Hello,
Yesterday I played with ba.tabulate() and I think we can have a better
helper which approximates an unary function using chebyshev polynomials
more accurately:
chebtab(CK, FX, NX,CD, X0,X1, x) = y with {
ck(0) = _;
ck(1) = max(0) : min(NX-1);
NC = CD + 1;
DX = (X1 - X0) / NX;
nx = (x - X0) / DX : int : ck(CK);
xc(n) = X0 + DX * (n + 1/2);
ch(i) = chtab(NC * nx + i);
y = (x - xc(nx)) * 2/DX <: sum(i,NC, ch(i) * ma.chebychev(i));
mapfx(nx, x) = FX(xc(nx) + DX/2 * x);
gench(nx, nc) = (1+(nc!=0))/NC * sum(k,NC,
mapfx(nx, cos(ma.PI*(k+1/2)/NC)) *
cos(ma.PI*nc*(k+1/2)/NC));
chtab = rdtable(NX*NC, (ba.time <: int(/(NC)), %(NC) : gench));
};
Usage:
_ : chebtab(CK, FX, NX,CD, X0,X1) : _
Where
CK: force the index in [X0, X1] range
FX: unary function
NX: number of segments
CD: degree of the chebyshev polynomials
X0: range minimal value
X1: range maximal value
chebtab() splits the interval [X0,X1] into NX segments, and for each segment
it calculates CD+1 coefficients for ma.chebychev(0) .. ma.chebychev(CD). IOW,
it uses rdtable as 2-dimensional array [NX][CD+1], so its size is NX * (CD+1).
Lets compare chebtab() with ba.tabulate().cub, test-case:
CK = 0;
FX = sin;
NX = 50;
CD = 3;
X0 = 0;
X1 = ma.PI;
// Both ch() and tb() create rdtable of size = 200, both use
// cubic polynomials, so this comparison is more or less fair.
ch(x) = chebtab(CK, FX, NX, CD, X0,X1, x);
tb(x) = ba.tabulate(CK, FX, NX*(CD+1), X0,X1, x).cub;
maxerr = abs : max ~ _;
line(n, x0,x1) = x0 + (ba.time%n)/n * (x1-x0);
process = line(50000, X0,X1) <: FX-tb,FX-ch : par(i,2,maxerr);
Now,
$ cheb-plot -n 50000 | tail -1
0.00349552557 5.07155429e-09
as you can see chebtab() is much more accurate. And according to my quick
testing it is even a little bit faster but I won't insist on that.
Just in case... tabulate().cub can be improved, it.interpolate_cubic() is
simply wrong near X0 or X1, but chebtab() will be more accurate anyway.
Do you think it can live in basics.lib ?
Oleg.
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