# polynomials - Man Page

Polynomial functions

## Synopsis

`package require `

**Tcl ?8.3?**

`package require `

**math::polynomials ?1.0.1?**

**::math::polynomials::polynomial** *coeffs*

**::math::polynomials::polynCmd** *coeffs*

**::math::polynomials::evalPolyn** *polynomial x*

**::math::polynomials::addPolyn** *polyn1 polyn2*

**::math::polynomials::subPolyn** *polyn1 polyn2*

**::math::polynomials::multPolyn** *polyn1 polyn2*

**::math::polynomials::divPolyn** *polyn1 polyn2*

**::math::polynomials::remainderPolyn** *polyn1 polyn2*

**::math::polynomials::derivPolyn** *polyn*

**::math::polynomials::primitivePolyn** *polyn*

**::math::polynomials::degreePolyn** *polyn*

**::math::polynomials::coeffPolyn** *polyn index*

**::math::polynomials::allCoeffsPolyn** *polyn*

## Description

This package deals with polynomial functions of one variable:

- the basic arithmetic operations are extended to polynomials
- computing the derivatives and primitives of these functions
- evaluation through a general procedure or via specific procedures)

## Procedures

The package defines the following public procedures:

**::math::polynomials::polynomial***coeffs*Return an (encoded) list that defines the polynomial. A polynomial

f(x) = a + b.x + c.x**2 + d.x**3

can be defined via:

set f [::math::polynomials::polynomial [list $a $b $c $d]

- list
*coeffs* Coefficients of the polynomial (in ascending order)

- list
**::math::polynomials::polynCmd***coeffs*Create a new procedure that evaluates the polynomial. The name of the polynomial is automatically generated. Useful if you need to evualuate the polynomial many times, as the procedure consists of a single [expr] command.

- list
*coeffs* Coefficients of the polynomial (in ascending order) or the polynomial definition returned by the

*polynomial*command.

- list
**::math::polynomials::evalPolyn***polynomial x*Evaluate the polynomial at x.

- list
*polynomial* The polynomial's definition (as returned by the polynomial command). order)

- float
*x* The coordinate at which to evaluate the polynomial

- list
**::math::polynomials::addPolyn***polyn1 polyn2*Return a new polynomial which is the sum of the two others.

- list
*polyn1* The first polynomial operand

- list
*polyn2* The second polynomial operand

- list
**::math::polynomials::subPolyn***polyn1 polyn2*Return a new polynomial which is the difference of the two others.

- list
*polyn1* The first polynomial operand

- list
*polyn2* The second polynomial operand

- list
**::math::polynomials::multPolyn***polyn1 polyn2*Return a new polynomial which is the product of the two others. If one of the arguments is a scalar value, the other polynomial is simply scaled.

- list
*polyn1* The first polynomial operand or a scalar

- list
*polyn2* The second polynomial operand or a scalar

- list
**::math::polynomials::divPolyn***polyn1 polyn2*Divide the first polynomial by the second polynomial and return the result. The remainder is dropped

- list
*polyn1* The first polynomial operand

- list
*polyn2* The second polynomial operand

- list
**::math::polynomials::remainderPolyn***polyn1 polyn2*Divide the first polynomial by the second polynomial and return the remainder.

- list
*polyn1* The first polynomial operand

- list
*polyn2* The second polynomial operand

- list
**::math::polynomials::derivPolyn***polyn*Differentiate the polynomial and return the result.

- list
*polyn* The polynomial to be differentiated

- list
**::math::polynomials::primitivePolyn***polyn*Integrate the polynomial and return the result. The integration constant is set to zero.

- list
*polyn* The polynomial to be integrated

- list
**::math::polynomials::degreePolyn***polyn*Return the degree of the polynomial.

- list
*polyn* The polynomial to be examined

- list
**::math::polynomials::coeffPolyn***polyn index*Return the coefficient of the term of the index'th degree of the polynomial.

- list
*polyn* The polynomial to be examined

- int
*index* The degree of the term

- list
**::math::polynomials::allCoeffsPolyn***polyn*Return the coefficients of the polynomial (in ascending order).

- list
*polyn* The polynomial in question

- list

## Remarks on the Implementation

The implementation for evaluating the polynomials at some point uses Horn's rule, which guarantees numerical stability and a minimum of arithmetic operations. To recognise that a polynomial definition is indeed a correct definition, it consists of a list of two elements: the keyword "POLYNOMIAL" and the list of coefficients in descending order. The latter makes it easier to implement Horner's rule.

## Bugs, Ideas, Feedback

This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category *math :: polynomials* of the *Tcllib Trackers* [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for enhancements you may have for either package and/or documentation.

When proposing code changes, please provide *unified diffs*, i.e the output of **diff -u**.

Note further that *attachments* are strongly preferred over inlined patches. Attachments can be made by going to the **Edit** form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar.

## Keywords

math, polynomial functions

## Category

Mathematics

## Copyright

Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>