# Fast and practical parallel polynomial interpolation

## Abstract

We present fast and practical parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms make use of fast parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. For n + 1 given input pairs the proposed interpolation algorithm requires 2 (log (n + 1)) + 2 parallel arithmetic steps and circuit size O(n/sup 2/). The algorithms are numerically stable and their floating-point implementation results in error accumulation similar to that of the widely used serial algorithms. This is in contrast to other fast serial and parallel interpolation algorithms which are subject to much larger roundoff. We demonstrate that in a distributed memory environment context, a cube connected system is very suitable for the algorithms' implementation, exhibiting very small communication cost. As further advantages we note that our techniques do not require equidistant points, preconditioning, or use of the Fast Fourier Transform. 21 refs., 4 figs.

- Authors:

- Publication Date:

- Research Org.:
- Illinois Univ., Urbana (USA). Center for Supercomputing Research and Development

- OSTI Identifier:
- 5545183

- Report Number(s):
- DOE/ER/25001-89; CSRD-646

ON: DE88003530

- DOE Contract Number:
- W-7405-ENG-48; FG02-85ER25001

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: Portions of this document are illegible in microfiche products

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; POLYNOMIALS; INTERPOLATION; PARALLEL PROCESSING; ALGORITHMS; ERRORS; NEWTON METHOD; USES; FUNCTIONS; ITERATIVE METHODS; MATHEMATICAL LOGIC; NUMERICAL SOLUTION; PROGRAMMING; 990230* - Mathematics & Mathematical Models- (1987-1989); 990210 - Supercomputers- (1987-1989)

### Citation Formats

```
Egecioglu, O., Gallopoulos, E., and Koc, C.K.
```*Fast and practical parallel polynomial interpolation*. United States: N. p., 1987.
Web.

```
Egecioglu, O., Gallopoulos, E., & Koc, C.K.
```*Fast and practical parallel polynomial interpolation*. United States.

```
Egecioglu, O., Gallopoulos, E., and Koc, C.K. Thu .
"Fast and practical parallel polynomial interpolation". United States.
```

```
@article{osti_5545183,
```

title = {Fast and practical parallel polynomial interpolation},

author = {Egecioglu, O. and Gallopoulos, E. and Koc, C.K.},

abstractNote = {We present fast and practical parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms make use of fast parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. For n + 1 given input pairs the proposed interpolation algorithm requires 2 (log (n + 1)) + 2 parallel arithmetic steps and circuit size O(n/sup 2/). The algorithms are numerically stable and their floating-point implementation results in error accumulation similar to that of the widely used serial algorithms. This is in contrast to other fast serial and parallel interpolation algorithms which are subject to much larger roundoff. We demonstrate that in a distributed memory environment context, a cube connected system is very suitable for the algorithms' implementation, exhibiting very small communication cost. As further advantages we note that our techniques do not require equidistant points, preconditioning, or use of the Fast Fourier Transform. 21 refs., 4 figs.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1987},

month = {1}

}