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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 
262 24 "Linear Algebra Powertool" }}{PARA 18 "" 0 "" {TEXT -1 30 "Poly
nomial Approximation - III" }}{PARA 256 "" 0 "" {TEXT -1 18 "Jacobi Po
lynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
42 "Worksheet by Mike May, S.J., maymk@slu.edu" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
8 "Overview" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 338 "We have been looki
ng at polynomial approximations to continuous functions defined on the
 domain [-1, 1] using orthogonal polynomials of various kinds.  In thi
s worksheet we use yet another norm, produing the  Jacobi polynomials,
 paying attention to how this changes the definition of best fit and t
he polynomials that can be approximated." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 187 "We begin by loading Maple's package for
 othogonal polynomials.  We also load the plots package so that we can
 graph our results.  We repeat the technical commands from the last wo
rksheet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "with(orthopoly)
; with(plots):\nassume('i',integer): assume('j',integer):\ninterface(s
howassumed=0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "For approximati
on by orthogonal polynomials, we define the norm by  " }}{PARA 257 "" 
0 "" {TEXT -1 15 "<f(x), g(x)> = " }{XPPEDIT 18 0 "int(wtfunc(x)*f(x)*
g(x),x = -1 .. 1);" "6#-%$intG6$*(-%'wtfuncG6#%\"xG\"\"\"-%\"fG6#F*F+-
%\"gG6#F*F+/F*;,$F+!\"\"F+" }{TEXT -1 7 ".,     " }}{PARA 0 "" 0 "" 
{TEXT -1 323 "where wtfunc(x) is a weight function.  For Legandre poly
nomials, the weight function is the constant 1, and all points in the \+
domain have equal weight in defining best fit approximation.  For Cheb
yshev polynomials of the first kind, the weight function is (1-x^2)^(-
1/2), so extra weight is added to the ends of the domain." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 348 "With Jacobi polyn
omials, with parameters a and b, the weight function is (1-x)^a*(1+x)^
b, where a and b are both > -1.  When a = b = 0 this reduces to the ca
se of Legandre polynomials.  If a and b are both positive the weight o
f the ends of the intervals is decreased.  If a and b are negative, th
e weight of the ends of the intervals is increased." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Consider the plot of t
he weight functions for Jacobi polynomials with a variety of values fo
r the parameters a and b." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
130 "plot(\{1, (1-x)*(1+x), (1-x)^2*(1+x)^3, (1-x)^6*(1+x)^8, \n     (
1-x)^(-.4)*(1+x)^(-.6), (1-x)^(-.9)*(1+x)^(-.9)\}, x=-1..1, y=0..3);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "We would also like to give a n
umeric measure to how the weighting shifts with different parameter va
lues.  One easy intuitive way is to compute the percent of the weight \+
that is in the middle 60 % of the domain." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 503 "\"w(0, 0)\" = int(1,x=-0.6..0.6)/int(1,x=-1..1);\n
\"w(1, 1)\" = int((1-x)*(1+x),x=-0.6..0.6)/int((1-x)*(1+x),x=-1..1);\n
\"w(2, 3)\" = int((1-x)^2*(1+x)^3,x=-0.6..0.6)/int((1-x)^2*(1+x)^3,x=-
1..1);\n\"w(6, 8)\" = int((1-x)^6*(1+x)^8,x=-0.6..0.6)/int((1-x)^6*(1+
x)^8,x=-1..1);\n\"w(-.4, -.6)\" = evalf(Int((1-x)^(-0.4)*(1+x)^(-0.6),
x=-0.6..0.6))/\n      evalf(Int((1-x)^(-0.4)*(1+x)^(-0.6),x=-1..1));\n
\"w(-.9, -.9)\" = evalf(Int((1-x)^(-0.9)*(1+x)^(-0.9),x=-0.6..0.6))/\n
      evalf(Int((1-x)^(-0.9)*(1+x)^(-0.9),x=-1..1));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 463 "
As mentioned above, one reason to change the parameter values is to ch
ange the way closeness is defined.  A second reason is to change the s
ubspace of functions over which the inner product makes sense.  In par
ticular,using positive values for a and b  produces an inner product u
nder which functions with poles at the end of the domain have finite n
