Solve System of Linear Equations by LU Decompose
One of the application of LU matrix decomposition is solving system of linear equations.
Ax = b
- A - coefficient matrix
- b - non-homogeneous vector
- x - unknown vector (solution vector)
The square matrix (coefficient matrix) A decomposed into LU (lower triangular and upper triangular) matrix by elementary-row-operations.
A = LU
the solution vector X of system found by forward and back substitution.- y = forward-substitute L and b
- X = back-substitute U and y
The two or more system of equations have same coefficient matrix A but have different non-homogeneous vectors (b1 ,b2,..), the solutions vectors (x1,x2,..) obtained by once applying LU decomposition over matrix A. System 1 Ax = b1 System 2 Ax = b2
- LU = LU-decompose(A)
- System 1
- y = forward-substitute (L , b1)
- x = back-substitute (U,y)
- System 2
- y = forward-substitute (L , b2)
- x= back-substitute (U,y)
Algorithm Steps for Solving System of Linear Equations by LU Decompose
Input : a square matrix, A,
non-homogeneous vector b
Output : Solution vector, X
read matrix A
read vector b
[L,U] = LUDecompose(A)
Y = forward-substitute(L,b)
X = back-substitute(U,Y)
print X, "is solution vector"
Java programming code for Solve System of Linear Equations by LU Decompose
import java.util.Arrays;
public class LUSolution {
public static void main(String[] args) {
double A [][]= {{2,4,6},{4,5,6},{3,1,-2} };
double b[] = {24,18,4};
LUMatrix lu=new LUMatrix(A);
lu.decompose();
Matrix L= lu.getL();
Matrix U = lu.getU();
System.out.println("Lower Triangular Matrix L");
System.out.println(L.toString());
System.out.println("Upper Triangular Matrix U");
System.out.println(U.toString());
System.out.println("verify that Matrix A = LU");
System.out.println(MatrixOpr.multiply(L, U).toString());
System.out.println(" Y = Lb");
double Y[]=Subsitution.forward(L, b);
System.out.println("Y =" + Arrays.toString(Y));
System.out.println(" X = UY");
double X[]=Subsitution.backward(U, Y);
System.out.println("Solution of Matrix A, X ="
+ Arrays.toString(X));
}
}
Java program output of Solve System of Linear Equations by LU Decompose
Matrix A 2 4 6 4 5 6 3 1 -2 b = [24 18 4] Lower Triangular Matrix L 1.0 0.0 0.0 0.5 1.0 0.0 0.75 -1.8333333 1.0 Upper Triangular Matrix U 4.0 5.0 6.0 0.0 1.5 3.0 0.0 0.0 -1.0 verify that Matrix A = LU 4.0 5.0 6.0 2.0 4.0 6.0 3.0 1.0 -2.0 Y = Lb Y =[24.0, 6.0, -3.0] X = UY Solution of Matrix A, X =[4.0, -2.0, 3.0]
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