Operators and solvers for high-performance computing.
The PyOperators package defines operators and solvers for high-performance computing. These operators are multi-dimensional functions with optimised and controlled memory management. If linear, they behave like matrices with a sparse storage footprint.
To define an operator, one needs to define a direct function which will replace the usual matrix-vector operation:
>>> def f(x, out): ... out[...] = 2 * x
Then, you can instantiate an Operator:
>>> A = pyoperators.Operator(direct=f, flags='symmetric')
An alternative way to define an operator is to define a subclass:
>>> from pyoperators import decorators, Operator ... @decorators.symmetric ... class MyOperator(Operator): ... def direct(x, out): ... out[...] = 2 * x ... ... A = MyOperator()
This operator does not have an explicit shape, it can handle inputs of any shape:
>>> A(ones(5)) array([ 2., 2., 2., 2., 2.]) >>> A(ones((2,3))) array([[ 2., 2., 2.], [ 2., 2., 2.]])
By setting the ‘symmetric’ flag, we ensure that A’s transpose is A:
>>> A.T is A True
To output a corresponding dense matrix, one needs to specify the input shape:
>>> A.todense(shapein=2) array([[2, 0], [0, 2]])
Operators do not have to be linear, but if they are not, they cannot be seen as matrices. Some operators are already predefined, such as the IdentityOperator, the DiagonalOperator or the nonlinear ClippingOperator.
The previous A matrix could be defined more easily like this :
>>> A = 2 * pyoperators.I
where I is the identity operator with no explicit shape.
Operators can be combined together by addition, element-wise multiplication or composition (note that the * sign stands for composition):
>>> B = 2 * pyoperators.I + pyoperators.DiagonalOperator(range(3)) >>> B.todense() array([[2, 0, 0], [0, 3, 0], [0, 0, 4]])
Algebraic rules are used to simplify an expression involving operators, so to speed up its execution:
>>> B DiagonalOperator(array([2, ..., 4], dtype=int64), broadcast='disabled', dtype=int64, shapein=3, shapeout=3) >>> C = pyoperators.Operator(flags='idempotent') >>> C * C is C True >>> D = pyoperators.Operator(flags='involutary') >>> D * D IdentityOperator()
List of requirements:
- python 2.6
- numpy >= 1.6
- scipy >= 0.9
- numexpr (>= 2.0 is better)
- PyWavelets : wavelet transforms