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besselj
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1994-09-23
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8KB
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301 lines
////-------------------------------------------------------------------//
// Syntax: besselj ( ALPHA, X )
// Description:
// Besselj is the bessel function of the first kind of order ALPHA.
// Either ALPHA or X may a vector in which case a vector result of
// the same dimension is returned. If both ALPHA and X are vectors
// a matrix result of dimension length(ALPHA) x length(X) is returned.
// If either ALPHA or X is a scalar, the other argument may be a matrix
// and a matrix result of the same dimension is returned.
//-------------------------------------------------------------------//
// See Numerical Recipes in C (second edition)
// Currently only integer order supported.
static (besseljn, besselj0, besselj1);
besselj = function(alph,x)
{
global (pi)
// Checking the class will automatically result in an error if
// the argument doesn't exist.
if(class(alph) != "num" || class(x) != "num") {
error("besselj: argument is non-numeric.");
}
if(min(size(alph)) > 1 && min(size(x)) > 1) {
// Both arguments are matrices.
error("besselj: only one argument may be a matrix.");
}
if(length(alph) == 1) {
// x is a matrix (or scalar), alph a scalar
if(alph == 0 || mod(alph,int(alph)) == 0) {
return besseljn(alph,x);
else
error("besselj: fractional orders not yet supported.");
// return besseljr(alph,x);
}
else if(length(x) == 1) {
// alph is a matrix (or vector), x a scalar
y = zeros(size(alph));
// Use integer order routines for integer alph
inte_ind = find(alph == 0 || mod(alph,int(alph)) == 0);
frac_ind = complement(inte_ind,1:alph.n);
if(inte_ind.n) {
for(index in inte_ind) {
y[index] = besseljn(alph[index],x);
}
}
if(frac_ind.n) {
error("besselj: fractional orders not yet supported.");
// for(index in frac_ind) {
// y[index] = besseljr(alph[index],x);
// }
}
return y;
else
// alph and x are both vectors
y = zeros(length(alph),length(x));
// Use integer order routines for integer alph
inte_ind = find(alph == 0 || mod(alph,int(alph)) == 0);
frac_ind = complement(inte_ind,1:alph.n);
if(inte_ind.n) {
for(index in inte_ind) {
y[index;] = besseljn(alph[index],x);
}
}
if(frac_ind.n) {
error("besselj: fractional orders not yet supported.");
// for(index in frac_ind) {
// y[index;] = besseljr(alph[index],x);
// }
}
return y;
}}
};
// In these functions x can be a vector (or matrix), but n
// must be a scalar.
// No argument checking is performed on static functions.
besseljn = function ( n, x ) {
local(abs_x,y,index,index_1,index_2,index_3,arg,p_1,p_2,p_3, ...
two_over_x,m,y_2,Norm,even,limit,lim_ind);
if(n == 0) {
return besselj0(x);
else if (n == 1) {
return besselj1(x);
else
// integer order greater than two
abs_x = abs(x);
y = zeros(size(x));
index_1 = find(abs_x > n);
index_2 = complement(index_1,1:x.n);
index_3 = find(abs_x == 0);
index_1 = complement(intersection(index_1,index_3),index_1);
index_2 = complement(intersection(index_2,index_3),index_2);
if(index_1.n) {
arg = x[index_1];
p_1 = besselj0(arg);
p_2 = besselj1(arg);
two_over_x = 2./arg;
for(index in 1:(n-1)) {
p_3 = index * (two_over_x .* p_2) - p_1;
p_1 = p_2;
p_2 = p_3;
}
if(mod(n,2) == 1) {
index = find(arg < 0.);
if(index.n) {
p_3[index] = -p_3[index];
}
}
y[index_1] = p_3;
}
if(index_2.n) {
limit=1.e20;
arg = x[index_2];
two_over_x = 2./arg;
// Downward recurrence from arbitrary starting values.
// Find a starting upper even index (this give 16 digits
// of accuracy).
m = 2*int((n + 16*sqrt(n))/2);
p_1 = zeros(size(arg));
p_2 = ones(size(arg));
Norm = zeros(size(arg));
y_2 = zeros(size(arg));
even = 0;
for(index in m:1:-1) {
p_3 = index * (two_over_x .* p_2) - p_1;
p_1 = p_2;
p_2 = p_3;
if(even) {
Norm = Norm + p_2;
}
lim_ind = find(abs(p_2) > limit);
if(!isempty(lim_ind)) {
p_2[lim_ind] = p_2[lim_ind] / limit;
p_1[lim_ind] = p_1[lim_ind] / limit;
Norm[lim_ind] = Norm[lim_ind] / limit;
if(index < n) {
y_2[lim_ind] = y_2[lim_ind] / limit;
}
}
even = !even;
if(index == n) {
// Save the unnormalized answer, continue looping to get
// normalization.
y_2 = p_1;
}
}
// Only need 1*j0(x) so subtract off p_2 after doubling.
