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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)

::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.

::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

::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

::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

::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

::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

::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

::math::polynomials::derivPolyn polyn

  Differentiate the polynomial and return the result.

  list polyn

    The polynomial to be differentiated

::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

::math::polynomials::degreePolyn polyn

  Return the degree of the polynomial.

  list polyn

    The polynomial to be examined

::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

::math::polynomials::allCoeffsPolyn polyn

  Return the coefficients of the polynomial (in ascending order).

  list polyn

    The polynomial in question

## 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.

## Keywords

math, polynomial functions

## Category

Mathematics

## Copyright

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