Author: bugman
Date: Fri Jun 25 14:16:55 2010
New Revision: 11248
URL: http://svn.gna.org/viewcvs/relax?rev=11248&view=rev
Log:
Added a function for the pseudo-elliptical cosine function.
This is a numerical approximation generated by series expansion.
Added:
1.3/maths_fns/pseudo_ellipse.py
Added: 1.3/maths_fns/pseudo_ellipse.py
URL:
http://svn.gna.org/viewcvs/relax/1.3/maths_fns/pseudo_ellipse.py?rev=11248&view=auto
==============================================================================
--- 1.3/maths_fns/pseudo_ellipse.py (added)
+++ 1.3/maths_fns/pseudo_ellipse.py Fri Jun 25 14:16:55 2010
@@ -1,0 +1,149 @@
+###############################################################################
+#
#
+# Copyright (C) 2010 Edward d'Auvergne
#
+#
#
+# This file is part of the program relax.
#
+#
#
+# relax is free software; you can redistribute it and/or modify
#
+# it under the terms of the GNU General Public License as published by
#
+# the Free Software Foundation; either version 2 of the License, or
#
+# (at your option) any later version.
#
+#
#
+# relax is distributed in the hope that it will be useful,
#
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
#
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
#
+# GNU General Public License for more details.
#
+#
#
+# You should have received a copy of the GNU General Public License
#
+# along with relax; if not, write to the Free Software
#
+# Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
#
+#
#
+###############################################################################
+
+# Module docstring.
+"""Module for the pseudo-elliptical functions."""
+
+# Python module import.
+from math import factorial, pi
+
+
+def pec(x, y):
+ """The 2D pseudo-elliptic cosine function.
+
+ This is defined as::
+
+ int_-pi^pi (1 - cos(theta_max)).dphi,
+
+ where::
+
+ theta_max = 1 / sqrt(cos**2(phi)/x**2 + sin**2(phi)/y**2).
+
+ The function is approximated as::
+
+ _inf
+ \ (-1)**n
+ 2*pi*x*y > ----------------- fn(x, y),
+ /___ 4**n * (2n + 2)!
+ n=0
+
+ and where the individual functions are::
+
+ n: 0; fn(x,y) = 1
+ n: 1; fn(x,y) = 2*a + 2*b
+ n: 2; fn(x,y) = 6*a^2 + 4*a*b + 6*b^2
+ n: 3; fn(x,y) = 20*a^3 + 12*a^2*b + 12*a*b^2 + 20*b^3
+ n: 4; fn(x,y) = 70*a^4 + 40*a^3*b + 36*a^2*b^2 + 40*a*b^3 + 70*b^4
+ n: 5; fn(x,y) = 252*a^5 + 140*a^4*b + 120*a^3*b^2 + 120*a^2*b^3 +
140*a*b^4 + 252*b^5
+ n: 6; fn(x,y) = 924*a^6 + 504*a^5*b + 420*a^4*b^2 + 400*a^3*b^3 +
420*a^2*b^4 + 504*a*b^5 + 924*b^6
+ n: 7; fn(x,y) = 3432*a^7 + 1848*a^6*b + 1512*a^5*b^2 + 1400*a^4*b^3
+ 1400*a^3*b^4 + 1512*a^2*b^5 + 1848*a*b^6 + 3432*b^7
+ n: 8; fn(x,y) = 12870*a^8 + 6864*a^7*b + 5544*a^6*b^2 + 5040*a^5*b^3
+ 4900*a^4*b^4 + 5040*a^3*b^5 + 5544*a^2*b^6 + 6864*a*b^7 + 12870*b^8
+ n: 9; fn(x,y) = 48620*a^9 + 25740*a^8*b + 20592*a^7*b^2 +
18480*a^6*b^3 + 17640*a^5*b^4 + 17640*a^4*b^5 + 18480*a^3*b^6 + 20592*a^2*b^7
+ 25740*a*b^8 + 48620*b^9
+ n: 10; fn(x,y) = 184756*a^10 + 97240*a^9*b + 77220*a^8*b^2 +
68640*a^7*b^3 + 64680*a^6*b^4 + 63504*a^5*b^5 + 64680*a^4*b^6 + 68640*a^3*b^7
+ 77220*a^2*b^8 + 97240*a*b^9 + 184756*b^10
+ n: 11; fn(x,y) = 705432*a^11 + 369512*a^10*b + 291720*a^9*b^2 +
257400*a^8*b^3 + 240240*a^7*b^4 + 232848*a^6*b^5 + 232848*a^5*b^6 +
240240*a^4*b^7 + 257400*a^3*b^8 + 291720*a^2*b^9 + 369512*a*b^10 + 705432*b^11
+
+
+ @param x: The theta_x cone angle (in rad).
