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 flags, Operator ... @flags.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(np.ones(5)) array([ 2., 2., 2., 2., 2.]) >>> A(np.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
For non-explicit shape operators, we get the corresponding dense matrix by specifying the input shape:
>>> A.todense(shapein=2) array([[2, 0], [0, 2]])
Operators do not have to be linear. Many operators are already predefined, such as the IdentityOperator, the DiagonalOperator or the nonlinear ClipOperator.
The previous A matrix could be defined more easily like this:
>>> from pyoperators import I >>> A = 2 * 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 operator * stands for matrix multiplication if the two operators are linear, or for element-wise multiplication otherwise:
>>> from pyoperators import I, DiagonalOperator >>> B = 2 * I + DiagonalOperator(range(3)) >>> B.todense() array([[2, 0, 0], [0, 3, 0], [0, 0, 4]])
Algebraic rules can easily be attached to operators. They are used to simplify expressions to speed up their execution. The B Operator has been reduced to:
>>> B DiagonalOperator(array([2, ..., 4], dtype=int64), broadcast='disabled', dtype=int64, shapein=3, shapeout=3)
Many simplifications are available. For instance:
>>> from pyoperators import Operator >>> C = Operator(flags='idempotent,linear') >>> C * C is C True >>> D = 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