A generalized non-square Cholesky decomposition algorithm with applications to finance
- Reiß, Oliver
2010 Mathematics Subject Classification
- 15-04 65F30 91-08
- Modified Cholesky decomposition, LDLT decomposition, Semi--positive matrices, Covariance matrix, Correlation matrix, Value at Risk
In several applications there is the need to compute a Cholesky decomposition of a given symmetric matrix. The usual Cholesky decomposition algorithm will fail if the given matrix is semi-positive, although such a decomposition exists. To overcome this problem there exists a LDLT decomposition for semi-positive matrices. In the case that the given symmetric matrix is not semi-positive, no Cholesky decomposition exists. In such a situation one aims to approximate this matrix by a (semi-)positive one and computes the Cholesky decomposition of the approximation. From the context of numerical optimization there exist algorithms by Gill, Murray and Wright and a refinement by Eskow and Schnabel. Both methods basicly return a Cholesky decomposition of a positive approximation of an indefinite input matrix. In this paper we extend the LDLT algorithm such that it coincides for a semi-definite input with the LDLT decomposition and for indefinite input it gives the decomposition of a semi-positive approximation. In contrast to the algorithms mentioned before, for indefinite input matrices our algorithm gives a decomposition, which has a lower rank. This gives the important opportunity to introduce a dimension reduction, if possible, and we will show that this algorithm can save computation time in several applications in finance, especially for risk management.