Specifications

Conditioning and Numerical Stability
11-7
Using row 1as the pivot row (i.e., subtracting 1000 times row 1from row 2)you
arrive at the equivalent triangular system.
Note that the coefficient multiplying in the second equation should be
, but because of roundoff, becomes . As a result, the second
equation yields , a good approximation, but now back-substitution
in the first equation
yields . This extremely bad approximation of is the result of
numerical instability. The problem itself can be shown to be quite
well-conditioned. Of course, MATLAB implements Gaussian elimination with
pivoting.
0.001 1.000
0 1000–
x
1
x
2
1.000
1000–
=
x
2
100
1
– 1000–
x
2
1.000=
0.001x
1
1.000 1.000
()
1.000
()
–=
x
1
0.000= x
1