Extreme Optimization > User's Guide > Vector and Matrix Library > Matrix Decompositions > Solving Linear Systems and Related Operations

Extreme Optimization User's Guide

User's Guide

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Solving Linear Systems and Related Operations

All decomposition classes inherit from the LinearTransformation abstract base class, which defines common methods for solving systems of simultaneous linear equations and related operations. All these methods call the Decompose method as needed. Unlike for matrix types, there is no performance penalty for calling several of these methods in succession. The decomposition is only calculated once.

The Solve method is overloaded. To solve for one right-hand side, pass this value as a Vector as the first parameter. To solve for multiple right-hand sides, store the vectors inthe columns of a GeneralMatrix. An optional second parameter specifies whether the right-hand sides should be overwritten with the solution. By default, a new vector or matrix instance is created of the same dimensions as the right-hand side.

A matrix is singular when not all of its rows or columns are linearly independent. The IsSingular method returns a Boolean value that indicates this condition.

The inverse matrix is returned by the GetInverse method.

The condition number of a matrix is defined as the ratio of its largest to its smallest singular value. Because the calculation of singular values is a very expensive operation, an estimate that is cheaper to calculate is usually preferred. The EstimateConditionNumber method returns such an estimate.

The condition number gives an indication ofthe worst case loss of precision when solving a system of simultaneous linear equations. The condition number of a singular matrix is infinite, which is returned as Double.PositiveInfinity.