Mostrando entradas con la etiqueta 6.1 Cholesky Method. Mostrar todas las entradas
Mostrando entradas con la etiqueta 6.1 Cholesky Method. Mostrar todas las entradas

27 julio 2010

Cholesky Method

Cholesky Method 

In linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a symmetric, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices and is an example of a square root of a matrix. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

U=Lt

AX=b

LLtX=b 


 L_{i,j} = \frac{1}{L_{j,j}} \left( A_{i,j} - 
\sum_{k=1}^{j-1} L_{i,k} L_{j,k}^* \right), \qquad\mbox{for } i>j.
 L_{i,i} = \sqrt{ A_{i,i} - \sum_{k=1}^{i-1} 
L_{i,k}L_{i,k}^* }.