Cholesky Decomposition

Cholesky decomposition is a matrix factorization technique that decomposes a symmetric positive-definite matrix into a product of a lower triangular matrix and its conjugate transpose.

Because of numerical stability and superior efficiency in comparison with other methods, Cholesky decomposition is widely used in numerical methods for solving symmetric linear systems. It is also used in non-linear optimization problems, Monte Carlo simulation, and Kalman filtration.

Details

Given a symmetric positive-definite matrix \(X\) of size \(p \times p\), the problem is to compute the Cholesky decomposition \(X = {LL}^T\), where \(L\) is a lower triangular matrix.

Batch Processing

Algorithm Input

Cholesky decomposition accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Input ID

Input

data

Pointer to the \(p \times p\) numeric table that represents the symmetric positive-definite matrix \(X\) for which the Cholesky decomposition is computed.

The input can be an object of any class derived from NumericTable that can represent symmetric matrices. For example, the PackedTriangularMatrix class cannot represent a symmetric matrix.

Algorithm Parameters

Cholesky decomposition has the following parameters:

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Performance-oriented computation method, the only method supported by the algorithm.

Algorithm Output

Cholesky decomposition calculates the result described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm. For more details, see Algorithms.

Result ID

Result

choleskyFactor

Pointer to the \(p \times p\) numeric table that represents the lower triangular matrix \(L\) (Cholesky factor).

By default, the result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except the PackedSymmetricMatrix class, СSRNumericTable class, and PackedTriangularMatrix class with the upperPackedTriangularMatrix layout.

Examples

Batch Processing:

Note

There is no support for Java on GPU.

Batch Processing:

Batch Processing:

Performance Considerations

To get the best overall performance when Cholesky decomposition:

  • If input data is homogeneous, for input matrix \(X\) and output matrix \(L\) use homogeneous numeric tables of the same type as specified in the algorithmFPType class template parameter.

  • If input data is non-homogeneous, use AOS layout rather than SOA layout.

Optimization Notice

Intel’s compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804