com.numericalmethod.suanshu.matrix.doubles.linearsystem
Class Solver

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.linearsystem.Solver

public class Solver
extends java.lang.Object

Solve a system of linear equations in the form:

Ax = b,

where A has #rows <= #columns. That is, the system must not be over-determined.

This linear system is solved in two steps.

1. We first solve Ax = 0 the homogeneous system for non trivial solutions.
2. We then solve Ax = b for a particular solution.

If A is square and det(A) != 0, we use the LU decomposition.

Otherwise, we find the particular solution 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

Nested Class Summary

static class

Solver.NoSolution RuntimeException thrown when it fails to solve a system of linear equations.

Field Summary

Matrix	A matrix A as in Ax = b
VectorList	basis the basis of A; solution for the homogeneous part, Ax = 0
double	epsilon a precision parameter: when a number |x| ≤ ε, it is considered 0
boolean	isFullRank true iff A has full rank

Constructor Summary

Solver(Matrix A)
Construct a Solver instance to solve for different Vector b.

Solver(Matrix A, double epsilon)
Construct a Solver instance to solve for different Vector b.

Method Summary

VectorList	homogeneousSoln() Get the basis of the homogeneous solution for the linear system, Ax = b.
Vector	solve(Vector b) 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

Field Detail

A

public final Matrix A

matrix A as in Ax = b

isFullRank

public final boolean isFullRank

true iff A has full rank

basis

public final VectorList basis

the basis of A; solution for the homogeneous part, Ax = 0

epsilon

public final double epsilon

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

Constructor Detail

Solver

public Solver(Matrix A,
              double epsilon)

Construct a Solver instance to solve for different Vector b.

Parameters:

A - a matrix

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

Solver

public Solver(Matrix A)

Construct a Solver instance to solve for different Vector b.

Parameters:

A - a matrix

Method Detail

solve

public Vector solve(Vector b)

Get a particular solution for the linear system,

Ax = b

Parameters:

b - a vector

Returns:

a particular solution

Throws:

Solver.NoSolution - if there is no solution to this system

homogeneousSoln

public VectorList homogeneousSoln()

Get the basis of the homogeneous solution for the linear system, Ax = b. That is, the solutions for

Ax = 0

Returns:

the homogeneous solution