math::polynomials - Tcl Math Library

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

## Name

math::polynomials - Polynomial functions

## 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.

Mathematics