1.7.4. Method.
The method is Gaussian elimination. If the problem is
ill-conditioned, then there are two relatively easy things one can do to reduce
the error. Either, one can increase the precision of the floating point
numbers, and this is easy to do in Maple. Or, one can avoid division by small
numbers, and this is known as row pivoting in the forward sweep of the
Gaussian elimination algorithm. The following remarkable example demonstrates
both these methods.
Example. We will consider three different versions of the Gaussian
elimination algorithm applied to the following system
Use real numbers:
Use floating point numbers with 3 digits:
The above row operation gives a different upper triangular matrix
Thus, the floating point solution is
Interchange rows of the system to avoid division by a small number:
The 3 digit floating point solution is X1 = .100 101
and X2 = .200 101!
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