Norm = 2*Norm - p_2;
y_2 = y_2 ./ Norm;
if(mod(n,2) == 1) {
index = find(arg < 0.);
if(index.n) {
y_2[index] = -y_2[index];
}
}
y[index_2] = y_2;
}
if(index_3.n) {
y[index_3] = zeros(size(index_3));
}
return y;
}}
};
besselj0 = function(x) {
local(abs_x,index_1,index_2,y,c_1,c_2,arg_1,arg_2,p_1,p_2);
abs_x = abs(x);
index_1 = find(abs_x < 8);
index_2 = complement(index_1,1:x.n);
// First evaluate using rational approx. for small arguments.
y = zeros(size(x));
if(index_1.n) {
c_1 = [ 57568490574, -13362590354, 651619640.7, ...
-11214424.18, 77392.33017, -184.9052456];
c_2 = [ 57568490411, 1029532985, 9494680.718, ...
59272.64853, 267.8532712];
arg_1 = x[index_1] .* x[index_1];
p_1 = c_1[1] + arg_1 .* (c_1[2] + arg_1 .* (c_1[3] + arg_1 .* ...
(c_1[4] + arg_1 .* (c_1[5] + arg_1 .* c_1[6]))));
p_2 = c_2[1] + arg_1 .* (c_2[2] + arg_1 .* (c_2[3] + arg_1 .* ...
(c_2[4] + arg_1 .* (c_2[5] + arg_1))));
y[index_1] = p_1./p_2;
}
// Now evaluate for large arguments using approximating form.
if(index_2.n) {
c_1 = [ -0.1098628627e-2, 0.2734510407e-4, ...
-0.2073370639e-5, 0.2093887211e-6 ];
c_2 = [ -0.1562499995e-1, 0.1430488765e-3, ...
-0.6911147651e-5, 0.7621095161e-6, 0.934935152e-7 ];
arg_1 = 8./x[index_2];
arg_2 = arg_1 .* arg_1;
p_1 = 1 + arg_2 .* (c_1[1] + arg_2 .* (c_1[2] + arg_2 .* (c_1[3] + ...
arg_2 .* c_1[4])));
p_2 = c_2[1] + arg_2 .* (c_2[2] + arg_2 .* (c_2[3] + arg_2 .* (c_2[4] + ...
arg_2 .* c_2[5])));
c_1 = abs_x[index_2];
c_2 = c_1 - pi/4;
y[index_2] = sqrt(2./(pi*c_1)).*(p_1.*cos(c_2) - arg_1.*p_2.*sin(c_2));
}
return y;
};
besselj1 = function(x) {
local(abs_x,index_1,index_2,y,c_1,c_2,arg_1,arg_2,p_1,p_2);
abs_x = abs(x);
index_1 = find(abs_x < 8);
index_2 = complement(index_1,1:x.n);
// First evaluate using rational approx. for small arguments.
y = zeros(size(x));
if(index_1.n) {
c_1 = [ 72362614232, -7895059235, 242396853.1, ...
-2972611.439, 15704.48260, -30.16036606 ];
c_2 = [ 144725228442, 2300535178, 18583304.74, ...
99447.43394, 376.9991397];
arg_1 = x[index_1];
arg_2 = arg_1 .* arg_1;
p_1 = arg_1.*(c_1[1] + arg_2 .* (c_1[2] + arg_2 .* (c_1[3] + arg_2 .* ...
(c_1[4] + arg_2 .* (c_1[5] + arg_2 .* c_1[6])))));
p_2 = c_2[1] + arg_2 .* (c_2[2] + arg_2 .* (c_2[3] + arg_2 .* ...
(c_2[4] + arg_2 .* (c_2[5] + arg_2))));
y[index_1] = p_1./p_2;
}
// Now evaluate for large arguments using approximating form.
if(index_2.n) {
c_1 = [ 0.183105e-2, -0.3516396496e-4, 0.2457520174e-5, ...
-0.240337019e-6 ];
c_2 = [ 0.04687499995, -0.2002690873e-3, 0.8449199096e-5, ...
-0.88228987e-6, 0.105787412e-6 ];
arg_1 = 8./x[index_2];
arg_2 = arg_1 .* arg_1;
p_1 = 1 + arg_2 .* (c_1[1] + arg_2 .* (c_1[2] + arg_2 .* (c_1[3] + ...
arg_2 .* c_1[4])));
p_2 = c_2[1] + arg_2 .* (c_2[2] + arg_2 .* (c_2[3] + arg_2 .* (c_2[4] + ...
arg_2 .* c_2[5])));
c_1 = abs_x[index_2];
c_2 = c_1 - 3*pi/4;
arg_2 = sqrt(2./(pi*c_1)).*(p_1.*cos(c_2) - arg_1.*p_2.*sin(c_2));
index_1 = find(arg_1 < 0);
if(!isempty(index_1)) {
arg_2[index_1] = -arg_2[index_1];
}
y[index_2] = arg_2;
}
return y;
};