n            P[1]/(2*a);\n        fi
;\n    end;\n\n    #get the horner-form coefficients\n    c := `polyno
mials/get_horner_coeffs`(S,x);\n    n := nops(c) - 1;\n    debug_print
(procname,\"horner coefficients\",5,c);\n\n    #starting with the inne
rmost horner nesting, replace x and T[n]*x with\n    #equivalents from
 the recursion formula for the polynomial type T\n    Q := 0;\n    for
 k from 1 to n do\n        if k=1 then\n            Q := c[n-k+1] + c[
n-k+2]*r1[ptype](pargs);\n        else\n            Q := c[n-k+1] + Q*
x;\n            Q := collect( Q, [seq(__T[j],j=0..k-1)] );\n          \+
  if type(Q,`+`) then\n                nterms := nops(Q);\n           \+
 else\n                nterms := 1;\n            fi;\n            tmp \+
:= Q;\n            for i from 1 to nterms do\n                if nterm
s=1 then\n                    p := tmp;\n                else\n       \+
             p := op(i,tmp);\n                fi;\n                if \+
has(p,__T) then\n                    m := op(1,select(has,p,__T));    \+
#order of T (i.e., T's subscript)\n                    Q := subs( p=al
gsubs( __T[m]*x=r[ptype](pargs,m), p ), Q );\n                    Q :=
 subs(__T[0]=1,Q);            #T[0]=1 for all orthogonal polynomials\n
                else\n                    Q := subs( p=subs(x=r1[ptype
](pargs),p), Q );\n                fi;\n            od;\n        fi;\n
    od;\n    Q := collect( Q, [seq(__T[j],j=0..n)], args[astart..nargs
] );\n\n    #replace T[n] with T[n](x)\n    for p in Q do\n        m :
= op(1,select(has,p,__T));\n        Q := subs( p=subs(__T[m]=__T[m](x)
,p), Q );\n    od;\n\n    Q;\n\nend:\n\n#-----------------------------
----------------------------------\n# Get the coefficients of the horn
er form of a polynomial in x.\n#\n# For example, 1 + 2*x - 3*x^2 becom
es 1 + (2 - 3*x)*x, and the\n# coefficients are [1,2,-3].\n#\n# Not in
tended for public use.\n#---------------------------------------------
------------------\n`polynomials/get_horner_coeffs` := proc( poly::\{`
+`,series\}, x::name )\n\n    local S, C, k, n, c, j, m, xpart;\n\n   \+
 if not type(poly,\{polynom(anything,x),series\}) then \n        ERROR
(cat(\"not a polynomial or series in \",x)); \n    fi;\n\n    if type(
poly,series) then\n        S := convert(poly,polynom);\n    else\n    \+
    S := poly;\n    fi;\n\n    S := convert( S, horner, x );\n    c :=
 [];\n    n := degree( collect(S,x), x );\n    for k from 1 to n do\n \+
       if type(S,`+`) then                    #S = a + (b + ...)*x\n  \+
          c := [ op(c), remove(has,S,x) ];\n        else              \+
                  #S = (a + ...)*x\n            c := [ op(c), 0 ];\n  \+
      fi;\n        xpart := select(has,S,x);\n        if xpart=x then
\n            S := 1;\n        else\n            if type(xpart,`^`) or
 nops(xpart) > 2 then    #xpart=a*b*...*x^m\n                m := degr
ee( xpart, x );\n            else                                     \+
   #xpart=(a + ...)*x^m\n                m := degree( op(2,xpart), x )
;\n            fi;\n            if m > 1 then\n                for j f
rom 2 to m do\n                    c     := [ op(c), 0 ];\n           \+
         xpart := xpart/x;\n                    k     := k + 1;\n     \+
           od;\n            fi;\n            S := xpart/x;\n        fi
;\n    od;\n    [ op(c), S ];\n\nend:\n\nend:\n\n\nsave( polynomials, \+
cat(POLY_PATH,\"polynomials.m\") );\n" }}}}{MARK "5 0 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 }