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# math::polynomials(n) 1.0.1 tcllib "Tcl Math Library"

## Name

math::polynomials - 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.
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.

## Category

Mathematics

## Copyright

Copyright © 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>