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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 23 "Newton-Cotes Quadrature
" }}{PARA 0 "" 0 "" {TEXT 256 19 "Mth 351 Summer 1999" }}{PARA 0 "" 0 
"" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 26 "Fil
ename: newton_cotes.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 71 "We investigate various compound (closed) Newton-Cote
s quadrature rules." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 80 "Newton-Cotes quadrature proceeds as follows - w
e wish to approximate an integral" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "Int(f(x),x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 48 "We subdivide the interval  [a,b]  into  m  (the " }{TEXT 260 6 
"degree" }{TEXT -1 82 ")  subintervals of equal length. The correspond
ing  m+1  endpoints are called the " }{TEXT 261 5 "nodes" }{TEXT -1 
322 ". Now let  p  be the unique interpolation polynomial of degree at
 most  m  through the points on the graph of  f  corresponding to the \+
nodes. Then the integral of  p  is our approximation to the integral o
f   f.  The Lagrange formula for the interpolation polynomial, a chang
e of variable and a quick calculation show that" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 71 "Int(p(x),x=a..b)= (b-a)*Sum(f(a+k*(b-a)/m),k=0
..m)*Int(p[k](x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6
$-%\"pG6#%\"xG/F*;%\"aG%\"bG*(,&F.\"\"\"F-!\"\"\"\"\"-%$SumG6$-%\"fG6#
,&F-F1*&*&%\"kGF1F0F1F3%\"mG!\"\"F1/F=;\"\"!F>F1-F%6$-&F(6#F=F)/F*;FBF
1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "where  p[k]  is the interp
olation polynomial which is 1 at " }{XPPEDIT 18 0 "k/m;" "6#*&%\"kG\"
\"\"%\"mG!\"\"" }{TEXT -1 33 " and 0 at the other multiples of " }
{XPPEDIT 18 0 "1/m;" "6#*&\"\"\"\"\"\"%\"mG!\"\"" }{TEXT -1 132 ".  Th
e integrals of the  p[k]  are called the Newton-Cotes coefficients of \+
degree n. Here's a routine to compute these coefficients:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "NCC:
=proc(m)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " local k,h,xx,y,yy,coef
,q,w;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " xx:=[seq(k/m,k=0..m)];" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " coef:=[];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 " y:=[seq(0,k=0..m)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "for k from 0 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "  yy:
=subsop(k+1=1,y); # make kth component in y 1 in yy" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "  q:=interp(xx,yy,x);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "  w:=int(q,x=0..1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 50 "  coef:=[op(coef),w];  # push onto list of weights" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "coef;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Let's check a few. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Tapezoida
l rule:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "NCC(1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$#\"\"\"\"\"#F$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Simpson's rule:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "NCC(2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%#\"\"\"\"\"'#\"\"#\"\"$F$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 19 "Simpson's 3/8 rule:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "NCC(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#\"\"\"\"
\")#\"\"$F&F'F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Boole's rule:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "NCC(4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7'#\"\"(\"#!*#\"#;\"#X#\"\"#\"#:F'F$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "NCC(5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7(#\"#>\"$)G#\"#D\"#'*#F(\"$W\"F*F'F$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "NCC(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#7)#\"#T\"$S)#\"\"*\"#N#F(\"$!G#\"#M\"$0\"F*F'F$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "NCC(7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*#
\"$^(\"&!G<#\"%xNF&#\"#\\\"$S'#\"%*)HF&F,F)F'F$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "NCC(8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+#
\"$*)*\"&]$G#\"%WH\"&vT\"#!$k%F)#\"%[_F)#!$a%\"%NGF,F*F'F$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 324 "Somethin
g important happened here. For the case of degree  8  we have some neg
ative weights. When the weights are all positive we can integrate nonn
egative functions without worrying about loss of significance errors. \+
