Accurate Computation of Divided Differences.
Abstract
The standard recurrence scheme does not always yield accurate divided differences in finite precision arithmetic. When the function of interest is known analytically and/or its values are easily calculated, methods other than the recurrence scheme can be used. In particular, a table of divided differences can be regarded as a function of a special bidiagonal matrix. Formulas and computational techniques suitable for computing matrix functions may, thus, be exploited for divided differences. Divided difference tables of the exponential function are profitably treated as the exponential of a special matrix. This approach is good precisely when the standard recurrence is bad, namely when the abscissae of the divided differences are close. When the abscissae are scaled down by powers of 2, the resulting scaled divided difference table may be squared to give the wanted table. For real abscissae this scaling and squaring technique, in combination with the standard recurrence where suitable, yields a hybrid algorithm which permits computation of any exponential divided difference to an accuracy dependent only on the order of the difference. For appropriate arrangements of complex abscissae, such as conjugate pairs, a similar result is established. A good way to compute the exponential of a real square matrix A is to use the Newton divided difference interpolating polynomial. Our algorithm finds an important application in computing accurately the coefficients of this polynomial. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- May 28, 1980
- Accession Number
- ADA088751
Entities
People
- Allan Charles Mccurdy
Organizations
- University of California, Berkeley