Altaxo.Calc.LinearAlgebra.Factorization Namespace

A class which encapsulates the functionality of a Cholesky factorization.

For a symmetric, positive definite matrix A, the Cholesky factorization is an lower triangular matrix L so that A = L*L'.

A class which encapsulates the functionality of the QR decomposition Modified Gram-Schmidt Orthogonalization.

Any real square matrix A may be decomposed as A = QR where Q is an orthogonal mxn matrix and R is an nxn upper triangular matrix.

A class which encapsulates the functionality of an LU factorization.

For a matrix A, the LU factorization is a pair of lower triangular matrix L and upper triangular matrix U so that A = L*U.

In the Math.NET implementation we also store a set of pivot elements for increased numerical stability. The pivot elements encode a permutation matrix P such that P*A = L*U.

A class which encapsulates the functionality of the QR decomposition.

Any real square matrix A (m x n) may be decomposed as A = QR where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning QTQ = I) and R is an upper triangular matrix (also called right triangular matrix).

A class which encapsulates the functionality of the singular value decomposition (SVD).

Suppose M is an m-by-n matrix whose entries are real numbers. Then there exists a factorization of the form M = UΣVT where: - U is an m-by-m unitary matrix; - Σ is m-by-n diagonal matrix with nonnegative real numbers on the diagonal; - VT denotes transpose of V, an n-by-n unitary matrix; Such a factorization is called a singular-value decomposition of M. A common convention is to order the diagonal entries Σ(i,i) in descending order. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not). The diagonal entries of Σ are known as the singular values of M.