package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Estimate the Taylor polynomial of f at x by polynomial fitting.

Parameters ---------- f : callable The function whose Taylor polynomial is sought. Should accept a vector of `x` values. x : scalar The point at which the polynomial is to be evaluated. degree : int The degree of the Taylor polynomial scale : scalar The width of the interval to use to evaluate the Taylor polynomial. Function values spread over a range this wide are used to fit the polynomial. Must be chosen carefully. order : int or None, optional The order of the polynomial to be used in the fitting; `f` will be evaluated ``order+1`` times. If None, use `degree`.

Returns ------- p : poly1d instance The Taylor polynomial (translated to the origin, so that for example p(0)=f(x)).

Notes ----- The appropriate choice of 'scale' is a trade-off; too large and the function differs from its Taylor polynomial too much to get a good answer, too small and round-off errors overwhelm the higher-order terms. The algorithm used becomes numerically unstable around order 30 even under ideal circumstances.

Choosing order somewhat larger than degree may improve the higher-order terms.

Examples -------- We can calculate Taylor approximation polynomials of sin function with various degrees:

>>> import matplotlib.pyplot as plt >>> from scipy.interpolate import approximate_taylor_polynomial >>> x = np.linspace(-10.0, 10.0, num=100) >>> plt.plot(x, np.sin(x), label='sin curve') >>> for degree in np.arange(1, 15, step=2): ... sin_taylor = approximate_taylor_polynomial(np.sin, 0, degree, 1, ... order=degree + 2) ... plt.plot(x, sin_taylor(x), label=f'degree=degree') >>> plt.legend(bbox_to_anchor=(1.05, 1), loc='upper left', ... borderaxespad=0.0, shadow=True) >>> plt.tight_layout() >>> plt.axis(-10, 10, -10, 10) >>> plt.show()

Convenience function for polynomial interpolation.

Constructs a polynomial that passes through a given set of points, then evaluates the polynomial. For reasons of numerical stability, this function does not compute the coefficients of the polynomial.

This function uses a 'barycentric interpolation' method that treats the problem as a special case of rational function interpolation. This algorithm is quite stable, numerically, but even in a world of exact computation, unless the `x` coordinates are chosen very carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon.

Parameters ---------- xi : array_like 1-D array of x coordinates of the points the polynomial should pass through yi : array_like The y coordinates of the points the polynomial should pass through. x : scalar or array_like Points to evaluate the interpolator at. axis : int, optional Axis in the yi array corresponding to the x-coordinate values.

Returns ------- y : scalar or array_like Interpolated values. Shape is determined by replacing the interpolation axis in the original array with the shape of x.

See Also -------- BarycentricInterpolator : Bary centric interpolator

Notes ----- Construction of the interpolation weights is a relatively slow process. If you want to call this many times with the same xi (but possibly varying yi or x) you should use the class `BarycentricInterpolator`. This is what this function uses internally.

Examples -------- We can interpolate 2D observed data using barycentric interpolation:

>>> import matplotlib.pyplot as plt >>> from scipy.interpolate import barycentric_interpolate >>> x_observed = np.linspace(0.0, 10.0, 11) >>> y_observed = np.sin(x_observed) >>> x = np.linspace(min(x_observed), max(x_observed), num=100) >>> y = barycentric_interpolate(x_observed, y_observed, x) >>> plt.plot(x_observed, y_observed, 'o', label='observation') >>> plt.plot(x, y, label='barycentric interpolation') >>> plt.legend() >>> plt.show()

The factorial of a number or array of numbers.

The factorial of non-negative integer `n` is the product of all positive integers less than or equal to `n`::

n! = n * (n - 1) * (n - 2) * ... * 1

Parameters ---------- n : int or array_like of ints Input values. If ``n < 0``, the return value is 0. exact : bool, optional If True, calculate the answer exactly using long integer arithmetic. If False, result is approximated in floating point rapidly using the `gamma` function. Default is False.

Returns ------- nf : float or int or ndarray Factorial of `n`, as integer or float depending on `exact`.

Notes ----- For arrays with ``exact=True``, the factorial is computed only once, for the largest input, with each other result computed in the process. The output dtype is increased to ``int64`` or ``object`` if necessary.

With ``exact=False`` the factorial is approximated using the gamma function:

.. math:: n! = \Gamma(n+1)

Examples -------- >>> from scipy.special import factorial >>> arr = np.array(3, 4, 5) >>> factorial(arr, exact=False) array( 6., 24., 120.) >>> factorial(arr, exact=True) array( 6, 24, 120) >>> factorial(5, exact=True) 120

Compute the factorial and return as a float

Returns infinity when result is too large for a double

Convenience function for polynomial interpolation.

See `KroghInterpolator` for more details.

Parameters ---------- xi : array_like Known x-coordinates. yi : array_like Known y-coordinates, of shape ``(xi.size, R)``. Interpreted as vectors of length R, or scalars if R=1. x : array_like Point or points at which to evaluate the derivatives. der : int or list, optional How many derivatives to extract; None for all potentially nonzero derivatives (that is a number equal to the number of points), or a list of derivatives to extract. This number includes the function value as 0th derivative. axis : int, optional Axis in the yi array corresponding to the x-coordinate values.

Returns ------- d : ndarray If the interpolator's values are R-D then the returned array will be the number of derivatives by N by R. If `x` is a scalar, the middle dimension will be dropped; if the `yi` are scalars then the last dimension will be dropped.

See Also -------- KroghInterpolator : Krogh interpolator

Notes ----- Construction of the interpolating polynomial is a relatively expensive process. If you want to evaluate it repeatedly consider using the class KroghInterpolator (which is what this function uses).

Examples -------- We can interpolate 2D observed data using krogh interpolation:

>>> import matplotlib.pyplot as plt >>> from scipy.interpolate import krogh_interpolate >>> x_observed = np.linspace(0.0, 10.0, 11) >>> y_observed = np.sin(x_observed) >>> x = np.linspace(min(x_observed), max(x_observed), num=100) >>> y = krogh_interpolate(x_observed, y_observed, x) >>> plt.plot(x_observed, y_observed, 'o', label='observation') >>> plt.plot(x, y, label='krogh interpolation') >>> plt.legend() >>> plt.show()