Sparse Incomplete Cholesky Decomposition
Implementation of the Incomplete Cholesky Decomposition with few methods.
The project includes a C implementation with a MATLAB MEX wrapper.
The aim is to have 3 variants of the incomplete decomposition:
- Threshold (
IC(\tau))
Using a threshold, $ \tau $ to define which elements will be kept from the decomposition.
It can be using global threshold or by a column.
Implemented - Pattern (
IC(l))
Filling elements which are up tolsteps in the graph of the matrixA. Forl = 0called Zero Fill where filling zeros in elements not defined by the pattern.
Also could be filled by a given pattern of sparsity (So givenAas the pattern it matchesl = 0).
Not Implemented - Number of Non Zero Elements (
IC(p))
Keeps the largestpelements per column.
Not Implemented
Generating MATLAB MEX
- Download the repository.
- Run
MakeMexin MATLAB with pre defined MATLAB MEX Compiler. - Go through the Unit Tests and the Run Time Analysis.
The MEX Wrapper supports only Sparse Real Matrices of Type Double.
Performance
Comparing the performance with MATLAB's functions.
Decomposition
The MEX file and MATLAB's ICT were the most memory efficient.
Pre Conditioning (Solving the Linear System)
To Do
- Move the array sorting related code to a dedicated repository with complete run time analysis.
References
- PyMatting.
TheCcode is basically a redo of the Pre Conditioner in the Python package. - MATLAB
ichol(). - Support Preconditioning Materials and Publications.
- An Incomplete Cholesky Factorization for Dense Symmetric Positive Definite Matrices.
- A Survey of Incomplete Factorization Preconditioners.
- Experimental Study of ILU Preconditioners for Indefinite Matrices.
- A Robust Limited Memory Incomplete Cholesky Factorization.
- Incomplete Cholesky Factorizations with Limited Memory.
- Modified Incomplete Cholesky Factorization for Solving Electromagnetic Scattering Problems.