Numerical Differentiation¶
The functions described in this chapter compute numerical derivatives by finite differencing. An adaptive algorithm is used to find the best choice of finite difference and to estimate the error in the derivative.
Again, the development of this module is inspired by the same present in GSL looking to adapt it completely to the practices and tools present in CML.
The functions described in this chapter are declared in the header
file cml/deriv.h
.
Functions¶
-
int
cml_deriv_central
(const cml_function_t *f, double x, double h, double *result, double *abserr)¶ This function computes the numerical derivative of the function
f
at the pointx
using an adaptive central difference algorithm with a step-size ofh
. The derivative is returned inresult
and an estimate of its absolute error is returned inabserr
.The initial value of
h
is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative is computed using a 5-point rule for equally spaced abscissae at , , , , , with an error estimate taken from the difference between the 5-point rule and the corresponding 3-point rule , , . Note that the value of the function at does not contribute to the derivative calculation, so only 4-points are actually used.
-
int
cml_deriv_forward
(const cml_function_t *f, double x, double h, double *result, double *abserr)¶ This function computes the numerical derivative of the function
f
at the pointx
using an adaptive forward difference algorithm with a step-size ofh
. The function is evaluated only at points greater thanx
, and never atx
itself. The derivative is returned inresult
and an estimate of its absolute error is returned inabserr
. This function should be used if has a discontinuity atx
, or is undefined for values less thanx
.The initial value of
h
is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative at is computed using an “open” 4-point rule for equally spaced abscissae at , , , , with an error estimate taken from the difference between the 4-point rule and the corresponding 2-point rule , .
-
int
cml_deriv_backward
(const cml_function_t *f, double x, double h, double *result, double *abserr)¶ This function computes the numerical derivative of the function
f
at the pointx
using an adaptive backward difference algorithm with a step-size ofh
. The function is evaluated only at points less thanx
, and never atx
itself. The derivative is returned inresult
and an estimate of its absolute error is returned inabserr
. This function should be used if has a discontinuity atx
, or is undefined for values greater thanx
.This function is equivalent to calling
cml_deriv_forward()
with a negative step-size.
Examples¶
The following code estimates the derivative of the function
at and at . The function is
undefined for so the derivative at is computed
using cml_deriv_forward()
.
#include <stdio.h>
#include <cml/math.h>
#include <cml/diff.h>
double
f(double x, void *params)
{
(void) params; /* avoid unused parameter warning */
return cml_pow(x, 1.5);
}
int
main(void)
{
cml_function_t F;
double result, abserr;
F.function = &f;
F.params = 0;
printf("f(x) = x^(3/2)\n");
cml_deriv_central(&F, 2.0, 1e-8, &result, &abserr);
printf("x = 2.0\n");
printf("f'(x) = %.10f +/- %.10f\n", result, abserr);
printf("exact = %.10f\n\n", 1.5 * sqrt(2.0));
cml_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
printf("x = 0.0\n");
printf("f'(x) = %.10f +/- %.10f\n", result, abserr);
printf("exact = %.10f\n", 0.0);
return 0;
}
Here is the output of the program,
f(x) = x^(3/2)
x = 2.0
f'(x) = 2.1213203120 +/- 0.0000005006
exact = 2.1213203436
x = 0.0
f'(x) = 0.0000000160 +/- 0.0000000339
exact = 0.0000000000
References and Further Reading¶
This work is a spiritual descendent of the Differentiation module in GSL.