Matrix A = LU Decompose
LU decompose - a square matrix A that can be expressed as product of L and U matrix, is called LU decomposition. A = LU
Matrix L - It is called Lower triangular Matrix, whose elements above the main diagonal is zero valued elements.
Matrix U - It is called Upper Triangular Matrix, whose elements below the main diagonal is zero valued elements.
How to LU decompose - a square matrix A is decomposed or factorized into L and U matrix by elementary row operation, such as swapping two rows and adding or subtracting row by a scalar value.
Application of LU decompose It is useful to find solution of system of linear equation and matrix inverse.
Algorithm steps for A= LU decompose
Read matrix A
U = copy (A)
L = Identity-matrix()
i,j - position of a element in the matrix
For i=1 To N
Eor j=i+1 To N
lambda=Uji/Uii
Rj <- Rj - lambda x Ri
Lji= lamda
End
End
Java programming - Matrix LU Decompose
public class LUMatrix {
Matrix L,U;
int rc[];
public LUMatrix(double A[][]) {
int nrow=A.length;
int ncol=A[0].length;
rc=new int[nrow];
U = new Matrix(nrow,ncol);
for(int r=0;r<nrow;r++) {
for(int c=0;c<ncol;c++)
U.setElement(r, c, A[r][c]);
}
L =new Matrix(nrow,ncol);
L.identity();
}
public void maxdiagonal() {
for(int r=0;r<U.getNrow();r++)
rc[r] = r;
for(int r=0;r<U.getNrow();r++)
{
double mx=0; int mr=r;
for (int c=r;c<U.getNrow();c++){
if ( Math.abs(U.getElement(c, r)) > mx ) {
mx = Math.abs(U.getElement(c, r));
mr = c;
}
}
if ( mr!=r ) {
rc[r] = mr;rc[mr] = r;
RowOperation.swap(U, r, mr);
}
}
}
public void decompose() {
this.maxdiagonal();
for (int row=0 ; row< U.getNrow() ; row++) {
for (int index= row+1 ; index< U.getNrow(); index++) {
if( U.getElement(index, row) != 0 ){
doublelamda=U.getElement(index, row)/U.getElement(row, row);
RowOperation.subtract(U, index, row,lamda);
L.setElement(index, row, lamda);
}
}
}
}
public Matrix getL() {
return L;
}
public Matrix getU(){
return U;
}
public int[] roworder() {
return rc;
}
public static void main(String[] args) {
double A[][] = { {2,1 ,2,3},{1,0 ,1,1},{-2,1,-1,1},{2,1,-1,0} };
LUMatrix de=new LUMatrix(A);
de.decompose();
Matrix L = de.getL();
Matrix U =de.getU();
System.out.println("Upper Triangular matrix");
System.out.println(U.toString());
System.out.println("Lower Triangular matrix");
System.out.println(L.toString());
System.out.println(" Verify that A =L*U");
Matrix LU = MatrixOpr.multiply(L, U);
System.out.println(LU.toString());
}
}
Java program output of Matrix LU Decompose
Matrix A 2.0 1.0 2.0 3.0 -2.0 1.0 -1.0 1.0 1.0 0.0 1.0 1.0 2.0 1.0 -1.0 0.0 Upper Triangular Matrix U 2.0 1.0 2.0 3.0 0.0 2.0 1.0 4.0 0.0 0.0 0.25 0.5 0.0 0.0 0.0 3.0 Lower Triangular Matrix L 1.0 0.0 0.0 0.0 -1.0 1.0 0.0 0.0 0.5 -0.25 1.0 0.0 1.0 0.0 -12.0 1.0 Verify that A =L*U 2.0 1.0 2.0 3.0 -2.0 1.0 -1.0 1.0 1.0 0.0 1.0 1.0 2.0 1.0 -1.0 0.0
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