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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 25 "Interpolation Polynomia
ls" }}{PARA 0 "" 0 "" {TEXT 256 19 "Mth 351 Spring 2001" }}{PARA 0 "" 
0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 35 "F
ilename: 351s2001_interp_polys.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "
In this worksheet I present a few procedures that are useful in experi
menting with interpolation polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 671 "finterp(
) interpolates a function, given the (distinct) nodes,  by computing t
he function values at the nodes and submitting the resulting lists of \+
abscissas and ordinates to Maple interp() function. Note that Maple al
lows sending any number of unspecified arguments to a procedure, so it
 is a simple matter to pass lists od data of arbitrary length to a pro
cedure. All the variables passed to the procedure may be accessed thro
ugh the list args - the nth entry in the list is args[n] - the sublist
 consisting of the kth to the nth entry is args[k..n]. Note args is no
t an array, in spite of appearances. The number of entries in the list
 of parameters is given by nargs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "finterp:=proc(f,x)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "  if nargs < 3 then ERROR(FAIL); fi;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "  interp([args[3..nargs]],map(f,[ar
gs[3..nargs]]),x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "He
re's a quick test of finterp()." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ps:=finterp(sin,x,0,Pi/4,Pi/
3,Pi/2,2*Pi/3,3*Pi/4,Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
98 "plot(sin(x)-ps,x=0..Pi, axes=NORMAL,thickness=3,title=\"Error in p
s\",numpoints=100,resolution=300);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
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\\[l$!1<?w!yde&H!#F-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"
x6\"%!G-%*AXESSTYLEG6#%'NORMALG-%*THICKNESSG6#\"\"$-%&TITLEG6#Q,Error~
in~psFafm-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 3 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 111 "We see here the typical error distri
bution for uniformly distributed nodes, so we have confidence in finte
rp()." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 401 
"If the coefficients in ps are evaluated in floating point above, the \+
accuracy will be reduced quite a bit because the values of ps are quit
e sensitive to roundoff (and other) errors in the coefficients. Thus I
 have written finterp(), and the other routines below to use symbolic \+
calculations (unless floating point data is received). This is ineffic
ient, of course, but allows us to maintain accuracy." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 218 "Here's a simple routin
e, Unodes(), which returns a list (not a formal list in the sense of d
ata types) of n+1 uniformly spaced nodes, in order,  for an interval [
a,b]. Thus the endpoints determine n equal subintervals." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Unod
es:=proc(n,a,b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "  local L, k;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "  if n < 1 then ERROR(FAIL); fi;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "  L:=[];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "for k from 0 to n do L:=[op(L),a+k*(b-a)/n]; od;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "op(L);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 382 "Note Unodes() returns the nodes symbolically i
f possible. We return op(L) in Unodes() so we get the actual nodes. Re
turning L would return a single data item, a list of the nodes. This i
s not just a matter of taste - it allows nargs in other routines to co
ntains the number of nodes plus other parameters, and saves us the tro
uble of counting the number of entries in a formal list." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Let's test Unodes(
)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "Unodes(6,a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%\"
aG,&F##\"\"&\"\"'%\"bG#\"\"\"F',&F##\"\"#\"\"$F(#F*F.,&F##F*F-F(F1,&F#
F/F(F,,&F#F)F(F%F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 63 "Cnodes() returns the n+1 Chebyshev nodes for an i
nterval [a,b]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "Cnodes:=proc(n,a,b)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "  local L, k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " \+
 if n < 1 then ERROR(FAIL); fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " \+
 L:=[];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "for k from 0 to n do L:=
[op(L),a+(b-a)*(1-cos((2*k+1)*Pi/(2*n+2)))/2]; od;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 6 "op(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 37 "Here's a couple of tests of Cnodes()." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Cnodes(4,
-1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6',$*&-%%sqrtG6#\"\"#\"\"\"-F&
6#,&\"\"&\"\"\"*$-F&6#F-F)F.F)#!\"\"\"\"%,$*&F%F)-F&6#,&F-F.F/F3F)F2\"
\"!,$F6#F.F4,$F$F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Cnode
s(4,a,b);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6',&%\"aG\"\"\"*&,&%\"bGF%F
$!\"\"F%,&F%F%*&-%%sqrtG6#\"\"#\"\"\"-F-6#,&\"\"&F%*$-F-6#F4F0F%F0#F)
\"\"%F%#F%F/,&F$F%*&F'F0,&F%F%*&F,F0-F-6#,&F4F%F5F)F0F8F%F:,&F$F:F(F:,
&F$F%*&F'F0,&F%F%F>#F%F9F%F:,&F$F%*&F'F0,&F%F%F+FFF%F:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Let's int
erpolate the sine function on " }{XPPEDIT 19 1 "[0, pi];" "6#7$\"\"!%#
piG" }{TEXT -1 14 " with 7 nodes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!%#piG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "pu:=finterp(sin,x,Unodes(6,0,Pi));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%#puG,:*&*$)%\"xG\"\"'\"\"\"F+*$)%#PiG\"\"'F+!
\"\"#!%C%)\"\"&*&*$)F)F3F+F+*$)F.\"\"&F+F0#\"&s_#F3*&*$)F)\"\"%F+F+*$)
F.\"\"%F+F0!%1d*&*$)F)\"\"$F+F+*$)F.\"\"$F+F0\"%))H*&*$)F)\"\"#F+F+*$)
F.\"\"#F+F0#!%mNF3*&F)F+F.F0#\"$3$F3*&*&-%%sqrtG6#FGF+F(F+F+*$)F.\"\"'
F+F0\"$s**&*&FZF+F6F+F+*$)F.\"\"&F+F0!%;H*&*&FZF+F>F+F+*$)F.\"\"%F+F0
\"%%H$*&*&FZF+FFF+F+*$)F.\"\"$F+F0!%G<*&*&FZF+FNF+F+*$)F.\"\"#F+F0#\"%
Z;F?*&*&FZF+F)\"\"\"F+F.F0#!$N\"F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "It's rather precise! Let's conv
ert it to floating point so we can see what it looks like." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eva
lf(pu,16);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"'\"\"\"$!.
8)R4o'H\"!#:*$)F&\"\"&F($\".qV*H4A7!#9*$)F&\"\"%F($!-o,=a7kF1*$)F&\"\"
$F($!/i*\\^]tg\"F1*$)F&\"\"#F($!-Qmr4XGF1F&$\"0.O%o(Q0+\"F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot(sin(x)-pu,x=0..Pi,axes=
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" {TEXT -1 395 "The symbolic Chebyshev nodes are fairly complicated he
re. If we use them to compute the exact interpolation polynomial we ob
tain an expression that covers many pages and consumes a great deal of
 RAM. Maple is likely to choke on this expression, or at least be very
 slow. Thus we convert the Chebyshev nodes to floating point with 16 d
ecimal digits before computing the interpolation polynomial. " }}
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" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Generally there is no \+
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" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "As we expected the Che
byshev nodes give a better distributed error. Moreover the maximum err
ors are smaller than in the uniform nodes case." }}{PARA 0 "" 0 "" 
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