Approximation Polynomial - II
Legandre vs Chebyshev Polynomials Worksheet by Mike May, S.J., [e-mail] O restart;
Overview
In the last worksheet we looked at polynomial approximation using Legandre polynomials. These polynomials form an orthogonal set with a least squares fit definition for the norm. In this worksheet we will look at other norms and the corresponding sets of orthogonal polynomials. 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 worksheet. O with(orthopoly); with(plots): assume('i',integer): assume('j',integer): interface(showassumed=0): [G, H, L, P, T, U ]
Warning, the name changecoords has been redefined
a1 For the Legangre polynomials we saw that the norm is defined by <f(x), g(x)> = b f (x ) g (x ) dx b cK1 . The other inner products we will look at are defined by <f(x), g(x)> = a1 b wtfunc(x ) f (x ) g (x ) dx where wtfunc(x) is a weight function. The use of different weight b cK1 functions gives us a way to make some parts of the domain more important than others. It also makes it possible to allow functions with poles into the set of functions we are trying to appoximate.
Chebyshev polynomials of the first kind
We will begin with Chebyshev polynomials of the first kind. O ?orthopoly,T The help page shows us that weight function of Chebyshev polynomials of the first kind is (1-x^2)^( -1/2), The effect of the weight function is to make the endpoints count more when defining what it means for function to be small. O plot({(1-x^2)^(-1/2),1}, x=-1..1, y=0..3);
3.0
2.5
2.0
y 1.5
1.0
0.5
0.0 -1.0 -0.8 -0.6 x -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Once we have loaded the orthopoly package, we obtain the nth Chebychev polynomial with T(n,x); O T(3,x); 4 x3 K 3 x (2.1) Exercise: 1) Find the 1st, second, and 5th Chebychev polynomials of the first kind. Verify that they are orthogonal under the oppropriate inner product. O
An extended example, back to f(x) = sin(3*Pi*x)
Time to look at an example to see how this works out in a particular case. Once again we will use sin(3x) as our function. O func := x -> sin(3*Pi*x); func := x/sin (3 p x ) (3.1)
Computing coefficients and approximating polynomials
Next we start computing coefficients. We will find the coefficients for both the Legangre polynomials and for Chebyshev polynomials of the first kind. Recall that if P(n,x) is the nth
Legandre polynomial, the projection coeffient of f(x) onto P(n,x) should be a1 a1 b f (x ) P (0, x ) dx divided by b P (n, x ) P (n, x ) dx . Similarly, if T(n,x) is the nth b b cK1 cK1 Chebyshev polynomial of the first kind, the projection coefficient of f(x) onto T(n,x) is a1 a1 b wtCheb (x ) f (x ) P (0, x ) dx divided by b wtCheb (x ) P (n, x ) P (n, x ) dx , where b b cK1 cK1 wtCheb(x) = (1-x^2)^(-1/2). O wtchev := (1-x^2)^(-1/2): for i from 0 to 15 do normP[i] := evalf(Int((P(i,x))^2, x=-1..1)); coeffP[i] := evalf(Int(simplify(P(i,x)*func(x)*1.0), x=-1..1) ); normT[i] := evalf(Int(wtchev*(T(i,x))^2, x=-1..1)); coeffT[i] := evalf(Int(wtchev*T(i,x)*func(x), x=-1..1)): od; normP0 := 2. coeffP0 := 0. normT0 := 3.141592654 coeffT0 := 0. normP1 := 0.6666666667 coeffP1 := 0.2122065907 normT1 := 1.570796327 coeffT1 := 0.5551985872 normP2 := 0.4000000000 coeffP2 := 0. normT2 := 1.570796327 coeffT2 := 0. normP3 := 0.2857142857 coeffP3 := 0.1763715524 normT3 := 1.570796327 coeffT3 := 0.2635803132 normP4 := 0.2222222222 coeffP4 := 0. normT4 := 1.570796327 coeffT4 := 0. normP5 := 0.1818181818 coeffP5 := K0.01322273781 normT5 := 1.570796327 coeffT5 := K.4620896712
