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## Accounting for floating point errors

The error bounds discussed in this section are subject to floating point errors, most of which are innocuous, but which deserve some discussion.

The infinity norm requires the fewest floating point operations to compute, and cannot overflow or cause other exceptions if the are themselves finite. On the other hand, computing in the most straightforward manner can easily overflow or lose accuracy to underflow even when the true result is far from either the overflow or underflow thresholds. For this reason, a careful implementation for computing without this danger is available (subroutine snrm2 in the BLAS [69] [141]), but it is more expensive than computing .

Now consider computing the residual by forming the matrix-vector product and then subtracting , all in floating point arithmetic with relative precision . A standard error analysis shows that the error in the computed is bounded by , where is typically bounded by , and usually closer to . This is why one should not choose in Criterion 1, and why Criterion 2 may not be satisfied by any method. This uncertainty in the value of induces an uncertainty in the error of at most . A more refined bound is that the error in the th component of is bounded by times the th component of , or more tersely . This means the uncertainty in is really bounded by . This last quantity can be estimated inexpensively provided solving systems with and as coefficient matrices is inexpensive (see the last paragraph of §). Both these bounds can be severe overestimates of the uncertainty in , but examples exist where they are attainable.

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