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import numpy as np
import scipy.linalg as la
from qutip.cy.interpolate import (interp, arr_interp,
zinterp, arr_zinterp)
__all__ = ['Cubic_Spline']
[docs]class Cubic_Spline(object):
'''
Calculates coefficients for a cubic spline
interpolation of a given data set.
This function assumes that the data is sampled
uniformly over a given interval.
Parameters
----------
a : float
Lower bound of the interval.
b : float
Upper bound of the interval.
y : ndarray
Function values at interval points.
alpha : float
Second-order derivative at a. Default is 0.
beta : float
Second-order derivative at b. Default is 0.
Attributes
----------
a : float
Lower bound of the interval.
b : float
Upper bound of the interval.
coeffs : ndarray
Array of coeffcients defining cubic spline.
Notes
-----
This object can be called like a normal function with a
single or array of input points at which to evaluate
the interplating function.
Habermann & Kindermann, "Multidimensional Spline Interpolation:
Theory and Applications", Comput Econ 30, 153 (2007).
'''
def __init__(self, a, b, y, alpha=0, beta=0):
y = np.asarray(y)
n = y.shape[0] - 1
h = (b - a)/n
coeff = np.zeros(n + 3, dtype=y.dtype)
# Solutions to boundary coeffcients of spline
coeff[1] = 1/6. * (y[0] - (alpha * h**2)/6) #C2 in paper
coeff[n + 1] = 1/6. * (y[n] - (beta * h**2)/6) #cn+2 in paper
# Compressed tridiagonal matrix
ab = np.ones((3, n - 1), dtype=float)
ab[0,0] = 0 # Because top row is upper diag with one less elem
ab[1, :] = 4
ab[-1,-1] = 0 # Because bottom row is lower diag with one less elem
B = y[1:-1].copy() #grabs elements y[1] - > y[n-2] for reduced array
B[0] -= coeff[1]
B[-1] -= coeff[n + 1]
coeff[2:-2] = la.solve_banded((1, 1), ab, B, overwrite_ab=True,
overwrite_b=True, check_finite=False)
coeff[0] = alpha * h**2/6. + 2 * coeff[1] - coeff[2]
coeff[-1] = beta * h**2/6. + 2 * coeff[-2] - coeff[-3]
self.a = a # Lower-bound of domain
self.b = b # Uppser-bound of domain
self.coeffs = coeff # Spline coefficients
self.is_complex = (y.dtype == complex) #Tells which dtype solver to use
def __call__(self, pnts, *args):
#If requesting a single return value
if isinstance(pnts, (int, float, complex)):
if self.is_complex:
return zinterp(pnts, self.a,
self.b, self.coeffs)
else:
return interp(pnts, self.a, self.b, self.coeffs)
#If requesting multiple return values from array_like
elif isinstance(pnts, (np.ndarray,list)):
pnts = np.asarray(pnts)
if self.is_complex:
return arr_zinterp(pnts, self.a,
self.b, self.coeffs)
else:
return arr_interp(pnts, self.a, self.b, self.coeffs)