math::polynomials - Polynomial functions

package require Tcl ?8.3? package require math::polynomials ?1.0? ::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

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)

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]

coeffs list 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.

coeffs list 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.

polynomial list The polynomial's definition (as returned by

the polynomial command). order)

x float 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.

polyn1 list The first polynomial operand

polyn2 list The second polynomial operand

::math::polynomials::subPolyn polyn1 polyn2

Return a new polynomial which is the difference of the two others.

polyn1 list The first polynomial operand

polyn2 list 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.

polyn1 list The first polynomial operand or a scalar

polyn2 list 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

polyn1 list The first polynomial operand

polyn2 list The second polynomial operand

::math::polynomials::remainderPolyn polyn1 polyn2

Divide the first polynomial by the second polynomial and return the remainder.

polyn1 list The first polynomial operand

polyn2 list The second polynomial operand

::math::polynomials::derivPolyn polyn

Differentiate the polynomial and return the result.

polyn list The polynomial to be differentiated

::math::polynomials::primitivePolyn polyn

Integrate the polynomial and return the result. The integration constant is set to zero.

polyn list The polynomial to be integrated

::math::polynomials::degreePolyn polyn

Return the degree of the polynomial.

polyn list The polynomial to be examined

::math::polynomials::coeffPolyn polyn index

Return the coefficient of the term of the index'th degree of the polynomial.

polyn list The polynomial to be examined

index int The degree of the term

::math::polynomials::allCoeffsPolyn polyn

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

polyn list The polynomial in question

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.

math, polynomial functions

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