GeographicLib::TrigfunExt Class Reference

A function defined by its derivative and its inverse. More...

#include <GeographicLib/Trigfun.hpp>

Public Member Functions
	TrigfunExt ()
	TrigfunExt (const std::function< real(real)> &fp, real halfp, bool sym=false, real scale=-1)
real	operator() (real x) const
real	deriv (real x) const
real	inv (real z) const
void	ComputeInverse ()
bool	Symmetric () const
int	NCoeffs () const
int	NCoeffsInv () const
real	Max () const
real	HalfPeriod () const
real	HalfRange () const
real	Slope () const

Detailed Description

A function defined by its derivative and its inverse.

This is an extension of Trigfun which allows a function to be defined by its derivative. In this case the derivative must be even, so that its integral is odd (and taken to be zero at the origin).

In addition, this class offers a flexible interface to computing the inverse of the function. If the inverse is only required at a few points

Example of use:

// Example of using the GeographicLib::TrigfunExt class.
 
#include <iostream>
#include <iomanip>
#include <exception>
#include <GeographicLib/Trigfun.hpp>
#include <GeographicLib/EllipticFunction.hpp>
 
using namespace std;
using namespace GeographicLib;
 
int main(int argc, const char* const argv[]) {
  try {
    // Integrate 1/sqrt(1 - k2*sin(x)^2) and compare this to the elliptic
    // integral of the first kine.
    double k2 = Math::sq(0.8);
    auto f = [k2] (double x) -> double
    { return 1/sqrt(1 - k2 * Math::sq(sin(x))); };
    TrigfunExt tfe(f, Math::pi()/2);
    cout << "Number of coefficients: " << tfe.NCoeffs() << "\n";
    EllipticFunction ell(k2);
    cout << "x tf(x) tf(x)-F(x)\n";
    for (int i = 0; i <= 20; ++i) {
      double x = i/10.0;
      cout << x << " " << tfe(x) << " " << tfe(x) - ell.F(x) << "\n";
    }
  }
  catch (const std::exception& e) {
    std::cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

Definition at line 498 of file Trigfun.hpp.

Constructor & Destructor Documentation

TrigfunExt() [1/2]

GeographicLib::TrigfunExt::TrigfunExt ( )

inline

Definition at line 532 of file Trigfun.hpp.

TrigfunExt() [2/2]

GeographicLib::TrigfunExt::TrigfunExt	(	const std::function< real(real)> &	fp,
		real	halfp,
		bool	sym = false,
		real	scale = -1
	)

Constructor specifying the derivative, an even periodic function

Parameters

[in]	fp	the derivative function, fp(x) = f'(x).
[in]	halfp	the half period.
[in]	sym	is fp symmetric about the quarter period point (so it contains only odd Fourier harmonics)?
[in]	scale;	this is passed to the Trigfun constructor when finding the Fourier series for fp.

Warning

fp must be an even periodic function. In addition fp must be nonnegative for the inverse of f to be computed (in this case, f is a monotonically increasing function). The inverse is undefined for sym = true.

Member Function Documentation

operator()()

real GeographicLib::TrigfunExt::operator() ( real  x ) const

inline

Evaluate the TrigfunExt.

Parameters

[in]

x

the function argument.

Returns

the function value f(x).

Definition at line 556 of file Trigfun.hpp.

deriv()

real GeographicLib::TrigfunExt::deriv ( real  x ) const

inline

Evaluate the derivative for TrigfunExt.

Parameters

[in]

x

the function argument.

Returns

the value of the derivative fp(x). This uses the function object passed to the constructor.

Definition at line 564 of file Trigfun.hpp.

inv()

real GeographicLib::TrigfunExt::inv ( real  z ) const

inline

Evaluate the inverse of f

Parameters

[in]

z

the value of f(x)

Returns

the value of x = f ⁻¹(z).

This compute the inverse using Newton's method with the derivative function fp supplied on construction. Initially, the starting guess is based on just the secular component of f(x). However, if ComputeInverse() is called, a rough Trigfun approximation to the inverse is found and this is used as the starting point for Newton's method.

Definition at line 577 of file Trigfun.hpp.

ComputeInverse()

void GeographicLib::TrigfunExt::ComputeInverse ( )

inline

Compute a coarse Fourier series approximation of the inverse.

This is used to provide a better starting guess for Newton's method in inv(). Because ComputeInverse() is fairly expensive, this only makes sense if inv() will be called many times. In order to limit the expense in computing this approximate inverse, the number of Fourier components in the Trigfun for the inverse is limited to 3/2 of the number of components for f and the tolerance is set to the square root of the machine epsilon.

Definition at line 591 of file Trigfun.hpp.

Symmetric()

bool GeographicLib::TrigfunExt::Symmetric ( ) const

inline

Returns

whether the function is symmetric about the quarter period point. If it is it, then the Fourier series has only odd harmonics.

Definition at line 602 of file Trigfun.hpp.

NCoeffs()

int GeographicLib::TrigfunExt::NCoeffs ( ) const

inline

Returns

the number of terms in the Fourier series for f.

Definition at line 606 of file Trigfun.hpp.

References GeographicLib::Trigfun::NCoeffs().

NCoeffsInv()

int GeographicLib::TrigfunExt::NCoeffsInv ( ) const

inline

Returns

the number of terms in the Fourier series for f⁻¹.

Definition at line 611 of file Trigfun.hpp.

References GeographicLib::Trigfun::NCoeffs().

Max()

real GeographicLib::TrigfunExt::Max ( ) const

inline

Returns

Max() for the underlying Trigfun.

Definition at line 617 of file Trigfun.hpp.

References GeographicLib::Trigfun::Max().

HalfPeriod()

real GeographicLib::TrigfunExt::HalfPeriod ( ) const

inline

Returns

HalfPeriod() for the underlying Trigfun.

Definition at line 621 of file Trigfun.hpp.

References GeographicLib::Trigfun::HalfPeriod().

HalfRange()

real GeographicLib::TrigfunExt::HalfRange ( ) const

inline

Returns

HalfRange() for the underlying Trigfun.

Definition at line 625 of file Trigfun.hpp.

References GeographicLib::Trigfun::HalfRange().

Slope()

real GeographicLib::TrigfunExt::Slope ( ) const

inline

Returns

Slope() for the underlying Trigfun.

Definition at line 629 of file Trigfun.hpp.

References GeographicLib::Trigfun::Slope().

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The documentation for this class was generated from the following file:

• Trigfun.hpp