orm.  Consider what happens to the function f(x) = 1/(1-x) with the Le
gandre norm and with the (3,3) Jacobi norm." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 92 "\"<f,f>_L\" = int(1*(1-x)^(-2), x=-1..1);\n\"<f,f>
_J(3,3)\" = int((1-x^2)^3*(1-x)^(-2), x=-1..1);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT 260 3 "Exe" }
{TEXT -1 0 "" }{TEXT 259 7 "rcises:" }{TEXT -1 2 "  " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 344 "1)  Describe the rational functions that have \+
a finite norm under the (a, b) Jacobi norm.  (To simplify the problem,
 assume that the rational functions are in a normal form with partial \+
fraction decomposition, i.e., each function is a sum of proper fractio
ns with the denominator a power of an irreducible polynomial with inte
ger coefficients.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "2)  For the (2,3) Jacobi norm, wh
at percentage of the weight is on the interval [0, 1]?  Repeat the que
stion with the (2, 5) Jacobi norm." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Jacobi poly
nomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "As we did in previous \+
worksheets we check the help page for the orthogonal polynomials we ar
e looking at in this worksheet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "?orthopoly,P" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Once we \+
have loaded the orthopoly package, we obtain the nth (a,b) Jacobi poly
nomial with P(n,a,b,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 
"\"P(3,1,1,x)\"=P(3,1,1,x);\n\"P(3,1,2,x)\"=P(3,1,2,x);\n\"P(3,2,1,x)
\"=P(3,2,1,x);\n\"P(3,-.4,-.6,x)\"=P(3,-.4,-.6,x);" }}}{EXCHG {PARA 5 
"" 0 "" {TEXT 257 1 "E" }{TEXT -1 0 "" }{TEXT 256 9 "xercise: " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "3)  Find the 2nd and 5th Jacobi p
olynomials with parameters (1,1), (2,3), and (6,6).  Verify that pair \+
associated with each parameter is orthogonal under the oppropriate inn
er product." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 19 "An extended example" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 124 "Time to look at an example to see how this works \+
out in a particular case.  Once again we will use sin(3\271x) as our f
unction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "func := x -> si
n(3*Pi*x);" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Computing coeffici
ents and approximating polynomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
280 "Next we  start computing coefficients and defining the approximat
ing polynomials up to degree 12 for the inner products of type (0,0), \+
(2,3), and (-.4, -.6).  Recall that if P(n,a,b,x) is the nth (a,b) Jac
obi polynomial, the projection coeffient of f(x) onto P(n,a,b,x) shoul
d be " }{XPPEDIT 18 0 "int((1-x)^a*(1+x)^b*f(x)*P(n,a,b,x),x = -1 .. 1
);" "6#-%$intG6$**),&\"\"\"F)%\"xG!\"\"%\"aGF)),&F)F)F*F)%\"bGF)-%\"fG
6#F*F)-%\"PG6&%\"nGF,F/F*F)/F*;,$F)F+F)" }{TEXT -1 14 "  divided by  \+
" }{XPPEDIT 18 0 "int((1-x)^a*(1+x)^b*P(n,a,b,x)*P(n,a,b,x),x = -1 .. \+
1);" "6#-%$intG6$**),&\"\"\"F)%\"xG!\"\"%\"aGF)),&F)F)F*F)%\"bGF)-%\"P
G6&%\"nGF,F/F*F)-F16&F3F,F/F*F)/F*;,$F)F+F)" }{TEXT -1 4 " .  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 888 "for i from 0 to 12 do\n nor
mP00[i] := evalf(Int((P(i,0,0,x))^2, x=-1..1)):\n coeffP00[i] := evalf
(Int(simplify(P(i,0,0,x)*func(x)*1.0), x=-1..1));\nod:\nJacobi00approx
 := proc(m,x) \n  local count:\n  sum(P(count,0,0,x)*coeffP00[count]/n
ormP00[count],count=0..m):\nend;\nfor i from 0 to 12 do\n normP23[i] :
= evalf(Int((1-x)^2*(1+x)^3*(P(i,2,3,x))^2, x=-1..1)):\n coeffP23[i] :
= evalf(Int(simplify((1-x)^2*(1+x)^3*P(i,2,3,x)*func(x)*1.0), x=-1..1)
):\nod:\nJacobi23approx := proc(m,x) \n  local count:\n  sum(P(count,2
,3,x)*coeffP23[count]/normP23[count],count=0..m):\nend;\nfor i from 0 \+
to 12 do\n normP46[i] := evalf(Int((1-x)^(-.4)*(1+x)^(-.6)*(P(i,-.4,-.
