"Improved" solution

[T Chellathurai]

Sasha Kasapinovic asked: " In computer assignment#2, we solve the
linear systems Az = r to obtain the "improved" solution x1+z for
the original system Ax = b.
What is the reason for improvement?"

ANSWER:
*******
Let x1 be the numerical(approximate) soln to the linear system
Ax=b.

Residual r1 = b-Ax1
Solving the linear system Az1=r1, the "improved" soln to the
system Ax=b is x2 = x1+z1.

x2 is the better soln than of x1 because,
Residual r2 = b - Ax2
= b - A(x1+z1)
= b - Ax1 - Az1
= b - (b-r1) - r1
= 0.

- C.Thamayanthi
tche...@engmail.uwaterloo.ca