I want to find a sympy function, that fulfills several arguments. It should go through several points and have a certain derivative at some points. How do I do that?
values = np.array([1.0, 2.0], [2.0, 3.0])
derivats = np.array([1.0, 3.0], [2.0, 5.0])
Please help me and my mathematical English is not that good, so please use words that I can google.
Here is a interpretation with SymPy's solve (docs) and polynomials, to give an idea of the possibilities:
from sympy import *
x = symbols('x', real=True)
n = 4 # degree of the polynomial
a = [symbols('a%d' % i, real=True) for i in range(n)]
print(a)
f = sum([a[i]*x**i for i in range(n)]) # f is a polynomial of degree n
g = f.diff(x)
sol = solve([Eq(f.subs(x, 1), 2), Eq(f.subs(x, 2), 3), Eq(g.subs(x, 1), 3), Eq(g.subs(x, 2), 5), ], a)
print(sol)
print(f.subs(sol))
This prints:
{a0: -15, a1: 37, a2: -26, a3: 6}
6*x**3 - 26*x**2 + 37*x - 15
Directly working with your arrays, this could be:
from sympy import *
x = symbols('x', real=True)
n = 4 # degree of the polynomial
a = [symbols('a%d' % i, real=True) for i in range(n)]
f = sum([a[i]*x**i for i in range(n)]) # f is a polynomial of degree n
g = f.diff(x)
values = [[1.0, 2.0], [2.0, 3.0]] # suppose an array of pairs [x, f(x)]
derivats = [[1.0, 3.0], [2.0, 5.0]] # suppose an array of pairs [x, f'(x)]
equations = [Eq(f.subs(x, xi), yi) for xi, yi in values] + [Eq(g.subs(x, xi), yi) for xi, yi in derivats]
sol = solve(equations)
print(sol)
print(f.subs(sol))
This outputs:
{a0: -15.0000000000000, a1: 37.0000000000000, a2: -26.0000000000000, a3: 6.00000000000000}
6.0*x**3 - 26.0*x**2 + 37.0*x - 15.0
Note that SymPy is less happy to work with reals than with integers and fractions.