The IMSL_FCN_DERIV function computes the first, second, or third derivative of a user-supplied function.
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The IMSL_FCN_DERIV function 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.
Examples
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 = IMSL_FCN_DERIV('fcn', 2.0)
PRINT, "f'(x) = ", deriv1
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 = IMSL_FCN_DERIV('fcn', 2.0, /Double)
deriv2 = IMSL_FCN_DERIV('fcn', 2.0, ORDER = 2, /Double)
deriv3 = IMSL_FCN_DERIV('fcn', 2.0, ORDER = 3, /Double)
PRINT, "f'(x) = ", deriv1, ', error =', $
ABS(deriv1 + 3.0*COS(1.5*2.0))
f'(x) = 2.9699775, error = 1.1094893e-07
PRINT, "f''(x) = ", deriv2, ', error =', $
ABS(deriv2 - 4.5*SIN(1.5*2.0))
f''(x) = 0.63504004, error = 5.1086361e-08
PRINT, "f'''(x) = ", deriv3, ', error =', $
ABS(deriv3 - 6.75*COS(1.5*2.0))
f'''(x) = -6.6824494, error = 1.1606068e-08
Syntax
Result = IMSL_FCN_DERIV(F, X [, /DOUBLE] [, ORDER=value] [, STEPSIZE=value] [, TOLERANCE=value])
Return Value
An estimate of the first, second or third derivative of f at x. If no value can be computed, NaN is returned.
Arguments
F
A scalar string specifying a user-supplied function whose derivative at X will be computed.
X
The point at which the derivative will be evaluated.
Keywords
DOUBLE (optional)
If present and nonzero, then double precision is used.
ORDER (optional)
Set this keyword equal to the order of the desired derivative (1, 2 or 3). Default: 1
STEPSIZE (optional)
Set this keyword equal to the 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: 0.01
TOLERANCE (optional)
Set this keyword equal to 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. The default is 
, where ε is machine epsilon.
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