On Error Analysis in Arithmetic With Varying Relative Precision
On Error Analysis in Arithmetic With Varying Relative Precision
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E. Is less than the underflow threshold (is denormalized) and differs from the true result by more than the usual . 5 units in the last place. Thus the underflow flag will be on only if underflow could actually have made the final result inaccurate [4, 6]. (The standard also permits raising the underflow flag if the underflowed result is merely inexact; this raises the flag somewhat more often). In any event, it is easy to tell when the possibility of a large relative error in the final result ...exists.
Therefore, the simple code above is unreliable because of roundoff with the Clenshaw/Olver arithmetic, over/underflow with conventional and IEEE floating point, and both roundoff and over/underflow with Matsui/Iri. In order to make it reliable, the following changes are necessary. By reliability, we mean prevention of any interrupts due to over/underflow, and either a reliable warning that the result is inaccurate or a guarantee of accuracy (assuming a reasonable bound on n).
For Matsui/Iri and Clenshaw/Olver the final error will clearly be bounded by n times the error in computing the largest intermediate p.
What to read after On Error Analysis in Arithmetic With Varying Relative Precision?
You can find similar books in the "Read Also" column, or choose other free books by James Demmel to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
E. Is less than the underflow threshold (is denormalized) and differs from the true result by more than the usual . 5 units in the last place. Thus the underflow flag will be on only if underflow could actually have made the final result inaccurate [4, 6]. (The standard also permits raising the underflow flag if the underflowed result is merely inexact; this raises the flag somewhat more often). In any event, it is easy to tell when the possibility of a large relative error in the final result ...exists.
Therefore, the simple code above is unreliable because of roundoff with the Clenshaw/Olver arithmetic, over/underflow with conventional and IEEE floating point, and both roundoff and over/underflow with Matsui/Iri. In order to make it reliable, the following changes are necessary. By reliability, we mean prevention of any interrupts due to over/underflow, and either a reliable warning that the result is inaccurate or a guarantee of accuracy (assuming a reasonable bound on n).
For Matsui/Iri and Clenshaw/Olver the final error will clearly be bounded by n times the error in computing the largest intermediate p.
What to read after On Error Analysis in Arithmetic With Varying Relative Precision?
You can find similar books in the "Read Also" column, or choose other free books by James Demmel to read onlineMoreLess
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