# Cholesky Factorization (G Dataflow)

Performs Cholesky factorization on a symmetric or Hermitian positive definite matrix.

## matrix A

A symmetric or Hermitian positive definite matrix.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.

If the input matrix is not symmetric or Hermitian, this node uses only the upper triangular portion of the input matrix. If the input matrix is not positive definite, this node returns an error.

Default: Empty array

## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error

## cholesky

The factored, upper triangular matrix.

## error out

Error information. The node produces this output according to standard error behavior.

## Algorithm for Performing Cholesky Factorization

The following equations show the factorization of the input matrix A for real cases and complex cases, respectively:

A = RTR

A = RHR

where A is the input matrix A, R is an upper triangular matrix, and all the diagonal elements of R are positive.

The Cholesky factorization exists only if the input matrix A is positive definite and either symmetric or Hermitian. If the input matrix A is not symmetric or Hermitian, this node uses only the upper triangular portion of the input matrix A. If the input matrix A is not positive definite, this node returns an error.

You can use Cholesky factorization to solve linear equations. For example, to solve the linear equation Ax = b, where A is a positive symmetric matrix and A = RTR, first derive the following equations: Rx = h and h = R-Tb. Then use the triangular property of R to solve the equations.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported