## org.apache.commons.math3.linear Class RectangularCholeskyDecomposition

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

`public class RectangularCholeskyDecompositionextends 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 1422313 2012-12-15 18:53:41Z psteitz \$
MathWorld, Wikipedia

Constructor Summary
`RectangularCholeskyDecomposition(RealMatrix matrix)`
Decompose a symmetric positive semidefinite matrix.
```RectangularCholeskyDecomposition(RealMatrix matrix, double small)```
Decompose a symmetric positive semidefinite matrix.

Method Summary
` 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)
throws NonPositiveDefiniteMatrixException```
Decompose a symmetric positive semidefinite matrix.

Note: this constructor follows the linpack method to detect dependent columns by proceeding with the Cholesky algorithm until a nonpositive diagonal element is encountered.

Parameters:
`matrix` - Symmetric positive semidefinite matrix.
Throws:
`NonPositiveDefiniteMatrixException` - if the matrix is not positive semidefinite.
Since:
3.1
Analysis of the Cholesky Decomposition of a Semi-definite Matrix

### 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 columns 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()`

### getRank

`public int getRank()`
Get the rank of the symmetric positive semidefinite matrix. The r is the number of independent rows in the symmetric positive semidefinite matrix, it is also the number of columns of the rectangular matrix of the decomposition.

Returns:
r of the square matrix.
`getRootMatrix()`