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+"""MIT License
+
+Copyright (c) 2019 David Luevano Alvarado
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+"""
+import numpy as np
+
+
+def cholesky_solve(K,
+ y):
+ """
+ Applies Cholesky decomposition to solve Ka=y. Where a are the alpha
+ coeficients.
+ K: kernel.
+ y: known parameters.
+ """
+ # Add a small lambda value.
+ K[np.diag_indices_from(K)] += 1e-8
+
+ # Get the Cholesky decomposition of the kernel.
+ L = np.linalg.cholesky(K)
+ size = K.shape[0]
+
+ # Solve Lx=y for x.
+ x = np.zeros(size, dtype=float)
+ x[0] = y[0] / L[0, 0]
+ for i in range(1, size):
+ temp_sum = 0.0
+ for j in range(i):
+ temp_sum += L[i, j] * x[j]
+ x[i] = (y[i] - temp_sum) / L[i, i]
+
+ # Now, solve LTa=x for a.
+ L2 = L.T
+ a = np.zeros(size, dtype=float)
+ a[size - 1] = x[size - 1] / L2[size - 1, size - 1]
+ for i in range(0, size - 1)[::-1]:
+ temp_sum = 0.0
+ for j in range(i, size)[::-1]:
+ temp_sum += L2[i, j] * a[j]
+ a[i] = (x[i] - temp_sum) / L2[i, i]
+
+ return a