Horner’s Method

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 138

Norman Thomson’s neat recursive formulation for calculating Stirling numbers (Vector Vol.8 No.1 p.95) has been gratefully added to my collection of teaching examples. Here is another one which readers of Education Vector may find useful:

Horner’s method for calculating values of a polynomial with coefficients C0,C1,C2,C3,…,Cn in ascending order, at points V, is:

C0 + V × C1 + V × C2 + V × C3 + ... + V × CN

Texts on numerical methods present this as the most efficient and accurate way of calculating the value of polynomials.

The exercise is to implement these operations, and one solution is:

Z←VS HORNER CS
→0 IF 1 = ⍴Z ← CS
Z ← CS[1] + VS × VS HORNER 1↓CS

which shows the nesting of linear functions that is involved here, and may be compared with the encoding

Z ← (⍉ ((⍴CS),⍴VS) ⍴ VS) ⊥ ⌽CS

or other algorithms for calculating polynomial values

Walter G Spunde
School of Information Technology
University of South Queensland
AUSTRALIA