com.numericalmethod.suanshu.matrix.doubles.linearsystem

Class LinearSystemSolver

java.lang.Object

com.numericalmethod.suanshu.matrix.doubles.linearsystem.LinearSystemSolver

All Implemented Interfaces:

java.io.Serializable

public class LinearSystemSolver
extends java.lang.Object
implements java.io.Serializable

Solve a system of linear equations in the form:

Ax = b,

We assume that, after row reduction, A has no more rows than columns. That is, the system must not be over-determined. For example, the following system is not over-determined. One of the rows is linearly dependent. \[ \begin{bmatrix} 1 & -1 & 0\\ 0 & -2 & 0\\ 0 & 0 & -1\\ 0 & 0 & -2 \end{bmatrix} x = \begin{bmatrix} -0.8\\ -1.6\\ 0.8\\ 1.6 \end{bmatrix} \] This linear system of equations is solved in two steps.

1. First, solve Ax = 0, the homogeneous system, for non trivial solutions.
2. Then, solve Ax = b for a particular solution.

If A has full rank, this implementation solves the system by LU decomposition. Otherwise, a particular solution is found by x = T * b. The final solution is:

x_particular + {x_null_space_of_A}

hence, the translation of the null space of A by the vector x_particular.

See Also:

• Wikipedia: System of linear equations
• Wikipedia: Nonhomogeneous equations
• Wikipedia: Homogeneous systems

, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	LinearSystemSolver.NoSolution This is the runtime exception thrown when it fails to solve a system of linear equations.
static interface	LinearSystemSolver.Solution This is the solution to a linear system of equations.

Constructor Summary

Constructors

Constructor and Description
LinearSystemSolver(double epsilon) Construct a solver for a linear system of equations.

Method Summary

Modifier and Type	Method and Description
LinearSystemSolver.Solution	solve(Matrix A0) Get a particular solution for the linear system, Ax = b.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

LinearSystemSolver

public LinearSystemSolver(double epsilon)

Construct a solver for a linear system of equations.

Parameters:

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

Method Detail

solve

public LinearSystemSolver.Solution solve(Matrix A0)

Get a particular solution for the linear system, Ax = b.

Parameters:

A0 - a matrix representing a linear system of equations (the homogeneous part)

Returns:

a particular solution