Tutorial: Differentiation
Ploblem
Estimate the velocity and acceleration from datasets of the position of an object.
t = 0:0.2:4;
y = [-5.87 -4.23 -2.55 -0.89 0.67 2.09 3.31 4.31 5.06 5.55 5.78 5.77 5.52 5.08 4.46 3.72 2.88 2.00 1.10 0.23 -0.59];
Tutorial: MATLAB
Example code
% asusume evenly spaced h
h=0.2;
vel = diff(y)./h;
tv=t(1:length(vel));
acc = diff(y,2)./h^2;
ta=t(1:length(acc));
Exercise (25 min)
Download the tutorial source file and fill in the blanks.
Run the code and validate your answer
Tutorial: C-Programming
Create a new folder under \tutorial Directory and name it as differentiation
e.g ) C:\Users\yourID\source\repos\NP\tutorial\TU_differentiation
Create a new empty project in Visual Studio Community. Name the project as TU_Differentiation
Create a new C/C++ source file for main()
Name the source file as TU_Differentiation_main.cpp
Paste from the following code
TU_differentiation_student.cpp
/*------------------------------------------------------------------------------\
@ Numerical Methods by Young-Keun Kim - Handong Global University
Author : Young-Keun Kim
Created : 01-04-2019
Modified : 11-27-2023
Language/ver : C in MSVS2017
Course : Numerical Programming
Description : [Tutorial]Differentiation.cpp
-------------------------------------------------------------------------------*/
#include "stdio.h"
#include "stdlib.h"
#include "../../include/myNP.h"
//#include "../../include/myNP_myID.h"
double myFunc(const double x);
int main(int argc, char* argv[])
{
/*==========================================================================*/
/* Part 1 - Differentiation from discrete dataset points */
/*==========================================================================*/
printf("\n**************************************************");
printf("\n| PART 1. |");
printf("\n**************************************************\n");
/************ Variables declaration & initialization ************/
int m = 21;
double t[21] = { 0 };
for (int i = 0; i < m; i++) t[i] = 0.2 * i;
double x[] = { -5.87, -4.23, -2.55, -0.89, 0.67, 2.09, 3.31, 4.31, 5.06, 5.55, 5.78, 5.77, 5.52, 5.08, 4.46, 3.72, 2.88, 2.00, 1.10, 0.23, -0.59 };
double dxdt[21] = { 0 };
/************ Solve & Show Output ************/
// Differentiation from discrete dataset points
// [YOUR CODE GOES HERE]
// gradient1D(t, x, dxdt, m);
// printVec(dxdt, m);
/*==========================================================================*/
/* Part 2 - Differentiation from a function */
/*==========================================================================*/
printf("\n**************************************************");
printf("\n| PART 2. |");
printf("\n**************************************************\n");
/************ Variables declaration & initialization ************/
double xin = 2.5;
double dydx[21] = { 0 }; // m=21 points
// User defined function F(x)
double y = myFunc(xin);
printf("\n y=myFun(xin) = %f \n\n", y);
/************ Solve & Show Output ************/
// Estimate differentiation from the user defined function
// [YOUR CODE GOES HERE]
// gradientFunc(myFunc, t, dydx, m);
// printVec(dydx, m);
system("pause");
return 0;
}
// User defined function: example y=x*x
// Modify to User Function
double myFunc(const double x) {
return x * x;
}
Exercise 1 : Differentiation from Discrete Points (25 min)
Create a C-program function for numerical differentiation from a set of discrete data. Read the instruction in the source code.
For m dataset, x[0] to x[m-1]
2-Point forward difference: for the first point x[0] to [m-2]
2-Point backward difference: for the last point x[m-1]
void gradient1D(double _x[], double _y[], double dydx[], int m);
The gradient1D function should be defined in your library header files, located in \include folder
function definitions: myNP_yourID.h
function declaration: myNP_yourID.cpp
// Example Code using 1D Array
void printVec(double _vec[], int _row);
int main()
{
// Static 1-D array with fixed array size and initialized
double a[4] = { 1, 2, 3, 4 };
// Print 1-D array element
printVec(a, 4);
return 0;
}
void printVec(double _vec[], int _row)
{
for (int i = 0; i<_row; i++)
printf("Vector[%d] = %f \n", i, _vec[i]);
}
Exercise 2 : Differentiation from a function (25 min)
Define a function named as myFunc() that defines the user's equation F(x).
For this tutorial, use y=x^3 , for x=0:0.2:4
double myFunc(const double _x);
Create a function named as gradientFunc() that returns the gradient of the user's equation.
It should receive myFunc() as the input argument.
It generates a discrete dataset of the user's F(x)
This function should return a 1D array of dydx[] .
void gradientFunc(double func(const double x), double _x[ ], double dydx[ ], int m);
This function should be defined in your library header files, located in \include folder
function definitions: myNP_yourID.h
function declaration: myNP_yourID.cpp
Validate the result