Differentiation.h File Reference

#include "../../Matrix.h"
#include "../../matrix_utils.h"
#include "../utils.h"

Go to the source code of this file.

Functions
Matrix< double >	forwardDiff (const Matrix< double > &x, const Matrix< double > &y)
Matrix< double >	backwardDiff (const Matrix< double > &x, const Matrix< double > &y)
Matrix< double >	centralDiff (const Matrix< double > &x, const Matrix< double > &y)
Matrix< double >	backwardDiff2 (const Matrix< double > &x, const Matrix< double > &y)
Matrix< double >	centralDiff4 (const Matrix< double > &x, const Matrix< double > &y)

Detailed Description

Numerical differentiation methods to approximate the first order derivative using support values.

$$f'(x) = ?$$

Implemented quotients:

• Forward $$ \frac{f(x+h)-f(x)}{h} $$
• Backward $$ \frac{f(x)-f(x-h)}{h} $$
• Center $$ \frac{f(x+h)-f(x-h)}{2h} $$

Usage:

// create a function
auto fun = [](const Matrix<double>& input){ return input.Apply([](const double& val){ return val*val; });};
 
// create support values
auto xk = linspace(-5, 5, 20).Transpose();
 
// calculate 1st derivative
centralDiff(xk, fun(xk));

Function Documentation

backwardDiff()

Matrix< double > backwardDiff ( const Matrix< double > &  x, const Matrix< double > &  y  )

inline

Computes backward difference quotient $$ f'(x) = \frac{f(x)-f(x-h)}{h} $$ Error O(dx)

Parameters

x	x-values
y	y-values @returnss approximated first differential evaluated on x-values

Examples

numerics/analysis/TestDifferentiation.cpp.

backwardDiff2()

Matrix< double > backwardDiff2 ( const Matrix< double > &  x, const Matrix< double > &  y  )

inline

Computes backward difference quotient for 1st differential uses 2nd order polynomial for approximation

Error O(dx)

Parameters

x	x-values
y	y-values @returnss approximated second differential evaluated on x-values

Examples

numerics/analysis/TestDifferentiation.cpp.

centralDiff()

Matrix< double > centralDiff ( const Matrix< double > &  x, const Matrix< double > &  y  )

inline

Computes central difference quotient $$ f'(x) = \frac{f(x+h)-f(x-h)}{2h} $$ Error O(dx^2)

Parameters

x	x-values
y	y-values @returnss approximated first differential evaluated on x-values

Examples

numerics/analysis/TestDifferentiation.cpp.

centralDiff4()

Matrix< double > centralDiff4 ( const Matrix< double > &  x, const Matrix< double > &  y  )

inline

Computes central difference quotient for 1st differential uses 2nd order polynomial for approximation

Error O(dx^2)

Parameters

x	x-values
y	y-values @returnss approximated second differential evaluated on x-values

Examples

numerics/analysis/TestDifferentiation.cpp.

forwardDiff()

Matrix< double > forwardDiff ( const Matrix< double > &  x, const Matrix< double > &  y  )

inline

Computes forward difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ Error O(dx)

Parameters

x	x-values
y	y-values @returnss approximated differential evaluated on x-values

Examples

numerics/analysis/TestDifferentiation.cpp.