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

RombergOne

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One DimensionalRomberg Integration

Syntax

# include <cppad/utility/romberg_one.hpp>

r = RombergOne ( F , a , b , n , e )

Description

Returns the Romberg integration estimate \(r\) for a one dimensional integral

\[r = \int_a^b F(x) {\bf d} x + O \left[ (b - a) / 2^{n-1} \right]^{2(p+1)}\]

Include

The file cppad/utility/romberg_one.hpp is included by cppad/cppad.hpp but it can also be included separately with out the rest of the CppAD routines.

r

The return value r has prototype

Float r

It is the estimate computed by RombergOne for the integral above.

F

The object F can be of any type, but it must support the operation

F ( x )

The argument x to F has prototype

const Float & x

The return value of F is a Float object (see description of Float below).

a

The argument a has prototype

const Float & a

It specifies the lower limit for the integration.

b

The argument b has prototype

const Float & b

It specifies the upper limit for the integration.

n

The argument n has prototype

size_t n

A total number of \(2^{n-1} + 1\) evaluations of F ( x ) are used to estimate the integral.

p

The argument p has prototype

size_t p

It must be less than or equal \(n\) and determines the accuracy order in the approximation for the integral that is returned by RombergOne . To be specific

\[r = \int_a^b F(x) {\bf d} x + O \left[ (b - a) / 2^{n-1} \right]^{2(p+1)}\]

e

The argument e has prototype

Float & e

The input value of e does not matter and its output value is an approximation for the error in the integral estimates; i.e.,

\[e \approx \left| r - \int_a^b F(x) {\bf d} x \right|\]

Float

The type Float must satisfy the conditions for a NumericType . The routine CheckNumericType will generate an error message if this is not the case. In addition, if x and y are Float objects,

x < y

returns the bool value true if x is less than y and false otherwise.

Example

The file romberg_one.cpp contains an example and test a test of using this routine.

Source Code

The source code for this routine is in the file cppad/romberg_one.hpp .