FCN_DERIV Function

Computes the first, second, or third derivative of a user-supplied function.

Usage

result = FCN_DERIV(f, x)

Input Parameters

f—Scalar string specifying a user-supplied function whose derivative at x will be computed.

x—Point at which the derivative will be evaluated.

Returned Value

result—An estimate of the first, second or third derivative of f at x. If no value can be computed, NaN is returned.

Input Keywords

Double—If present and nonzero, double precision is used.

Order—The order of the desired derivative (1, 2, or 3). Default: Order = 1

Stepsize—Beginning value used to compute the size of the interval for approximating the derivative. Stepsize must be chosen small enough that f is defined and reasonably smooth in the interval (x – 4.0*Stepsize, x + 4.0*Stepsize), yet large enough to avoid roundoff problems. Default: Stepsize = 0.01

Tolerance—The relative error desired in the derivative estimate. Convergence is assumed when (2/3) |d₂ – d₁| < Tolerance, for two successive derivative estimates, d₁ and d₂. Default: Tolerance = where e is machine epsilon.

Discussion

The function FCN_DERIV produces an estimate to the first, second, or third derivative of a function. The estimate originates from first computing a spline interpolant to the input function using values within the interval (x – 4.0*Stepsize, x + 4.0*Stepsize), then differentiating the spline at x.

Example 1

This example obtains the approximate first derivative of the function f(x) = –2sin(3x/2) at the point x = 2.

FUNCTION fcn, x

   f  =  -2*SIN(1.5*x)   

   RETURN,  f

END

deriv1 = FCN_DERIV('fcn', 2.0)

PRINT, 'f''(x)   = ', deriv1

; PV-WAVE prints: f'(x)   =       2.97008

Example 2

This example obtains the approximate first, second, and third derivative of the function f(x) = –2sin(3x/2) at the point x = 2.

FUNCTION fcn,  x

   f  =  -2*SIN(1.5*x)   

   RETURN,  f

END

deriv1  =  FCN_DERIV('fcn', 2.0, /Double)

deriv2  =  FCN_DERIV('fcn', 2.0, ORDER = 2, /Double)

deriv3  =  FCN_DERIV('fcn', 2.0, ORDER = 3, /Double)

PRINT, 'f''(x)   = ', deriv1, ',  error =', $

   ABS(deriv1 + 3.0*COS(1.5*2.0))

; PV-WAVE prints: f'(x) = 2.9699775,  error =   1.1094893e-07

PRINT, 'f''''(x)  = ', deriv2, ',  error =', $

   ABS(deriv2 - 4.5*SIN(1.5*2.0))

; PV-WAVE prints: f''(x)  = 0.63504004,  error =   5.1086361e-08

PRINT, 'f''''''(x) = ', deriv3, ',  error =', $

   ABS(deriv3 - 6.75*COS(1.5*2.0))

; PV-WAVE prints: f'''(x) = -6.6824494,  error =   1.1606068e-08