User`s guide

Conditioning and Numerical Stability
10-7
Using row 1 as the pivot row (i.e., subtracting 1000 times row 1 from 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 ca n 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
1001
–
1000
–
x
2
1.000
=
0.001x
1
1.000 1.000()1.000()–=
x
1
0.000
=
x
1