Suppose a numerical algorithm requires the solution of a 56

Suppose a numerical algorithm requires the solution of a 56 times 56 linear system of equations 236 times, where the only thing changing each time is the right hand side vector. Effective use of LU factorization can reduce the computational cost of this step by approximately percent compared to using MATLAB left division alone. (Enter your answer in percent to two decimal places.)

Solution

LU factorization involves tw steps :

favtorization to decompose the A matrix into a product of lower traingular matrix L and an upper triangular matrix U. L has 1 for each entry on the diagonal

Substituation to solve for X

gauss elimination can be implemented using LU factorization

A X = b can be rewritten as L.U X = b using LU factorization

The LU factorization alogorithm requiresthe same total flops as for Gauss elimination.

the main advantage is oncce A isdecomposed the same L and U can be used for multiple vecors.

MaTLAB lu function can be used to generate the L and U matrices.

MATLAB is left division operator examines the system to see which method will most effeciently solve the problem. this includes trying banded solvers, back nd forward substituions cholesky factorization for symmentric systems.