yroots.approximate¶
-Approximator¶
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- -yroots.ChebyshevApproximator.chebApproximate(f, a, b, absApproxTol=1e-10, relApproxTol=1e-10)¶ -
Generate and return an approximation for the function f on the interval [a,b].
-Uses properties of Chebyshev polynomials and the FFT to quickly generate a reliable -approximation. Examines approximation one dimension at a time to determine the degree at which -the coefficients geometrically converge to 0 in each dimension, then calculates and returns a -final approximation of these degree values along with the associated approximation error.
-NOTE: The approximate function is only guaranteed to work well on functions that are continuous -and smooth on the approximation interval. If the input function is not continuous and smooth on -the interval, the approximation may get stuck in recursion.
-Examples
--->>> f = lambda x,y,z: x**2 - y**2 + 3*x*y ->>> approx, error = yroots.approximate(f,[-1,-1,-1],[1,1,1]) ->>> print(approx) -[[[ 0.00000000e+00] - [ 1.11022302e-16] - [-5.00000000e-01]] - [[ 1.11022302e-16] - [ 3.00000000e+00] - [-1.11022302e-16]] - [[ 5.00000000e-01] - [-1.11022302e-16] - [ 0.00000000e+00]]] ->>> print(error) -2.8014584982224306e-24 -
-->>> g = np.sqrt ->>> approx = yroots.approximate(g,[0],[5])[0] ->>> print(approx) -[ 1.42352509e+00 9.49016725e-01 -1.89803345e-01 ... -1.24418041e-10 - 1.24418045e-10 -6.22090244e-11] -
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- Parameters: -
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f (function) – The function to be approximated. NOTE: Valid input is restricted to callable Python functions -(including user-created functions) and yroots Polynomial (MultiCheb and MultiPower) objects. -String representations of functions are not valid input.
-a (list or numpy array) – An array containing the lower bound of the approximation interval in each dimension, listed in -dimension order
-b (list or numpy array) – An array containing the upper bound of the approximation interval in each dimension, listed in -dimension order.
-absApproxTol (float) – The absolute tolerance used to determine at what degree the Chebyshev coefficients have -converged to zero. If all coefficients after degree n are within absApproxTol from zero, -the coefficients will be considered to have converged at degree n. Defaults to 1e-10.
-relApproxTol (float) – The relative tolerance used to determine at what degree the Chebyshev coefficients have -converged to zero. If all coefficients after degree n are within relApproxTol * supNorm -(the maximum function evaluation on the interval) of zero, the coefficients will be -considered to have converged at degree n. Defaults to 1e-10.
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-- Returns: -
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coefficient_matrix (numpy array) – The coefficient matrix of the Chebyshev approximation.
-error (float) – The error associated with the approximation.
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