\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

Poly

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Evaluate a Polynomial or its Derivative

Syntax

# include <cppad/utility/poly.hpp>

p = Poly ( k , a , z )

Description

Computes the k-th derivative of the polynomial

\[P(z) = a_0 + a_1 z^1 + \cdots + a_d z^d\]

If k is equal to zero, the return value is \(P(z)\).

Include

The file cppad/utility/poly.hpp is included by cppad/cppad.hpp but it can also be included separately with out the rest of the CppAD routines. Including this file defines Poly within the CppAD namespace.

k

The argument k has prototype

size_t k

It specifies the order of the derivative to calculate.

a

The argument a has prototype

const Vector & a

(see Vector below). It specifies the vector corresponding to the polynomial \(P(z)\).

z

The argument z has prototype

const Type & z

(see Type below). It specifies the point at which to evaluate the polynomial

p

The result p has prototype

Type p

(see Type below) and it is equal to the k-th derivative of \(P(z)\); i.e.,

\[p = \frac{k !}{0 !} a_k + \frac{(k+1) !}{1 !} a_{k+1} z^1 + \ldots + \frac{d !}{(d - k) !} a_d z^{d - k}\]

If \(k > d\), p = Type (0) .

Type

The type Type is determined by the argument z . It is assumed that multiplication and addition of Type objects are commutative.

Operations

The following operations must be supported where x and y are objects of type Type and i is an int :

x = i	assignment
x = y	assignment
x \* = y	multiplication compound assignment
x += y	addition compound assignment

Vector

The type Vector must be a SimpleVector class with elements of type Type . The routine CheckSimpleVector will generate an error message if this is not the case.

Operation Sequence

The Type operation sequence used to calculate p is Independent of z and the elements of a (it does depend on the size of the vector a ).

Example

The file poly.cpp contains an example and test of this routine.

Source

The file poly.hpp contains the current source code that implements these specifications.