Summary: ESTIMATION OF ERROR
Let bx denote an approximate solution for Ax = b;
perhaps bx is obtained by Gaussian elimination. Let x
denote the exact solution. Then introduce
r = b - Abx
a quantity called the residual for bx. Then
r = b - Abx
= Ax - Abx
= A (x - bx)
x - bx = A-1r
or the error e = x - bx is the exact solution of
Ae = r
Thus we can solve this to obtain an estimate be of our
error e.
EXAMPLE. Recall the linear system
.729x1 + .81x2 + .9x3 = .6867
x1 + x2 + x3 = .8338
1.331x1 + 1.21x2 + 1.1x3 = 1.000
The true solution, rounded to four significant digits,
is

Source: Atkinson, Kendall - Departments of Computer Science & Mathematics, University of Iowa

Collections: Computer Technologies and Information Sciences