294 lines
4.1 KiB
C++
294 lines
4.1 KiB
C++
//-----------------------------------------------------------------------------
|
|
//
|
|
// Legendre polynomial
|
|
//
|
|
// Input: tos-3 x (can be a symbol or expr)
|
|
//
|
|
// tos-2 n
|
|
//
|
|
// tos-1 m
|
|
//
|
|
// Output: Result on stack
|
|
//
|
|
// Uses the recurrence relation
|
|
//
|
|
// P(x,0) = 1
|
|
//
|
|
// P(x,1) = x
|
|
//
|
|
// n*P(x,n) = (2*(n-1)+1)*x*P(x,n-1) - (n-1)*P(x,n-2)
|
|
//
|
|
// In the "for" loop we have i = n-1 so the recurrence relation becomes
|
|
//
|
|
// (i+1)*P(x,n) = (2*i+1)*x*P(x,n-1) - i*P(x,n-2)
|
|
//
|
|
// For m > 0
|
|
//
|
|
// P(x,n,m) = (-1)^m * (1-x^2)^(m/2) * d^m/dx^m P(x,n)
|
|
//
|
|
//-----------------------------------------------------------------------------
|
|
|
|
#include "stdafx.h"
|
|
#include "defs.h"
|
|
|
|
#define X p1
|
|
#define N p2
|
|
#define M p3
|
|
#define Y p4
|
|
#define Y0 p5
|
|
#define Y1 p6
|
|
|
|
static void __legendre(void), __legendre2(int, int), __legendre3(int);
|
|
|
|
void
|
|
legendre(void)
|
|
{
|
|
save();
|
|
__legendre();
|
|
restore();
|
|
}
|
|
|
|
static void
|
|
__legendre(void)
|
|
{
|
|
int m, n;
|
|
|
|
M = pop();
|
|
N = pop();
|
|
X = pop();
|
|
|
|
push(N);
|
|
n = pop_integer();
|
|
|
|
push(M);
|
|
m = pop_integer();
|
|
|
|
if (n < 0 || m < 0) {
|
|
push_symbol(LEGENDRE);
|
|
push(X);
|
|
push(N);
|
|
push(M);
|
|
list(4);
|
|
return;
|
|
}
|
|
|
|
if (issymbol(X))
|
|
__legendre2(n, m);
|
|
else {
|
|
Y = X; // do this when X is an expr
|
|
X = symbol(SECRETX);
|
|
__legendre2(n, m);
|
|
X = Y;
|
|
push(symbol(SECRETX));
|
|
push(X);
|
|
subst();
|
|
eval();
|
|
}
|
|
|
|
__legendre3(m);
|
|
}
|
|
|
|
static void
|
|
__legendre2(int n, int m)
|
|
{
|
|
int i;
|
|
|
|
push_integer(1);
|
|
push_integer(0);
|
|
|
|
Y1 = pop();
|
|
|
|
// i=1 Y0 = 0
|
|
// Y1 = 1
|
|
// ((2*i+1)*x*Y1 - i*Y0) / i = x
|
|
//
|
|
// i=2 Y0 = 1
|
|
// Y1 = x
|
|
// ((2*i+1)*x*Y1 - i*Y0) / i = -1/2 + 3/2*x^2
|
|
//
|
|
// i=3 Y0 = x
|
|
// Y1 = -1/2 + 3/2*x^2
|
|
// ((2*i+1)*x*Y1 - i*Y0) / i = -3/2*x + 5/2*x^3
|
|
|
|
for (i = 0; i < n; i++) {
|
|
|
|
Y0 = Y1;
|
|
|
|
Y1 = pop();
|
|
|
|
push_integer(2 * i + 1);
|
|
push(X);
|
|
multiply();
|
|
push(Y1);
|
|
multiply();
|
|
|
|
push_integer(i);
|
|
push(Y0);
|
|
multiply();
|
|
|
|
subtract();
|
|
|
|
push_integer(i + 1);
|
|
divide();
|
|
}
|
|
|
|
for (i = 0; i < m; i++) {
|
|
push(X);
|
|
derivative();
|
|
}
|
|
}
|
|
|
|
// tos = tos * (-1)^m * (1-x^2)^(m/2)
|
|
|
|
static void
|
|
__legendre3(int m)
|
|
{
|
|
if (m == 0)
|
|
return;
|
