python中的预处理共轭梯度和LinearOperator

Question

[Homework] I am going to solve the linear system Ax=b by the Preconditioned Conjugate Gradient method, and I use spilu function from scipy.sparse.linalg for the preconditioner. A is a sparse symmetric 162*162 matrix. Since the spilu gives an approximation to the inverse of A, say M approximates A, and so spilu(A) gives M^-1, which is the preconditioner. I find that we can directly gives the preconditioner in the python Conjugate Gradient function, but my code below does not work.

M_inverse=scipy.sparse.linalg.spilu(A)
M2=scipy.sparse.linalg.LinearOperator((162,162),M_inverse.solve)
x3=scipy.sparse.linalg.cg(A,b,M2)
TypeError                                 Traceback (most recent call last)
<ipython-input-84-86f8f91df8d2> in <module>()
----> 1 x3=scipy.sparse.linalg.cg(A,b,M2)

/Users/ruobinghan/anaconda/lib/python3.4/site-packages/scipy/sparse/linalg/isolve/iterative.py in cg(A, b, x0, tol, maxiter, xtype, M, callback)

/Users/ruobinghan/anaconda/lib/python3.4/site-packages/scipy/sparse/linalg/isolve/iterative.py in non_reentrant(func, *a, **kw)
     83     try:
     84         d['__entered'] = True
---> 85         return func(*a, **kw)
     86     finally:
     87         d['__entered'] = False

/Users/ruobinghan/anaconda/lib/python3.4/site-packages/scipy/sparse/linalg/isolve/iterative.py in cg(A, b, x0, tol, maxiter, xtype, M, callback)
    219 @non_reentrant
    220 def cg(A, b, x0=None, tol=1e-5, maxiter=None, xtype=None, M=None, callback=None):
--> 221     A,M,x,b,postprocess = make_system(A,M,x0,b,xtype)
    222 
    223     n = len(b)

/Users/ruobinghan/anaconda/lib/python3.4/site-packages/scipy/sparse/linalg/isolve/utils.py in make_system(A, M, x0, b, xtype)
    108         x = zeros(N, dtype=xtype)
    109     else:
--> 110         x = array(x0, dtype=xtype)
    111         if not (x.shape == (N,1) or x.shape == (N,)):
    112             raise ValueError('A and x have incompatible dimensions')

TypeError: float() argument must be a string or a number, not 'LinearOperator' 

Also, the question hints I will need to use LinearOperator interface, I do not understand what is exactly LinearOperator doing and why we need it here.

Any suggestion would be appreciated! Thanks in advance!

Answer 1

I think the parameters are in wrong order,

x3=scipy.sparse.linalg.cg(A,b,M2)

In the Error message:

220 def cg(A, b, x0=None, tol=1e-5, maxiter=None, xtype=None, M=None, 
callback=None):
--> 221     A,M,x,b,postprocess = make_system(A,M,x0,b,xtype)

M2 is in the place of x0 - the initial guess of the solution but not the preconditioner. In my host, with correct order, class- LinearOperator is functioning well.

correct version

x3=scipy.sparse.linalg.cg(A,b,M=M2)

Please use "key word" arguments as often as possible.