org.apache.commons.math3.linear

## Class RectangularCholeskyDecomposition

• java.lang.Object
• org.apache.commons.math3.linear.RectangularCholeskyDecomposition

• ```public class RectangularCholeskyDecomposition
extends Object```
Calculates the rectangular Cholesky decomposition of a matrix.

The rectangular Cholesky decomposition of a real symmetric positive semidefinite matrix A consists of a rectangular matrix B with the same number of rows such that: A is almost equal to BBT, depending on a user-defined tolerance. In a sense, this is the square root of A.

The difference with respect to the regular `CholeskyDecomposition` is that rows/columns may be permuted (hence the rectangular shape instead of the traditional triangular shape) and there is a threshold to ignore small diagonal elements. This is used for example to generate ```correlated random n-dimensions vectors``` in a p-dimension subspace (p < n). In other words, it allows generating random vectors from a covariance matrix that is only positive semidefinite, and not positive definite.

Rectangular Cholesky decomposition is not suited for solving linear systems, so it does not provide any `decomposition solver`.

Since:
2.0 (changed to concrete class in 3.0)
Version:
\$Id: RectangularCholeskyDecomposition.java 1244107 2012-02-14 16:17:55Z erans \$
MathWorld, Wikipedia
• ### Constructor Summary

Constructors
Constructor and Description
```RectangularCholeskyDecomposition(RealMatrix matrix, double small)```
Decompose a symmetric positive semidefinite matrix.
• ### Method Summary

Methods
Modifier and Type Method and Description
`int` `getRank()`
Get the rank of the symmetric positive semidefinite matrix.
`RealMatrix` `getRootMatrix()`
Get the root of the covariance matrix.
• ### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ### Constructor Detail

• #### RectangularCholeskyDecomposition

```public RectangularCholeskyDecomposition(RealMatrix matrix,
double small)
throws NonPositiveDefiniteMatrixException```
Decompose a symmetric positive semidefinite matrix.
Parameters:
`matrix` - Symmetric positive semidefinite matrix.
`small` - Diagonal elements threshold under which column are considered to be dependent on previous ones and are discarded.
Throws:
`NonPositiveDefiniteMatrixException` - if the matrix is not positive semidefinite.
• ### Method Detail

• #### getRootMatrix

`public RealMatrix getRootMatrix()`
Get the root of the covariance matrix. The root is the rectangular matrix `B` such that the covariance matrix is equal to `B.BT`
Returns:
root of the square matrix
`getRank()`
`public int getRank()`
`getRootMatrix()`