Tutorial - Sine Taylor

Tutorial - Sine Taylor

Tutorial - Programming Sin(x) with Taylor Series

Download Supplementary PPT

Part 1

Q1 a) Create sinTaylor(x) that returns the output of sine x, where x is in [rad].

b) Create sindTaylor(x) that returns the output of sine x, where x in in [deg].

You must use your own function of power() and factorial() from Assignment 0

Procedure

1. Create a new project “ TU_TaylorSeries” with Visual Studio

2. Create the new source file and name it as “C_taylorSeries_exercise.cpp”

3. Copy the source code from this link: C_taylorSeries_exercise.c

4. Fill in the definition of sinTaylor(rad) in the main source.

5. Compare your answer and calculate the absolute error

• sin(π/3)= 0.86602540378

6. Create sindTaylor(deg) for degree unit input and output.

• Hint: re-use sinTaylor(rad) definition

TIP

Approximation of Sine with Taylor series

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Pseudocode for Programming Sine with Taylor series

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Pseudocode for Programming power()

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See here for the TA Tutorial Video

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Part 2

Q. Define your sinTaylor(x) in a header file

Procedure

1. Create a new project “ TU_TaylorSeries_Part2” with Visual Studio

2. Create the new source file and name it as “C_taylorSeries_exercise_part2.cpp”

3. Copy the source code from this link: C_taylorSeries_exercise_part2.c

4. Create a new header file named as myNP_tutorial.h and myNP_tutorial.c

• These files can be downloaded from the link

• These files should be saved in “ \include\” folder.

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5. Your sinTaylor(rad) of Exercise 1 should be declared and defined in the header file.

6. Run and check the answer

See here for the TA Tutorial Video

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Extra Work

1. Create double cosTaylor(double rad)

2. Create double expTaylor(double x)

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Exercises and solution

Part 1

C_taylorSeries_exercise.c

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define		PI		3.14159265358979323846264338327950288419716939937510582


double factorial(int _x);
double sinTaylor(double _x);
double sindTaylor(double _x);

double power(double _x, int N);
double sinTaylor2(double _x);

int main(int argc, char* argv[])
{

	double x = PI / 3;
	//double x = 60;

	double S_N = 0;

	/*===== Select the function to call =====*/
	S_N = sinTaylor(x);
	//S_N = sindTaylor(x);
	
	printf("\n\n");
	printf("=======================================\n");
	printf("    sin( %f[rad] ) Calculation   \n", x);
	printf("=======================================\n");	
	printf("   -  My     result = %3.12f    \n", S_N);
	printf("   -  Math.h result = %3.12f    \n", sin(x));
	printf("   -  absolute err. = %3.12f    \n", S_N - sin(x));
	printf("=======================================\n");
	
	system("pause");
	return 0;
}


// factorial function
double factorial(int N)
{
	int y = 1;
	for (int k = 2; k <= N; k++)
		y = y * k;

	return y;
}


//  Taylor series approximation for sin(x) using pre-defined functions (input unit: [rad])
double sinTaylor(double _x)
{	
	int N_max = 10;
	double S_N = 0;			

	for (int k = 0; k < N_max; k++)
		// [TODO] add your algorithm here
	
	return S_N;
}


// Taylor series approximation for sin(x) using pre-defined functions (input unit: [deg])
double sindTaylor(double _x)
{
	// [TODO] add your algorithm here
}

// power fuction
double power(double _x, int N)
{
	// [TODO] add your algorithm here
}

//  Taylor series approximation for sin(x) using pre-defined functions (input unit: [rad])
double sinTaylor2(double _x)
{
	// [TODO] add your algorithm here
}

solution

/*----------------------------------------------------------------\
@ C-Tutorial by Young-Keun Kim - Handong Global University
Author           : YOUR NAME
Created          : 09-01-2022
Modified         : 09-01-2022
Language/ver     : C in MSVS2022
Description      : C_taylorSeries_exercise_solution.c
----------------------------------------------------------------*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define		PI		3.14159265358979323846264338327950288419716939937510582

double factorial(int _x);
double sinTaylor(double _x);
double sindTaylor(double _x);

double power(double _x, int N);
double sinTaylor2(double _x);

int main(int argc, char* argv[])
{

	double x = PI / 3;
	//double x = 60;
	
	double S_N = 0;

