math4610
USU Math 4610
Derivative Approximation Software Manual
Routine Name: deriv_approx
Author: Philip Nelson
Language: C++. The code can be compiled using the GNU C++ compiler (gcc). A make file is included to compile an example program
For example,
make
will produce an executable ./release.out that can be executed.
Description/Purpose: This routine approximates the derivative of a function \(f\). The approximation of the derivative \[\frac{d}{dx} f(x) \approx \frac{f(x+h) - f(x)}{h}\] is used in the routine.
Input: There are three inputs, a function, the location to approximate the derivative, and the h to use in the approximation.
@tparam T The type of x, h, and the function f's param and return
@param f The T(T) function to approximate
@param x The point to approximate the function at
@param h The value of h to use in the approximation
Output: The approximation of the function f at x using h in the equation above.
Usage/Example:
int main()
{
auto x = 1.0;
auto h = .25;
auto value = cos(x);
auto f = sin;
auto approx = deriv_approx(f, x, h);
std::cout << approx << std::endl;
}
Output from the lines above
0.4300545382
explanation of output: The routine output the approximation of the derivative of \(sin(x)\) at \(x=1\) using \(h=.25\). The correct value should be \(cos(1) = 0.5403023059\)
Implementation/Code: The following is the code for deriv_approx
template <typename T, typename F>
T deriv_approx(F f, T x, T h)
{
return (f(x + h) - f(x)) / h;
}
Experimentation
The first function tested was \(\frac{d}{dx}sin(x)\) at \(x=1\). Below are the results:
h d/dx sin(1) approx Error Abs Error Rel
-------------------------------------------------------------------------
1 0.5403023059 0.0678264420 0.4724758639 0.874465755
0.5 0.5403023059 0.3120480036 0.2282543023 0.4224566503
0.25 0.5403023059 0.4300545382 0.1102477677 0.2040483013
0.125 0.5403023059 0.4863728743 0.05392943154 0.09981343954
0.0625 0.5403023059 0.5136632057 0.02663910012 0.04930406521
0.03125 0.5403023059 0.5270674561 0.01323484972 0.02449526789
0.015625 0.5403023059 0.5337064629 0.00659584301 0.01220768992
0.0078125 0.5403023059 0.5370098303 0.003292475538 0.006093765477
0.00390625 0.5403023059 0.5386574359 0.001644869986 0.003044351224
0.001953125 0.5403023059 0.5394802136 0.0008220922623 0.00152154128
0.0009765625 0.5403023059 0.5398913455 0.0004109603504 0.0007606118759
0.00048828125 0.5403023059 0.5400968472 0.0002054587178 0.0003802662242
0.000244140625 0.5403023059 0.5401995819 0.000102723993 0.0001901231807
0.0001220703125 0.5403023059 0.5402509452 5.136065484e-05 9.505910724e-05
6.103515625e-05 0.5403023059 0.5402766259 2.567999236e-05 4.75289335e-05
3.051757812e-05 0.5403023059 0.540289466 1.283991169e-05 2.376431037e-05
1.525878906e-05 0.5403023059 0.5402958859 6.419934111e-06 1.188211496e-05
7.629394531e-06 0.5403023059 0.5402990959 3.209956235e-06 5.941037452e-06
3.814697266e-06 0.5403023059 0.5403007009 1.604967296e-06 2.970498699e-06
1.907348633e-06 0.5403023059 0.5403015034 8.024873792e-07 1.485256255e-06
9.536743164e-07 0.5403023059 0.5403019047 4.012037649e-07 7.42554234e-07
4.768371582e-07 0.5403023059 0.5403021052 2.006201654e-07 3.713109553e-07
2.384185791e-07 0.5403023059 0.5403022054 1.005029886e-07 1.86012511e-07
1.192092896e-07 0.5403023059 0.5403022561 4.974590828e-08 9.207050894e-08
5.960464478e-08 0.5403023059 0.5403022803 2.553152134e-08 4.7254141e-08
2.980232239e-08 0.5403023059 0.5403022952 1.063036015e-08 1.967483765e-08
1.490116119e-08 0.5403023059 0.540302299 6.905069849e-09 1.278001181e-08
7.450580597e-09 0.5403023059 0.5403023064 5.455107477e-10 1.009639866e-09
3.725290298e-09 0.5403023059 0.5403023064 5.455107477e-10 1.009639866e-09
Here are graphs of the real vs the approximate solution, and the errors
The second function tested was \(x^3 + 2x^2 + 10x + 7\) at \(x=3\). Below are the results:
h d/dx f(3) approx Error Abs Error Rel
-------------------------------------------------------------------------
1 49 61 12 0.2448979592
0.5 49 54.75 5.75 0.1173469388
0.25 49 51.8125 2.8125 0.05739795918
0.125 49 50.390625 1.390625 0.02838010204
0.0625 49 49.69140625 0.69140625 0.01411033163
0.03125 49 49.34472656 0.3447265625 0.007035235969
0.015625 49 49.17211914 0.1721191406 0.003512635523
0.0078125 49 49.08599854 0.08599853516 0.001755072146
0.00390625 49 49.04298401 0.04298400879 0.0008772246692
0.001953125 49 49.02148819 0.0214881897 0.0004385344836
0.0009765625 49 49.01074314 0.01074314117 0.0002192477791
0.00048828125 49 49.00537133 0.005371332169 0.0001096190238
0.000244140625 49 49.00268561 0.00268560648 5.48082955e-05
0.0001220703125 49 49.00134279 0.001342788339 2.740384365e-05
6.103515625e-05 49 49.00067139 0.000671390444 1.37018458e-05
3.051757812e-05 49 49.00033569 0.0003356942907 6.850903892e-06
1.525878906e-05 49 49.00016785 0.0001678466797 3.425442443e-06
7.629394531e-06 49 49.00008392 8.392333984e-05 1.712721221e-06
3.814697266e-06 49 49.00004196 4.196166992e-05 8.563606107e-07
1.907348633e-06 49 49.00002098 2.098083496e-05 4.281803053e-07
9.536743164e-07 49 49.00001049 1.049041748e-05 2.140901527e-07
4.768371582e-07 49 49.00000525 5.24520874e-06 1.070450763e-07
2.384185791e-07 49 49.00000262 2.62260437e-06 5.352253817e-08
1.192092896e-07 49 49.00000131 1.311302185e-06 2.676126908e-08
5.960464478e-08 49 49.00000072 7.152557373e-07 1.459705586e-08
2.980232239e-08 49 49 0 0
1.490116119e-08 49 49 0 0
Last Modified: September 2018