LU Decomposition

Like Gauss elimination, LU decomposition method is a kind of exact solution of system of linear algebraic equations. This method attempts to decompose coefficient matrix into two lower and upper triangular matrices. Assume that the main system of algebraic equation can be written as follows:

[A]{X} = {B}

After LU decomposition, the system of algebraic equations is transformed to the following form:

[L][U]{X} = [L]{D}


[L][U] = [A] ;         [L]{D}={B}


It can be seen that [L] is a lower triangular matrix and [U] is an upper triangular matrix. After decomposition [L] can be eliminated form both side of equation then the main equation can be written as follows:

[U]{X} = {D}

Because [U] is an upper triangular matrix, the solution can easily be found by back substitution as follows:


Multiplying [L]’s rows by [U]’s columns, [L] and [U] matrices can be determined from following relations:


System of Linear Equations                                   
         (Source Code in C++)                                                     

o        Gauss Elimination Method                                                                                

o        Gauss-Seidel Method                                                                                           

o        Jacobi Method                                                                

o        LU Decomposition Method                                                                               

o        Relaxation Methods