factorization A = LU

Question

Suppose you have a LU factorization A = LU for a given matrix A . How could this be used to efficiently solve a system of the form A*x = b ?

Answer 1

Assuming A^-1 exists,

 

A*x = b

 

A = LU

 

(LU)*x = b

 

L^-1(LU)*x = L^-1*b

 

(L^-1L)U*x = L^-1*b

 

IU*x = U*x = L^-1*b

 

Similarly, multiply by U^-1 on both sides:

 

x = (U^-1L^-1)*b