|
|
|
if (car(X) == symbol(COS)) {
|
|
push(cadr(X));
|
|
sine();
|
|
square();
|
|
} else if (car(X) == symbol(SIN)) {
|
|
push(cadr(X));
|
|
cosine();
|
|
square();
|
|
} else {
|
|
push_integer(1);
|
|
push(X);
|
|
square();
|
|
subtract();
|
|
}
|
|
|
|
push_integer(m);
|
|
push_rational(1, 2);
|
|
multiply();
|
|
power();
|
|
multiply();
|
|
|
|
if (m % 2)
|
|
negate();
|
|
}
|
|
|
|
#if SELFTEST
|
|
|
|
static char *s[] = {
|
|
|
|
"legendre(x,n)",
|
|
"legendre(x,n,0)",
|
|
|
|
"legendre(x,n,m)",
|
|
"legendre(x,n,m)",
|
|
|
|
"legendre(x,0)-1",
|
|
"0",
|
|
|
|
"legendre(x,1)-x",
|
|
"0",
|
|
|
|
"legendre(x,2)-1/2*(3*x^2-1)",
|
|
"0",
|
|
|
|
"legendre(x,3)-1/2*(5*x^3-3*x)",
|
|
"0",
|
|
|
|
"legendre(x,4)-1/8*(35*x^4-30*x^2+3)",
|
|
"0",
|
|
|
|
"legendre(x,5)-1/8*(63*x^5-70*x^3+15*x)",
|
|
"0",
|
|
|
|
"legendre(x,6)-1/16*(231*x^6-315*x^4+105*x^2-5)",
|
|
"0",
|
|
|
|
"legendre(x,0,0)-1",
|
|
"0",
|
|
|
|
"legendre(x,1,0)-x",
|
|
"0",
|
|
|
|
"legendre(x,1,1)+(1-x^2)^(1/2)",
|
|
"0",
|
|
|
|
"legendre(x,2,0)-1/2*(3*x^2-1)",
|
|
"0",
|
|
|
|
"legendre(x,2,1)+3*x*(1-x^2)^(1/2)",
|
|
"0",
|
|
|
|
"legendre(x,2,2)-3*(1-x^2)",
|
|
"0",
|
|
|
|
"legendre(x,3,0)-1/2*x*(5*x^2-3)",
|
|
"0",
|
|
|
|
"legendre(x,3,1)-3/2*(1-5*x^2)*(1-x^2)^(1/2)",
|
|
"0",
|
|
|
|
"legendre(x,3,2)-15*x*(1-x^2)",
|
|
"0",
|
|
|
|
"legendre(x,3,3)+15*(1-x^2)^(3/2)",
|
|
"0",
|
|
|
|
"legendre(x,4,0)-1/8*(35*x^4-30*x^2+3)",
|
|
"0",
|
|
|
|
"legendre(x,4,1)-5/2*x*(3-7*x^2)*(1-x^2)^(1/2)",
|
|
"0",
|
|
|
|
"legendre(x,4,2)-15/2*(7*x^2-1)*(1-x^2)",
|
|
"0",
|
|
|
|
"legendre(x,4,3)+105*x*(1-x^2)^(3/2)",
|
|
"0",
|
|
|
|
"legendre(x,4,4)-105*(1-x^2)^2",
|
|
"0",
|
|
|
|
"legendre(x,5,0)-1/8*x*(63*x^4-70*x^2+15)",
|
|
"0",
|
|
|
|
"legendre(cos(theta),0,0)-1",
|
|
"0",
|
|
|
|
"legendre(cos(theta),1,0)-cos(theta)",
|
|
"0",
|
|
|
|
"legendre(cos(theta),1,1)+sin(theta)",
|
|
"0",
|
|
|
|
"legendre(cos(theta),2,0)-1/2*(3*cos(theta)^2-1)",
|
|
"0",
|
|
|
|
"legendre(cos(theta),2,1)+3*sin(theta)*cos(theta)",
|
|
"0",
|
|
|
|
"legendre(cos(theta),2,2)-3*sin(theta)^2",
|
|
"0",
|
|
|
|
"legendre(cos(theta),3,0)-1/2*cos(theta)*(5*cos(theta)^2-3)",
|
|
"0",
|
|
|
|
"legendre(cos(theta),3,1)+3/2*(5*cos(theta)^2-1)*sin(theta)",
|
|
"0",
|
|
|
|
"legendre(cos(theta),3,2)-15*cos(theta)*sin(theta)^2",
|
|
"0",
|
|
|
|
"legendre(cos(theta),3,3)+15*sin(theta)^3",
|
|
"0",
|
|
|
|
"legendre(a-b,10)-eval(subst(a-b,x,legendre(x,10)))",
|
|
"0",
|
|
};
|
|
|
|
void
|
|
test_legendre(void)
|
|
{
|
|
test(__FILE__, s, sizeof s / sizeof (char *));
|
|
}
|
|
|
|
#endif
|