110 lines
2.7 KiB
Python
110 lines
2.7 KiB
Python
"""Numerical Methods for Holonomic Functions"""
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from sympy.core.sympify import sympify
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from sympy.holonomic.holonomic import DMFsubs
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from mpmath import mp
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def _evalf(func, points, derivatives=False, method='RK4'):
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"""
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Numerical methods for numerical integration along a given set of
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points in the complex plane.
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"""
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ann = func.annihilator
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a = ann.order
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R = ann.parent.base
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K = R.get_field()
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if method == 'Euler':
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meth = _euler
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else:
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meth = _rk4
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dmf = []
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for j in ann.listofpoly:
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dmf.append(K.new(j.rep))
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red = [-dmf[i] / dmf[a] for i in range(a)]
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y0 = func.y0
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if len(y0) < a:
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raise TypeError("Not Enough Initial Conditions")
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x0 = func.x0
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sol = [meth(red, x0, points[0], y0, a)]
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for i, j in enumerate(points[1:]):
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sol.append(meth(red, points[i], j, sol[-1], a))
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if not derivatives:
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return [sympify(i[0]) for i in sol]
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else:
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return sympify(sol)
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def _euler(red, x0, x1, y0, a):
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"""
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Euler's method for numerical integration.
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From x0 to x1 with initial values given at x0 as vector y0.
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"""
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A = sympify(x0)._to_mpmath(mp.prec)
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B = sympify(x1)._to_mpmath(mp.prec)
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y_0 = [sympify(i)._to_mpmath(mp.prec) for i in y0]
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h = B - A
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f_0 = y_0[1:]
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f_0_n = 0
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for i in range(a):
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f_0_n += sympify(DMFsubs(red[i], A, mpm=True))._to_mpmath(mp.prec) * y_0[i]
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f_0.append(f_0_n)
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sol = []
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for i in range(a):
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sol.append(y_0[i] + h * f_0[i])
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return sol
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def _rk4(red, x0, x1, y0, a):
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"""
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Runge-Kutta 4th order numerical method.
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"""
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A = sympify(x0)._to_mpmath(mp.prec)
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B = sympify(x1)._to_mpmath(mp.prec)
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y_0 = [sympify(i)._to_mpmath(mp.prec) for i in y0]
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h = B - A
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f_0_n = 0
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f_1_n = 0
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f_2_n = 0
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f_3_n = 0
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f_0 = y_0[1:]
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for i in range(a):
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f_0_n += sympify(DMFsubs(red[i], A, mpm=True))._to_mpmath(mp.prec) * y_0[i]
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f_0.append(f_0_n)
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f_1 = [y_0[i] + f_0[i]*h/2 for i in range(1, a)]
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for i in range(a):
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f_1_n += sympify(DMFsubs(red[i], A + h/2, mpm=True))._to_mpmath(mp.prec) * (y_0[i] + f_0[i]*h/2)
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f_1.append(f_1_n)
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f_2 = [y_0[i] + f_1[i]*h/2 for i in range(1, a)]
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for i in range(a):
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f_2_n += sympify(DMFsubs(red[i], A + h/2, mpm=True))._to_mpmath(mp.prec) * (y_0[i] + f_1[i]*h/2)
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f_2.append(f_2_n)
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f_3 = [y_0[i] + f_2[i]*h for i in range(1, a)]
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for i in range(a):
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f_3_n += sympify(DMFsubs(red[i], A + h, mpm=True))._to_mpmath(mp.prec) * (y_0[i] + f_2[i]*h)
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f_3.append(f_3_n)
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sol = []
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for i in range(a):
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sol.append(y_0[i] + h * (f_0[i]+2*f_1[i]+2*f_2[i]+f_3[i])/6)
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return sol
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