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Concept
Polynomial evaluation
Summary
In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the polynomial at consists of computing See also
For evaluating the univariate polynomial the most naive method would use multiplications to compute , use multiplications to compute and so on for a total of multiplications and additions.
Using better methods, such as Horner's rule, this can be reduced to multiplications and additions. If some preprocessing is allowed, even more savings are possible.
This problem arises frequently in practice. In computational geometry, polynomials are used to compute function approximations using Taylor polynomials. In cryptography and hash tables, polynomials are used to compute k-independent hashing.
In the former case, polynomials are evaluated using floating-point arithmetic, which is not exact. Thus different schemes for the evaluation will, in general, give slightly different answers. In the latter case, the polynomials are usually evaluated in a finite field, in which case the answers are always exact.
Horner's method
Horner's method evaluates a polynomial using repeated bracketing:
This method reduces the number of multiplications and additions to just
Horner's method is so common that a computer instruction "multiply–accumulate operation" has been added to many computer processors, which allow doing the addition and multiplication operations in one combined step.
If the polynomial is multivariate, Horner's rule can be applied recursively over some ordering of the variables.
E.g.
can be written as
An efficient version of this approach was described by Carnicer and Gasca.
Estrin's scheme
While it's not possible to do less computation than Horner's rule (without preprocessing), on modern computers the order of evaluation can matter a lot for the computational efficiency.
A method known as Estrin's scheme computes a (single variate) polynomial in a tree like pattern:
Combined by Exponentiation by squaring, this allows parallelizing the computation.
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Horner's method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials.