edu.princeton.cs.algs4.GaussianElimination Maven / Gradle / Ivy
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/******************************************************************************
* Compilation: javac GaussianElimination.java
* Execution: java GaussianElimination m n
* Dependencies: StdOut.java
*
* Gaussian elimination with partial pivoting for m-by-n system.
*
* % java GaussianElimination m n
* -1.000000
* 2.000000
* 2.000000
*
* 3.000000
* -1.000000
* -2.000000
*
* System is infeasible
*
* -6.250000
* -4.500000
* 0.000000
* 0.000000
* 1.000000
*
* System is infeasible
*
* -1.375000
* 1.625000
* 0.000000
*
******************************************************************************/
package edu.princeton.cs.algs4;
/**
* The {@code GaussianElimination} data type provides methods
* to solve a linear system of equations Ax = b,
* where A is an m-by-n matrix
* and b is a length n vector.
*
* This is a bare-bones implementation that uses Gaussian elimination
* with partial pivoting.
* See GaussianEliminationLite.java
* for a stripped-down version that assumes the matrix A is square
* and nonsingular. See {@link GaussJordanElimination} for an alternate
* implementation that uses Gauss-Jordan elimination.
* For an industrial-strength numerical linear algebra library,
* see JAMA.
*
* For additional documentation, see
* Section 9.9
* Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class GaussianElimination {
private static final double EPSILON = 1e-8;
private final int m; // number of rows
private final int n; // number of columns
private double[][] a; // m-by-(n+1) augmented matrix
/**
* Solves the linear system of equations Ax = b,
* where A is an m-by-n matrix and b
* is a length m vector.
*
* @param A the m-by-n constraint matrix
* @param b the length m right-hand-side vector
* @throws IllegalArgumentException if the dimensions disagree, i.e.,
* the length of {@code b} does not equal {@code m}
*/
public GaussianElimination(double[][] A, double[] b) {
m = A.length;
n = A[0].length;
if (b.length != m) throw new IllegalArgumentException("Dimensions disagree");
// build augmented matrix
a = new double[m][n+1];
for (int i = 0; i < m; i++)
for (int j = 0; j < n; j++)
a[i][j] = A[i][j];
for (int i = 0; i < m; i++)
a[i][n] = b[i];
forwardElimination();
assert certifySolution(A, b);
}
// forward elimination
private void forwardElimination() {
for (int p = 0; p < Math.min(m, n); p++) {
// find pivot row using partial pivoting
int max = p;
for (int i = p+1; i < m; i++) {
if (Math.abs(a[i][p]) > Math.abs(a[max][p])) {
max = i;
}
}
// swap
swap(p, max);
// singular or nearly singular
if (Math.abs(a[p][p]) <= EPSILON) {
continue;
}
// pivot
pivot(p);
}
}
// swap row1 and row2
private void swap(int row1, int row2) {
double[] temp = a[row1];
a[row1] = a[row2];
a[row2] = temp;
}
// pivot on a[p][p]
private void pivot(int p) {
for (int i = p+1; i < m; i++) {
double alpha = a[i][p] / a[p][p];
for (int j = p; j <= n; j++) {
a[i][j] -= alpha * a[p][j];
}
}
}
/**
* Returns a solution to the linear system of equations Ax = b.
*
* @return a solution x to the linear system of equations
* Ax = b; {@code null} if no such solution
*/
public double[] primal() {
// back substitution
double[] x = new double[n];
for (int i = Math.min(n-1, m-1); i >= 0; i--) {
double sum = 0.0;
for (int j = i+1; j < n; j++) {
sum += a[i][j] * x[j];
}
if (Math.abs(a[i][i]) > EPSILON)
x[i] = (a[i][n] - sum) / a[i][i];
else if (Math.abs(a[i][n] - sum) > EPSILON)
return null;
}
// redundant rows
for (int i = n; i < m; i++) {
double sum = 0.0;
for (int j = 0; j < n; j++) {
sum += a[i][j] * x[j];
}
if (Math.abs(a[i][n] - sum) > EPSILON)
return null;
}
return x;
}
/**
* Returns true if there exists a solution to the linear system of
* equations Ax = b.
*
* @return {@code true} if there exists a solution to the linear system
* of equations Ax = b; {@code false} otherwise
*/
public boolean isFeasible() {
return primal() != null;
}
// check that Ax = b
private boolean certifySolution(double[][] A, double[] b) {
if (!isFeasible()) return true;
double[] x = primal();
for (int i = 0; i < m; i++) {
double sum = 0.0;
for (int j = 0; j < n; j++) {
sum += A[i][j] * x[j];
}
if (Math.abs(sum - b[i]) > EPSILON) {
StdOut.println("not feasible");
StdOut.println("b[" + i + "] = " + b[i] + ", sum = " + sum);
return false;
}
}
return true;
}
/**
* Unit tests the {@code GaussianElimination} data type.
*/
private static void test(String name, double[][] A, double[] b) {
StdOut.println("----------------------------------------------------");
StdOut.println(name);
StdOut.println("----------------------------------------------------");
GaussianElimination gaussian = new GaussianElimination(A, b);
double[] x = gaussian.primal();
if (gaussian.isFeasible()) {
for (int i = 0; i < x.length; i++) {
StdOut.printf("%.6f\n", x[i]);
}
}
else {
StdOut.println("System is infeasible");
}
StdOut.println();
StdOut.println();
}
// 3-by-3 nonsingular system
private static void test1() {
double[][] A = {
{ 0, 1, 1 },
{ 2, 4, -2 },
{ 0, 3, 15 }
};
double[] b = { 4, 2, 36 };
test("test 1 (3-by-3 system, nonsingular)", A, b);
}
// 3-by-3 nonsingular system
private static void test2() {
double[][] A = {
{ 1, -3, 1 },
{ 2, -8, 8 },
{ -6, 3, -15 }
};
double[] b = { 4, -2, 9 };
test("test 2 (3-by-3 system, nonsingular)", A, b);
}
// 5-by-5 singular: no solutions
private static void test3() {
double[][] A = {
{ 2, -3, -1, 2, 3 },
{ 4, -4, -1, 4, 11 },
{ 2, -5, -2, 2, -1 },
{ 0, 2, 1, 0, 4 },
{ -4, 6, 0, 0, 7 },
};
double[] b = { 4, 4, 9, -6, 5 };
test("test 3 (5-by-5 system, no solutions)", A, b);
}
// 5-by-5 singular: infinitely many solutions
private static void test4() {
double[][] A = {
{ 2, -3, -1, 2, 3 },
{ 4, -4, -1, 4, 11 },
{ 2, -5, -2, 2, -1 },
{ 0, 2, 1, 0, 4 },
{ -4, 6, 0, 0, 7 },
};
double[] b = { 4, 4, 9, -5, 5 };
test("test 4 (5-by-5 system, infinitely many solutions)", A, b);
}
// 3-by-3 singular: no solutions
private static void test5() {
double[][] A = {
{ 2, -1, 1 },
{ 3, 2, -4 },
{ -6, 3, -3 },
};
double[] b = { 1, 4, 2 };
test("test 5 (3-by-3 system, no solutions)", A, b);
}
// 3-by-3 singular: infinitely many solutions
private static void test6() {
double[][] A = {
{ 1, -1, 2 },
{ 4, 4, -2 },
{ -2, 2, -4 },
};
double[] b = { -3, 1, 6 };
test("test 6 (3-by-3 system, infinitely many solutions)", A, b);
}
// 4-by-3 full rank and feasible system
private static void test7() {
double[][] A = {
{ 0, 1, 1 },
{ 2, 4, -2 },
{ 0, 3, 15 },
{ 2, 8, 14 }
};
double[] b = { 4, 2, 36, 42 };
test("test 7 (4-by-3 system, full rank)", A, b);
}
// 4-by-3 full rank and infeasible system
private static void test8() {
double[][] A = {
{ 0, 1, 1 },
{ 2, 4, -2 },
{ 0, 3, 15 },
{ 2, 8, 14 }
};
double[] b = { 4, 2, 36, 40 };
test("test 8 (4-by-3 system, no solution)", A, b);
}
// 3-by-4 full rank system
private static void test9() {
double[][] A = {
{ 1, -3, 1, 1 },
{ 2, -8, 8, 2 },
{ -6, 3, -15, 3 }
};
double[] b = { 4, -2, 9 };
test("test 9 (3-by-4 system, full rank)", A, b);
}
/**
* Unit tests the {@code GaussianElimination} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
test1();
test2();
test3();
test4();
test5();
test6();
test7();
test8();
test9();
// n-by-n random system
int n = Integer.parseInt(args[0]);
double[][] A = new double[n][n];
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
A[i][j] = StdRandom.uniform(1000);
double[] b = new double[n];
for (int i = 0; i < n; i++)
b[i] = StdRandom.uniform(1000);
test(n + "-by-" + n + " (probably nonsingular)", A, b);
}
}
/******************************************************************************
* Copyright 2002-2018, Robert Sedgewick and Kevin Wayne.
*
* This file is part of algs4.jar, which accompanies the textbook
*
* Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
* Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
* http://algs4.cs.princeton.edu
*
*
* algs4.jar is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* algs4.jar is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with algs4.jar. If not, see http://www.gnu.org/licenses.
******************************************************************************/