6,x))^2, x=-1..1));\n coeffP46[i] := evalf(Int(simplify((1-x)^(-.4)*(1
+x)^(-.6)*\n       P(i,-.4,-.6,x)*func(x)*1.0), x=-1..1));\nod:\nJacob
i46approx := proc(m,x) \n  local count:\n  sum(P(count,-.4,-.6,x)*coef
fP46[count]/normP46[count],count=0..m):\nend;" }}}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 28 "Comparing the approximations" }}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 186 "Our past experience with this function is that the ap
proximation start to be interesting with the ninth degree. Note first \+
that the polynomial approximations are significantly different." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "n:= 9:\nsort(Jacobi00approx(
n,x));\nsort(Jacobi23approx(n,x));\nsort(Jacobi46approx(n,x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Next compare the graphs of the app
roximations and of the errors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 619 "lowy := -1.2:  highy := 1.6:  n:= 7:\npl1 := plot(func(x), x=
-1..1,y=lowy..highy, color=red):\npl2 := plot(Jacobi00approx(n,x), x=-
1..1,y=lowy..highy, color=green):\npl3 := plot(Jacobi23approx(n,x), x=
-1..1,y=lowy..highy, color=blue):\npl4 := plot(Jacobi46approx(n,x), x=
-1..1,y=lowy..highy, color=black):\nB := textplot(\{[-0.8,highy-(highy
-lowy)/20, \n         `The function and the `||n||` term Jacobi approx
imations`],\n      [-0.8,highy-(highy-lowy)/9, \n      `(0,0) (green) \+
vs (2,3) (blue), vs (-.4, -.6) (black)`]\},\n    align=RIGHT, font = [
TIMES, BOLD, 12] ):\ndisplay(\{pl1, pl2, pl3, pl4, B\},view=[-1..1, lo
wy..highy]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "As expected, the
 (2, 3) approximation tends to be best in the center of the graph, whi
le the (-.4, -.6) approximation tends to be best on the ends.  This be
comes even clearer when we look at the graphs of the errors." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 552 "lowy := -1.0:  highy := 1.4
:  n:= 10:\npl1 := plot(func(x) - Jacobi00approx(n,x), x=-1..1,y=lowy.
.highy, color=green):\npl2 := plot(func(x) - Jacobi23approx(n,x), x=-1
..1,y=lowy..highy, color=blue):\npl3 := plot(func(x) - Jacobi46approx(
n,x), x=-1..1,y=lowy..highy, color=black):\nB := textplot(\{[-0.8,high
y-(highy-lowy)/20, \n          `The `||n||` term Jacobi error `],\n   \+
   [-0.8,highy-(highy-lowy)/9, \n       `(0,0) (green) vs (2,3) (blue)
, vs (-.4, -.6) (black)`]\}, \n         align=RIGHT, font = [HELVETICA
, BOLD, 12] ):\ndisplay(\{pl1, pl2, pl3, B\});\n " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 130 "It is clear that the (2,3) Jacobi approximation i
s best in the middle and the (-.4, -.6) Jacobi approximation is best o
n the ends." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 5 "" 0 "" {TEXT 258 10 "Exercises:" }{TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 90 "4)  Describe the kind kind of error you would expect with a (-.
6, 5) Jacobi approximation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "5)  Plot the function y=c
os(2*Pi*x) along with the degree 10 (-.6, 5) Jacobi approximation.  In
 a separate graph plot the error of the approxiamtion." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0 2" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }