LU Class

LU factorization of a matrix of type double.

For a list of all members of this type, see LU Members.

public class LU

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

Remarks

LU performs an LU factorization of a real general coefficient matrix. The Condition method estimates the reciprocal of the condition number of the matrix. The LU factorization is done using scaled partial pivoting. Scaled partial pivoting differs from partial pivoting in that the pivoting strategy is the same as if each row were scaled to have the same infinity norm.

The condition number of the matrix A is defined to be $\kappa(A)=||A||_1 ||A^{-1}||_1$ . Since it is expensive to compute $||A^{-1}||_1$ , the condition number is only estimated. The estimation algorithm is the same as used by LINPACK and is described in a paper by Cline et al. (1979).

Note that A is not retained for use by other methods of this class, only the factorization of A is retained. Thus, A is a required parameter to the condition method.

An estimated condition number greater than $1/\epsilon$ (where $\epsilon$ is machine precision) indicates that very small changes in A can cause very large changes in the solution x. Iterative refinement can sometimes find the solution to such a system. If there is conern about the input matrix being ill-conditioned, the user of this class should check the condition number of the input matrix using the condition method before using one of the other class methods.

LU fails if U, the upper triangular part of the factorization, has a zero diagonal element. This can occur only if A either is singular or is very close to a singular matrix.

Use the Solve method to solve systems of equations. The Determinant method can be called to compute the determinant of the coefficient matrix.

LU is based on the LINPACK routine SGECO; see Dongarra et al. (1979). SGECO uses unscaled partial pivoting.

Requirements

Namespace: Imsl.Math

Assembly: ImslCS (in ImslCS.dll)

LU Class

Thread Safety

Remarks

Requirements

See Also