performance - Why is fitting a polynomial faster than changing polynomial basis? -

given data points on interval [–1, 1] , best-fit chebyshev polynomial points, want convert chebyshev polynomial legendre polynomial.

there 2 ways it, shown in code below. direct way call convert(kind = legendre) on chebyshev polynomial, took 19.591 seconds. alternative call legendre.fit on data points, took 3.356 seconds.

import numpy np numpy.polynomial import chebyshev, legendre x = np.linspace(-1, 1, 1000) y = 1.0 / (1 + x ** 2) + 1e-3 * np.random.random(1000) t = chebyshev.fit(x, y, 99)  timeit import timeit timeit("t.convert(kind = legendre)", setup = "from __main__ import x, y, t, legendre",        number = 200) timeit("legendre.fit(x, y, 99)", setup = "from __main__ import x, y, legendre",        number = 200) 

question: why legendre.fit faster convert(kind = legendre)? doing wrongly?