Source Code for Module minfx.hessian_mods.cholesky

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 2  #                                                                             # 
 3  # Copyright (C) 2003, 2006, 2008 Edward d'Auvergne                            # 
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 5  # This file is part of the minfx optimisation library.                        # 
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21   
22  # Python module imports. 
23  from numpy import dot, sqrt, trace, transpose 
24  from numpy.linalg import LinAlgError, cholesky, solve 
25   
26   
27 -def cholesky(dfk, d2fk, I, n, print_prefix, print_flag, return_matrix=0): 
28      """Cholesky with added multiple of the identity. 
29   
30      Algorithm 6.3 from page 145 of 'Numerical Optimization' by Jorge Nocedal and Stephen J. 
31      Wright, 1999, 2nd ed. 
32   
33      Returns the modified Newton step. 
34      """ 
35   
36      # Find the minimum diagonal value of the Hessian. 
37      min_aii = 1e99 
38      for i in xrange(n): 
39          min_aii = min(d2fk[i, i], min_aii) 
40   
41      # Calculate the Frobenius norm of the Hessian. 
42      norm = sqrt(trace(dot(d2fk, d2fk))) 
43      half_norm = norm / 2.0 
44   
45      # Choose the initial tk value. 
46      if min_aii > 0.0: 
47          tk = 0.0 
48      else: 
49          tk = half_norm 
50   
51      # Debugging. 
52      if print_flag >= 3: 
53          print print_prefix + "Frobenius norm: " + `norm` 
54          print print_prefix + "min aii: " + `min_aii` 
55          print print_prefix + "tk: " + `tk` 
56   
57      # Loop until the matrix is positive definite. 
58      while 1: 
59          if print_flag >= 3: 
60              print print_prefix + "Iteration" 
61   
62          # Calculate the matrix A + tk.I 
63          matrix = d2fk + tk * I 
64   
65          # Attempt the Cholesky decomposition. 
66          try: 
67              L = cholesky(matrix) 
68              if print_flag >= 3: 
69                  print print_prefix + "\tCholesky matrix L:" 
70                  for i in xrange(n): 
71                      print print_prefix + "\t\t" + `L[i]` 
72          except LinAlgError: 
73              if print_flag >= 3: 
74                  print print_prefix + "\tLinearAlgebraError, matrix is not positive definite." 
75   
76              # Update of tk. 
77              tk = max(2.0*tk, half_norm) 
78          else: 
79              # Success, therefore break out of the while loop. 
80              break 
81   
82      # Calculate the Newton direction. 
83      y = solve(L, dfk) 
84      import sys; sys.exit() 
85      if return_matrix: 
86          return -solve(transpose(L), y), matrix 
87      else: 
88          return -solve(transpose(L), y) 
89