When some weights are negative this is no longer the case. As a result
 the degree 8 method is not popular." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "NCC(9);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7,#\"%dG\"&+'*)#\"&Td\"F&#\"#F\"%SA#\"%47\"%+c#\"%*)G
\"&+[%F/F,F)F'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "NCC(10);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-#\"&ng\"\"'_()f#\"&vl#\"')o\\\"#
!&vh\"\"'%e*>#\"%vc\"&uC\"#!%D[\"&)36#\"&2y\"\"&[\\#F0F-F*F'F$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "De
gree 9 is well-behaved but degree 10 again has a sign problem. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "This pro
blem with the sign of the weights makes it useless to increase the deg
ree in an effort to increase accuracy. We may see a decrease in trunca
tion error, but an increase in roundoff error. Thus compound methods a
re usually used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 64 "The idea in the compound methods is we break the interval
 into  " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 22 "  subintervals wh
ere  " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 30 "  is divisible by t
he degree  " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 80 "  that we pla
n to use. Then we use the Newton-Cotes method on the first set of  " }
{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 124 "  subintervals, then on th
e second set, and so on, and then add the reuslts together. Let's writ
e a routine to do it for us:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "NCCint:=proc(f,r::range,m,n)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  local k,j,a,b,q,C,h,hh,S;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "if type(n, numeric) then  " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "  if not type(n,posint) then ERROR(
\"n must be positive\"); fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "  i
f not type(m,posint) then ERROR(\"m must be positive\"); fi;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 72 "  if not irem(n,m)=0 then ERROR(\"m (degr
ee) must divide n (order)\"); fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 
"fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "a:=op(1,r); b:=op(2,r); q:=
n/m; h:=(b-a)/n; hh:=(b-a)/q;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "C:
=NCC(m); S:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for j from 0 to q
-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  for k from 0 to m do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "    S:=S+C[k+1]*f(a+j*hh+k*h);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "od; od;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "S:=hh*S;  # rescale, recall intervals here have lengt
h (b-a)/q " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "For example her
e is the (compound) Simpson's rule with 6 subintervals on [a,b]." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "NCCint(f,a..b,2,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"bG
#\"\"\"\"\"$%\"aG#!\"\"F(F',0-%\"fG6#F)#F'\"\"'-F.6#,&F)#\"\"&F1F%F0#
\"\"#F(-F.6#,&F)F7F%F&F&-F.6#,&F)#F'F8F%F?F7-F.6#,&F)F&F%F7F&-F.6#,&F)
F0F%F5F7-F.6#F%F0F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "If we mult
iply by 6 and simplify we see the familiar 1,4,2,4,2,....,2,4,1 patter
n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Now
 let's compare some order 36 methods for" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "rng:=2..5: ex:=Int(exp(x),x=rng);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#exG-%$IntG6$-%$expG6#%\"xG/F+;\"\"#\"\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`(1,36) rel_error (trapezoid
al)`; evalf((ex-NCCint(exp,rng,1,36))/ex,40);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%?(1,36)~rel_error~(trapezoidal)G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$!IJPLwDTVlpb]>Q6\"=Nnjy&!#V" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 71 "`(2,36) rel_error (Simpson's)`; evalf((ex-NCCint(e
xp,rng,2,36))/ex,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=(2,36)~rel_
error~(Simpson's)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!IPl9O5/1]g)yjM
7XO\\qpn#!#Y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`(3,36) rel
_error (Simpson 3/8)`; evalf((ex-NCCint(exp,rng,3,36))/ex,40);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%?(3,36)~rel_error~(Simpson~3/8)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$!I\\\")o))\\B)ogEkOgR5057#=g!#Y" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "`(4,36) rel_error (Boole)`; \+
