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Iterative Solvers

# Preconditioners¶

## The precon Module¶

The precon module provides preconditioners, which can be used e.g.for the iterative methods implemented in the in the itsolvers module or the JDSYM eigensolver (in the jdsym module).

In the Pysparse framework, any Python object that has the following properties can be used as a preconditioner:

• a shape attribute, which returns a 2-tuple describing the dimension of the preconditioner,
• a precon method, that accepts two vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, and applies the preconditioner to $$\mathbf{x}$$ and stores the result in $$\mathbf{y}$$. Both $$\mathbf{x}$$ and $$\mathbf{y}$$ are double precision, rank-1 NumPy arrays of appropriate size.

The precon module implements two new object types jacobi and ssor, representing Jacobi and SSOR preconditioners.

### precon Module Functions¶

precon.jacobi(A, omega=1.0, steps=1)

Creates a jacobi object, representing the Jacobi preconditioner. The parameter $$\mathbf{A}$$ is the system matrix used for the Jacobi iteration. The matrix needs to be subscriptable using two-dimensional indices, so e.g.an ll_mat object would work. The optional parameter $$\omega$$, which defaults to 1.0, is the weight parameter. The optional steps parameter (defaults to 1) specifies the number of iteration steps.

precon.ssor(A, omega=1.0, steps=1)

Creates a ssor object, representing the SSOR preconditioner. The parameter $$\mathbf{A}$$ is the system matrix used for the SSOR iteration. The matrix $$\mathbf{A}$$ has to be an object of type sss_mat. The optional parameter $$\omega$$, which defaults to 1.0, is the relaxation parameter. The optional steps parameter (defaults to 1) specifies the number of iteration steps.

### jacobi and ssor Objects¶

Both jacobi and ssor objects provide the shape attribute and the precon method, that every preconditioner object in the PySparse framework must implement. Apart from that, there is nothing noteworthy to say about these objects.

### Example: Diagonal Preconditioner¶

The diagonal preconditioner is just a special case of the Jacobi preconditioner, with omega=1.0 and steps=1, which happen to be the default values of these parameters.

It is however easy to implement the diagonal preconditioner using a Python class:

class diag_prec:
def __init__(self, A):
self.shape = A.shape
n = self.shape[0]
self.dinv = numpy.empty(n)
for i in xrange(n):
self.dinv[i] = 1.0 / A[i,i]
def precon(self, x, y):
numpy.multiply(x, self.dinv, y)


So:

>>> D1 = precon.jacobi(A, 1.0, 1)


and:

>>> D2 = diag_prec(A)


yield functionally equivalent preconditioners. D1 is probably faster than D2, because it is fully implemented in C.