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# NAME

math::polynomials - Polynomial functions

# Table Of Contents

# SYNOPSIS

package require Tcl ?8.5?

package require math::polynomials ?1.0.2?

**::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.

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

# CATEGORY

Mathematics

# COPYRIGHT

Copyright © 2004 Arjen Markus