evalf((ex-NCCint(exp,rng,4,36))/ex,18);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%9(4,36)~rel_error~(Boole)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
!3rA'\\#eQ-iq!#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`(6,36)
 rel_error`; evalf((ex-NCCint(exp,rng,6,36))/ex,40);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%1(6,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$!IuX!GPT-i\\A!)>ncs)3.5qC!#^" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "`(9,36) rel_error`; evalf((ex-NCCint(exp,rng,9,36))/e
x,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1(9,36)~rel_errorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$!Ic(yITfoG$))R+(*pwq0je$3#!#`" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`(12,36) rel_error`; evalf((
ex-NCCint(exp,rng,12,36))/ex,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
2(12,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!Iwp;A**>s(\\3
g:73Fw$*4$\\@!#e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`(18,36
) rel_error`; evalf((ex-NCCint(exp,rng,18,36))/ex,40);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%2(18,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$!IqbW\\J*[)4*=7&)e(fJWIunH!#l" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 60 "Note each of these methods required 37 function evaluations." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "In comp
uting the error we use the actual value of the integral  and both ex  \+
and NCCint() are computed " }{TEXT 262 14 "symbolically. " }{TEXT -1 
49 "The difference is found and divided by  ex, also " }{TEXT 263 12 "
symbolically" }{TEXT -1 197 ". We then convert to floating point at a \+
prescribed precision. By using a very high precision we are able to co
mpute the error with minimal roundoff - so we are seeing mostly trunca
tion error here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 31 "Let's consider another example:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "rng:=0..Pi: ex:=Int(sin(x),x=rng);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#exG-%$IntG6$-%$sinG6#%\"xG/F+;\"\"!
%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`(1,36) rel_error \+
(trapezoidal)`; evalf((ex-NCCint(sin,rng,1,36))/ex,40);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%?(1,36)~rel_error~(trapezoidal)G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"I+]=SeMcW,Gezz-xv=+Zj!#V" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 71 "`(2,36) rel_error (Simpson's)`; evalf((ex-NC
Cint(sin,rng,2,36))/ex,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=(2,36
)~rel_error~(Simpson's)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!I+++b.SL
W<s6JFtU'))f[A$!#Y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`(3,3
6) rel_error (Simpson 3/8)`; evalf((ex-NCCint(sin,rng,3,36))/ex,40);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%?(3,36)~rel_error~(Simpson~3/8)G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!I+++lC\">*)ei!\\Dy^*4cCDE(!#Y" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "`(4,36) rel_error (Boole)`; \+
evalf((ex-NCCint(sin,rng,4,36))/ex,18);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%9(4,36)~rel_error~(Boole)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"3+++++Fr%Q*!#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`(6,36)
 rel_error`; evalf((ex-NCCint(sin,rng,6,36))/ex,40);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%1(6,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$!I+++++]Z)fd%z6ca#)e#f&QO!#^" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "`(9,36) rel_error`; evalf((ex-NCCint(sin,rng,9,36))/e
x,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1(9,36)~rel_errorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I++++++]kvf_?)[B-mF%[M!#`" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`(12,36) rel_error`; evalf((
ex-NCCint(sin,rng,12,36))/ex,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
2(12,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I+++++++++X6
f/$)zX**QSW!#e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`(18,36) \+
rel_error`; evalf((ex-NCCint(sin,rng,18,36))/ex,40);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%2(18,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$!I++++++++++++]\"Ga`Lw)*)!#l" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "One more set of examples:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "f:=x->exp(x): rng:=3..8: ex:=Int(f(x),x=rng); " }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#exG-%$IntG6$-%$expG6#%\"xG/F+;\"\"$\"\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`(1,36) rel_error (trapezoid
