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By Nicholas J. Higham

A remedy of the behaviour of numerical algorithms in finite precision mathematics that mixes algorithmic derivations, perturbation thought, and rounding errors research. software program practicalities are emphasised all through, with specific connection with LAPACK and MATLAB.

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In fact, for any x, the calculator computes, in place of f ( x ) = x, the 16 PRINCIPLES OF FINITE PRECISION COMPUTATION function The calculator is producing a completely inaccurate approximation to f ( x ) in just 120 operations on nonnegative numbers. How can this happen? 999... x 10499. If we define r(x) — x1/2 then, for any machine number x > 1, which rounds to 1, since the HP 48G works to about 12 decimal digits. 0, which the squarings leave unchanged. 99... 9. 99... 9. The 60 squarings are represented by s(x) — x 9 , and 60 Because it is smaller than the smallest positive representable number, this result is set to zero on the calculator—a process known as underflow.

Try to avoid subtracting quantities contaminated by error (though such subtractions may be unavoidable). 2. Minimize the size of intermediate quantities relative to the final solution. The reason is that if intermediate quantities are very large then the final answer may be the result of damaging subtractive cancellation. Looked at another way, large intermediate numbers swamp the initial data, resulting in loss of information. 2). 3. Look for different formulations of a computation that are mathematically but not numerically equivalent.

Accumulation of Rounding Errors Since the first electronic computers were developed in the 1940s, comments along the following lines have often been made: "The enormous speed of current machines means that in a typical problem many millions of floating point operations are performed. " This sentiment is true, but misleading. Most often, instability is caused not by the accumulation of millions of rounding errors, but by the insidious growth of just a few rounding errors. As an example, let us approximate e = exp(l) by taking finite n in the definition e := limn_+00(l + l/n) n .

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