The Cholesky Decomposition

The Cholesky decomposition or Cholesky factorization of a matrix is defined only for positive-definite symmetric or Hermitian matrices. It expresses a matrix as the product of a lower triangular matrix and its transpose.

The Cholesky decomposition algorithm exploits the special structure of symmetric matrices. As a result, it is about twice as fast as the Cholesky decomposition. Moreover, the decomposition is fairly stable, so it is suitable for use in a broader range of problems.

Working with Cholesky decompositions

The Cholesky decomposition of a matrix is implemented by the CholeskyDecomposition<T> class. It has no constructors. Instead, it is created by calling the GetCholeskyDecomposition method on the matrix. This method has two overloads. The first overload has no arguments. The second overload takes a Boolean value that specifies whether the contents of the matrix may be overwritten by the Cholesky decomposition. The default is false.

var A = Matrix.CreateSymmetric(4, new double[] {
    4.16,-3.12, 0.56,-0.10,
    0   , 5.03,-0.83, 1.18,
    0   , 0   , 0.76, 0.34,
    0   , 0   , 0   , 1.18
}, MatrixTriangle.Upper, MatrixElementOrder.RowMajor);
var c = A.GetCholeskyDecomposition();

The Decompose method performs the actual decomposition. This method copies the matrix if necessary. It then calls the appropriate LAPACK routine to perform the actual decomposition. This method is called by other methods as needed. You will rarely need to call it explicitly.

If the matrix is not positive definite a MatrixNotPositiveDefiniteException is thrown. The TryDecompose method performs the same operation, but never throws an exception. Instead, it returns true if the decomposition succeeded, and false otherwise.

Once the decomposition is computed, a number of operations can be performed in much less time. You can repeatedly solve a system of linear equations with different right-hand sides. You can also calculate the determinant and the inverse of the matrix:

var b = Matrix.Create(4, 2, new double[] {
    8.70,-13.35, 1.89,-4.14,
    8.30,  2.13, 1.61, 5.00
}, MatrixElementOrder.ColumnMajor);
var x = c.Solve(b);
var invA = c.GetInverse();
var detA = c.GetDeterminant();

The LowerTriangularFactor property returns a TriangularMatrix<T> containing the lower triangular matrix, G, of the decomposition.

var L = c.LowerTriangularFactor;
var LT = c.UpperTriangularFactor;

Band Cholesky Decomposition

The Cholesky decomposition of a symmetric band matrix is made up of a lower band matrix with lower bandwidth the same as the original matrix, and its transpose. The band structure allows for significant savings in computation time over a full Cholesky decomposition. No specific action is needed to take advantage of this: the most suitable form of the decomposition is chosen automatically.