al)`; evalf((ex-NCCint(f,rng,1,36))/ex,40);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%?(1,36)~rel_error~(trapezoidal)G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$!I>$)3(G_#3x>5NnhZX2P*pg\"!#U" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 69 "`(2,36) rel_error (Simpson's)`; evalf((ex-NCCint
(f,rng,2,36))/ex,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=(2,36)~rel_
error~(Simpson's)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!IAzt<lCp**[`ag
OB1'oLD1#!#X" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "`(3,36) rel
_error (Simpson 3/8)`; evalf((ex-NCCint(f,rng,3,36))/ex,40);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%?(3,36)~rel_error~(Simpson~3/8)G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$!IDagq:(3(yYBJrtF^d44IY!#X" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`(4,36) rel_error (Boole)`; evalf((
ex-NCCint(f,rng,4,36))/ex,18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9(4
,36)~rel_error~(Boole)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3A4g%QK+R
]\"!#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`(6,36) rel_error
`; evalf((ex-NCCint(f,rng,6,36))/ex,40);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%1(6,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!IPgH_qO
GMKVd'eO!RuZ4[9!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`(9,
36) rel_error`; evalf((ex-NCCint(f,rng,9,36))/ex,40);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%1(9,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$!I\"e30l&p(pu(3+t)Q[l*R#fK$!#^" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "`(12,36) rel_error`; evalf((ex-NCCint(f,rng,12,36))/e
x,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2(12,36)~rel_errorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$!IarxFwm`AB*3)f4G)*3o$)oD!#b" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`(18,36) rel_error`; evalf((
ex-NCCint(f,rng,18,36))/ex,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2(
18,36)~rel_errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!IzBM(4Z)RJz^'y;
0!QiC&=,(!#h" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=evaln(f)
: # unassign" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 225 "For our (nice and smooth) examples the higher degree \+
methods are much superior. The composite Boole's method is no harder t
o apply than Simpson's rule, even for hand calculations, but has small
er error (for smooth functions)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "Compound Boole, 12 subintervals" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "N
CCint(f,a..b,4,12): algsubs(b-a=12*h,%): (1/90)*simplify(90*%);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,<-%\"fG6#%\"aG\"\"(-F'6#,&F)\"\"
\"%\"hGF.\"#K-F'6#,&F)F.F/\"\"#\"#7-F'6#,&F)F.F/\"\"$F0-F'6#,&F)F.F/\"
\"%\"#9-F'6#,&F)F.F/\"\"&F0-F'6#,&F)F.F/\"\"'F5-F'6#,&F)F.F/F*F0-F'6#,
&F)F.F/\"\")F>-F'6#,&F)F.F/\"\"*F0-F'6#,&F)F.F/\"#5F5-F'6#,&F)F.F/\"#6
F0-F'6#,&F)F.F/F5F*F.F/F.#F4\"#X" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Compound Boole, 8 subintervals" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 62 "NCCint(f,a..b,4,8): algsubs(b-a=8*h,%): (1/90)*simplify(90*%);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,4-%\"fG6#%\"aG\"\"(-F'6#,&F)\"
\"\"%\"hGF.\"#K-F'6#,&F)F.F/\"\"#\"#7-F'6#,&F)F.F/\"\"$F0-F'6#,&F)F.F/
\"\"%\"#9-F'6#,&F)F.F/\"\"&F0-F'6#,&F)F.F/\"\"'F5-F'6#,&F)F.F/F*F0-F'6
#,&F)F.F/\"\")F*F.F/F.#F4\"#X" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 34 "We see we have a coefficient of   " }
{XPPEDIT 18 0 "2*h/45;" "6#*&*&\"\"#\"\"\"%\"hGF&F&\"#X!\"\"" }{TEXT 
-1 178 "  and then the sequence  7  32  12  32  14  32  12  32  14 ...
.... 14  32  12  32  7.  The  14's  come from two  7's  at the points \+
where two of the interpolation intervals meet." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 239 "Since Boole's rule depen
ds on interpolation of degree 4 it is obviously exact for polynomials \+
of degree at most 4. However, just as for Simpson's rule, it is actual
ly exact for one degree more, that is, for polynomials of degree at mo
st 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "collect(NCCint(x->x^5,a..b,4,4),b);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,&*$)%\"bG\"\"'\"\"\"#\"\"\"F'*$)%\"aGF'F(#!\"\"F'" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
187 "The degree  9  rule is one of the higher degree rules which has p
ositive weights. We can expect it to be fairly well-behaved. Why don't
 you experiment a bit with compound degree  9  rule?" }}}}{MARK "64 1 \+
0" 104 }{VIEWOPTS 1 1 0 1 1 1803 }