Developer Guide and Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

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 LaTex Math image. , the problem is to compute the Cholesky decomposition LaTex Math image. , 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 LaTex Math image. 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 LaTex Math image. 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

C++ (CPU)
Batch Processing:
Java*
There is no support for Java on GPU.
Batch Processing:
Python*
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

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.