Python 中的移位切比雪夫多项式

Question

I am trying to extract the coefficients of a shifted Chebyshev Polynomial in Python, but I couldn't find the function to do that.

There is this function:

scipy.special.eval_sh_chebyt

scipy.special.eval_sh_chebyt(n, x, out=None) = <ufunc 'eval_sh_chebyt'>

but I can't extract only the coefficients.

The shifted Chebyshev polynomials are: T^{*}_n(x) = T_n (2x -1)

and then:

T_1^{*}(x) = 2x - 1

T_2^{*}(x) = 8x^2 - 8x -1

I would like to extract a matrix only with the coefficients, like 2 and 1 or 8,8 and 1.

Answer 1

scipy.special.eval_sh_chebyt evaluates the function T* at a given point x . Unless you are very smart in choosing your arguments, this is not an appropriate way to get the coeffcients.

You can instead calculate the coefficients directly from the recurrence formula T0 = 1, T1 = x, T{n} = 2xT{n-1} - T{n-2} . And then calculate the shifted polynomials from _T*{n} = T{n}(2x - 1).

degree = 10

coeffs = []
# for T_0
coeffLine = [1]
coeffs.append(coeffLine)
# for T_1
coeffLine = [0, 1]
coeffs.append(coeffLine)

for i in range(2, degree + 1):
    coeffLine = [0] * (1 + i)
    coeffLine[0] = -coeffs[i - 2][0]
    for j in range(1, i - 1):
        coeffLine[j] = 2 * coeffs[i - 1][j - 1] - coeffs[i - 2][j]
    coeffLine[-2] = 2 * coeffs[i - 1][-2]
    coeffLine[-1] = 2 * coeffs[i - 1][-1]
    coeffs.append(coeffLine)        

print("T_%d" % degree, coeffs[-1])

shiftedCoeffs = [0] * (degree + 1);
for i in range(degree + 1):
    binom = 1
    for j in range(i + 1):
        shiftedCoeffs[i - j] += coeffs[-1][i] * 2 ** (i - j) * (-1) ** j * binom
        binom *= (i - j) / (j + 1)


print("T*_%d" % degree, shiftedCoeffs)

Edit: I originally misread the formula for T*{n} as _T{n}(x) * (2x - 1). However it is T*{n} = T{n}(2x - 1) . We therefore need to calculate the binomial coefficients sum them up depending on the order of x .