User`s guide

10 Reliable Computations
10-4
Conditioning and Numerical Stability
Two of the key concepts in numerical anal ysis are the conditioning of problems
and the stability of algorithms.
Conditioning
Consider the linear system given by
A =
0.7800 0.5630
0.9130 0.6590
b =
0.2170
0.2540
Thetruesolutionisx = [1, –1]' and youcan calculateitapproximatelyusing
MATLAB.
x = A\b
x =
1.0000
–1.0000
format long, x
x =
0.99999999991008
–0.99999999987542
Of course, in real problems you almost never have the luxury of knowing the
true solution. This problem is very ill-conditioned. To see this, add a small
perturbation to
A
E =
0.0010 0.0010
–0.0020 –0.0010
and solve the perturbed system
xe = (A+E)\b
xe =
–5.0000
7.3085
Ax b
=
AE
+()
xb
=