normP6 := 0.1538461538 coeffP6 := 0. normT6 := 1.570796327 coeffT6 := 0. normP7 := 0.1333333333 coeffP7 := K.2657941222 normT7 := 1.570796327 coeffT7 := K.9263350154 normP8 := 0.1176470588 coeffP8 := 0. normT8 := 1.570796327 coeffT8 := 0. normP9 := 0.1052631579 coeffP9 := 0.1669531813 normT9 := 1.570796327 coeffT9 := 0.7906766020 normP10 := 0.09523809524 coeffP10 := 0. normT10 := 1.570796327 coeffT10 := 0. normP11 := 0.08695652174 coeffP11 := K0.04841458486 normT11 := 1.570796327 coeffT11 := K.2675504440 normP12 := 0.08000000000 coeffP12 := 0. normT12 := 1.570796327 coeffT12 := 0. normP13 := 0.07407407407 coeffP13 := 0.008597245012 normT13 := 1.570796327 coeffT13 := 0.05294405753 := normP14 0.06896551724 coeffP14 := 0. normT14 := 1.570796327 coeffT14 := 0. normP15 := 0.06451612903
(3.1.1)
coeffP15 := K0.001053256856 normT15 := 1.570796327 coeffT15 := K0.007062371922 Now we define the approximating polynomials. O Lapprox := proc(m,x) local count: sum(P(count,x)*coeffP[count]/normP[count],count=0..m): end; Chebapprox := proc(m,x) local count: sum(T(count,x)*coeffT[count]/normT[count],count=0..m): end; Lapprox := proc (m, x ) localcount; sum ( (P (count, x ) * coeffP [count ] ) / normP [count ], count = 0 ..m ) end proc Chebapprox := proc (m, x ) localcount; sum ( (T (count, x ) * coeffT[count ] ) / normT [count ], count = 0 ..m ) end proc
(3.1.1)
(3.1.2)
Comparing the approximations
Since our function has 6 bumps and looks like it could be a 7th degreee polynomial, we want to compare the 7th degree approximations. Note first that the polynomial approximations are significantly different. O n:= 7: Lapprox(n,x); Chebapprox(n,x); 3.616684570 x K 37.06656803 x3 C 85.76884958 x5 K 53.44953677 x7 2.507234281 x K 26.46978826 x3 C 61.34219016 x5 K 37.74228394 x7 (3.2.1) Next compare the graphs of the approximations and of the errors. O lowy := -1.2: highy := 1.6: n:= 7: pl1 := plot(func(x), x=-1..1,y=lowy..highy, color=red): pl2 := plot(Lapprox(n,x), x=-1..1,y=lowy..highy, color=green) : pl3 := plot(Chebapprox(n,x), x=-1..1,y=lowy..highy, color= blue): B := textplot({[-0.8,highy-(highy-lowy)/20, `The function and the `||n||` term approximations`] , [-0.8,highy-(highy-lowy)/9, `Chebyshev (blue) vs Legandre (green)`]}, align=RIGHT, font = [TIMES, BOLD, 12] ): display({pl1, pl2, pl3, B},view=[-1..1, lowy..highy]);
1.5 The function and the 7 term approximations
Chebyshev (blue) vs Legandre (green)
1.0 y 0.5
0.0 -1.0 -0.8 -0.6 x -0.5 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
-1.0
As expected, the Legandre approximation tends to be better in the center of the graph, while the Chebyshev approximation tends to be better on the ends. This becomes even clearer when we look at the graphs of the errors. O lowy := -1.0: highy := 1.4: n:= 7: pl1 := plot(func(x) - Chebapprox(n,x), x=-1..1,y=lowy..highy, color=green): pl2 := plot(func(x) - Lapprox(n,x), x=-1..1,y=lowy..highy, color=blue): B := textplot({[-0.8,highy-(highy-lowy)/20, `The `||n||` term Legandre error (blue)`], [-0.8,highy-(highy-lowy)/9, `and Chebychev error (green)`]}, align=RIGHT, font = [HELVETICA, BOLD, 12] ): display({pl1, pl2, B});
1.4
The 7 term Legandre error (blue) and Chebychev error (green)
1.0 0.8 y 0.6 0.4 0.2 0.0 -1.0 -0.8 -0.6 x -0.4 -0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0.2 0.4 0.6 0.8 1...