some new features

.venv/lib/python3.12/site-packages/scipy/interpolate/_pade.py

@@ -0,0 +1,67 @@
+from numpy import zeros, asarray, eye, poly1d, hstack, r_
+from scipy import linalg
+
+__all__ = ["pade"]
+
+def pade(an, m, n=None):
+    """
+    Return Pade approximation to a polynomial as the ratio of two polynomials.
+
+    Parameters
+    ----------
+    an : (N,) array_like
+        Taylor series coefficients.
+    m : int
+        The order of the returned approximating polynomial `q`.
+    n : int, optional
+        The order of the returned approximating polynomial `p`. By default,
+        the order is ``len(an)-1-m``.
+
+    Returns
+    -------
+    p, q : Polynomial class
+        The Pade approximation of the polynomial defined by `an` is
+        ``p(x)/q(x)``.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.interpolate import pade
+    >>> e_exp = [1.0, 1.0, 1.0/2.0, 1.0/6.0, 1.0/24.0, 1.0/120.0]
+    >>> p, q = pade(e_exp, 2)
+
+    >>> e_exp.reverse()
+    >>> e_poly = np.poly1d(e_exp)
+
+    Compare ``e_poly(x)`` and the Pade approximation ``p(x)/q(x)``
+
+    >>> e_poly(1)
+    2.7166666666666668
+
+    >>> p(1)/q(1)
+    2.7179487179487181
+
+    """
+    an = asarray(an)
+    if n is None:
+        n = len(an) - 1 - m
+        if n < 0:
+            raise ValueError("Order of q <m> must be smaller than len(an)-1.")
+    if n < 0:
+        raise ValueError("Order of p <n> must be greater than 0.")
+    N = m + n
+    if N > len(an)-1:
+        raise ValueError("Order of q+p <m+n> must be smaller than len(an).")
+    an = an[:N+1]
+    Akj = eye(N+1, n+1, dtype=an.dtype)
+    Bkj = zeros((N+1, m), dtype=an.dtype)
+    for row in range(1, m+1):
+        Bkj[row,:row] = -(an[:row])[::-1]
+    for row in range(m+1, N+1):
+        Bkj[row,:] = -(an[row-m:row])[::-1]
+    C = hstack((Akj, Bkj))
+    pq = linalg.solve(C, an)
+    p = pq[:n+1]
+    q = r_[1.0, pq[n+1:]]
+    return poly1d(p[::-1]), poly1d(q[::-1])
+