Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 42

Another way to gauge the accuracy of the solutionistocompute Ax and compare
the result to b :

>> A*x
ans =
-0.00000000000091
0.99999999999909
-0.00000000000819
0.99999999998272
-0.00000000005366
0.99999999994998

The result seemstoconfirm our previousconclusion.

2.3 LU Decomposition Methods

Introduction

It is possible to show that any square matrix A can beexpressedas aproduct of a lower
triangular matrix L and an upper triangular matrix U :

A = LU (2.11)

The process of computing L and U for a given A isknown as LU decomposition or
LU factorization .LU decompositionis not unique (the combinationsof L and U for
aprescribed A areendless), unless certain constraints are placed on L or U . These
constraints distinguish onetypeofdecomposition fromanother. Three commonly
useddecompositions arelistedinTable 2.2.

Name Constraints

Doolittle's decomposition L ii = 1, i = 1 , 2 ,..., n

Crout's decomposition U ii = 1, i = 1 , 2 ,..., n

U T

Choleski's decomposition L =

Table 2.2

=

Afterdecomposing A , it iseasy to solve the equations Ax b , as pointed out in

Art. 2.1. We first rewrite the equations as LUx = b . Uponusing the notation Ux = y ,

the equations become

Ly = b