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

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

public class BackwardSubstitution
extends java.lang.Object

Backward substitution solves a matrix equation in the form

Ux = b

by an iterative process for an upper triangular matrix U.

The process is so called because for an upper triangular matrix, one first computes x_n, then substitutes that backward into the next equation to solve for x_n-1, and repeats onto x₁.

Note that some diagonal entries in U can be 0s, provided that the system of equations is consistent. E.g.,

 | 1 2 3 |     | 10 |   | 10 |
 | 0 0 5 | %*% |  0 | = |  0 |
 | 0 0 0 |     |  0 |   |  0 |
 

See Also:

Wikipedia: Forward and Back Substitution

Field Summary

UpperTriangularMatrix

U the system of linear equations (the homogeneous part)

Constructor Summary

BackwardSubstitution(UpperTriangularMatrix U)
Construct a BackwardSubstitution instance to solve for different Vector b's.

Method Summary

Vector

solve(Vector b) Solve Ux = b

Methods inherited from class java.lang.Object

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

Field Detail

U

public final UpperTriangularMatrix U

the system of linear equations (the homogeneous part)

Constructor Detail

BackwardSubstitution

public BackwardSubstitution(UpperTriangularMatrix U)

Construct a BackwardSubstitution instance to solve for different Vector b's.

Parameters:

U - an upper triangular matrix

Method Detail

solve

public Vector solve(Vector b)

Solve

Ux = b

Parameters:

b - a vector

Returns:

x such that

Ux = b

Throws:

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