+ @type x: float
+ @param y: The theta_y cone angle (in rad).
+ @type y: float
+ @return: The approximate function value.
+ @rtype: float
+ """
+
+ # Init.
+ fn = 0.0
+ fact = 1.0
+
+ # The x powers.
+ a = x**2
+ a2 = a**2
+ a3 = a**3
+ a4 = a**4
+ a5 = a**5
+ a6 = a**6
+ a7 = a**7
+ a8 = a**8
+ a9 = a**9
+ a10 = a**10
+
+ # The y powers.
+ b = y**2
+ b2 = b**2
+ b3 = b**3
+ b4 = b**4
+ b5 = b**5
+ b6 = b**6
+ b7 = b**7
+ b8 = b**8
+ b9 = b**9
+ b10 = b**10
+
+ # Term 0.
+ f0 = 1.0
+ fn = fn + f0 / factorial(2)
+
+ # Term 1.
+ f1 = 2*a + 2*b
+ fn = fn - f1 / 4.0 / factorial(4)
+
+ # Term 2.
+ f2 = 6*a2 + 4*a*b + 6*b2
+ fn = fn + f2 / 16.0 / factorial(6)
+
+ # Term 3.
+ f3 = 20*a3 + 12*a2*b + 12*a*b2 + 20*b3
+ fn = fn - f3 / 64.0 / factorial(8)
+
+ # Term 4.
+ f4 = 70*a4 + 40*a3*b + 36*a2*b2 + 40*a*b3 + 70*b4
+ fn = fn + f4 / 256.0 / factorial(10)
+
+ # Term 5.
+ f5 = 252*a5 + 140*a4*b + 120*a3*b2 + 120*a2*b3 + 140*a*b4 + 252*b5
+ fn = fn - f5 / 1024.0 / factorial(12)
+
+ # Term 6.
+ f6 = 924*a6 + 504*a5*b + 420*a4*b2 + 400*a3*b3 + 420*a2*b4 + 504*a*b5 +
924*b6
+ fn = fn + f6 / 4096.0 / factorial(14)
+
+ # Term 7.
+ f7 = 3432*a7 + 1848*a6*b + 1512*a5*b2 + 1400*a4*b3 + 1400*a3*b4 +
1512*a2*b5 + 1848*a*b6 + 3432*b7
+ fn = fn - f7 / 16384.0 / factorial(16)
+
+ # Term 8.
+ f8 = 12870*a8 + 6864*a7*b + 5544*a6*b2 + 5040*a5*b3 + 4900*a4*b4 +
5040*a3*b5 + 5544*a2*b6 + 6864*a*b7 + 12870*b8
+ fn = fn + f8 / 65536.0 / factorial(18)
+
+ # Term 9.
+ f9 = 48620*a9 + 25740*a8*b + 20592*a7*b2 + 18480*a6*b3 + 17640*a5*b4 +
17640*a4*b5 + 18480*a3*b6 + 20592*a2*b7 + 25740*a*b8 + 48620*b9
+ fn = fn - f9 / 262144.0 / factorial(20)
+
+ # Term 10.
+ f10 = 184756*a10 + 97240*a9*b + 77220*a8*b2 + 68640*a7*b3 + 64680*a6*b4
+ 63504*a5*b5 + 64680*a4*b6 + 68640*a3*b7 + 77220*a2*b8 + 97240*a*b9 +
184756*b10
+ fn = fn + f10 / 1048576.0 / factorial(22)
+
+ # Common factor.
+ fn = 2.0 * pi * x * y * fn
+
+ # Return the approximate function value.
+ return fn
r11248 - /1.3/maths_fns/pseudo_ellipse.py