Example 3.  Solve the linear system  AX = B  by finding the  A = LU  factorization with the Doolittle method, where:
        ,  and  .  
Then solve the lower-triangular system  LY = B  for  Y,  then solve the upper-triangular system  UX = Y  for  X.
Use the forward substitution and back substitution subroutines.  

Solution 3.

Enter the matrix and vector and invoke the subroutine Doolittle.  

First, solve the lower-triangular system    LY = B  for  Y.

Verify that  LY = B.

Second, solve the upper-triangular system    UX = Y  for  X.

Verify that  UX = Y.  

Therefore X is the solution to  LUX = B. and hence AX = B
And we can verify that it is the solution.

(c) John H. Mathews 2003