	/*===== Select the function to call =====*/
	S_N = sinTaylor2(x);
	//S_N = sindTaylor(x);
	
	printf("\n\n");
	printf("=======================================\n");
	printf("    sin( %f[rad] ) Calculation   \n", x);
	printf("=======================================\n");	
	printf("   -  My     result = %3.12f    \n", S_N);
	printf("   -  Math.h result = %3.12f    \n", sin(x));
	printf("   -  absolute err. = %3.12f    \n", S_N - sin(x));
	printf("=======================================\n");
	
	system("pause");
	return 0;
}

// factorial function
double factorial(int N)
{
	int y = 1;
	for (int k = 2; k <= N; k++)
		y = y * k;

	return y;
}

//  Taylor series approximation for sin(x) using pre-defined functions (input unit: [rad])
double sinTaylor(double _x)
{	
	int N_max = 10;
	double S_N = 0;			

	for (int k = 0; k < N_max; k++)
		S_N = S_N + pow(-1, k) * pow(_x, 2 * k + 1) / factorial(2 * k + 1);
		
	return S_N;
}
	
// Taylor series approximation for sin(x) using pre-defined functions (input unit: [deg])
double sindTaylor(double _x)
{
	return sinTaylor(_x * PI / 180);
}

// power fuction
double power(double _x, int N)
{
	double y = 1;

	for (int k = 1; k <= N; k++)
		y = y * _x;

	return y;
}


// Taylor series approximation for sin(x) without using pre-defined functions (input unit: [rad])
double sinTaylor2(double _x)
{
	int N_max = 10;
	double S_N = 0;

	for (int k = 0; k < N_max; k++)
		S_N = S_N + power(-1, k) * power(_x, 2 * k + 1) / factorial(2 * k + 1);

	return S_N;
}

---

Part 2

C_taylorSeries_exercise_part2.c

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
//#define		PI		3.14159265358979323846264338327950288419716939937510582

#include "../../../include/myNP_tutorial.h"

int main(int argc, char* argv[])
{

	double x = PI / 3;
	//double x = 60;

	double S_N = 0;

	/*===== Select the function to call =====*/
	S_N = sinTaylor(x);
	//S_N = sindTaylor(x);

	printf("\n\n");
	printf("=======================================\n");
	printf("    sin( %f[rad] ) Calculation   \n", x);
	printf("=======================================\n");
	printf("   -  My     result = %3.12f    \n", S_N);
	printf("   -  Math.h result = %3.12f    \n", sin(x));
	printf("   -  absolute err. = %3.12f    \n", S_N - sin(x));
	printf("=======================================\n");

	system("pause");
	return 0;
}

myNP_tutorial.h

/*----------------------------------------------------------------\
@ Numerical Methods by Young-Keun Kim - Handong Global University

Author           : SSS Lab
Created          : 05-03-2021
Modified         : 05-03-2021
Language/ver     : C in MSVS2019

Description      : myNP_tutorial.h
/----------------------------------------------------------------*/

#ifndef		_MY_NM_H		// use either (#pragma once) or  (#ifndef ...#endif)
#define		_MY_NM_H
#define		PI		3.14159265358979323846264338327950288419716939937510582

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

// Factorial function
extern double factorial(double _x);

// Taylor series approximation for sin(x) using pre-defined functions (input unit: [rad])
extern double sinTaylor(double _x);

// Taylor series approximation for sin(x) using pre-defined functions (input unit: [deg])
extern double sindTaylor(double _x);

// Taylor series approximation for sin(x) without using pre-defined functions (input unit: [rad])
extern double sinTaylor2(double _x);

// Function that reduced the computation cost of sinTaylor2 (input unit: [rad])
extern double sinTaylor3(double _x);

#endif

myNP_tutorial.c

/*----------------------------------------------------------------\
@ Numerical Methods by Young-Keun Kim - Handong Global University

Author           : SSS Lab
Created          : 05-03-2021
Modified         : 05-03-2021
Language/ver     : C in MSVS2019

Description      : myNP_tutorial.c
/----------------------------------------------------------------*/

#include "myNP_tutorial.h"

// factorial function
double factorial(int N)
{
	int y = 1;
	for (int k = 2; k <= N; k++)
		y = y * k;

	return y;
}

//  Taylor series approximation for sin(x) using pre-defined functions (input unit: [rad])
double sinTaylor(double _x)
{
	int N_max = 10;
	double S_N = 0;

	for (int k = 0; k < N_max; k++)
		S_N = S_N + pow(-1, k) * pow(_x, 2 * k + 1) / factorial(2 * k + 1);

	return S_N;
}

// Taylor series approximation for sin(x) using pre-defined functions (input unit: [deg])
double sindTaylor(double _x)
{
	return sinTaylor(_x * PI / 180);
}

// power fuction
double power(double _x, int N)
{
	double y = 1;

	for (int k = 1; k <= N; k++)
		y = y * _x;

	return y;
}

// Taylor series approximation for sin(x) without using pre-defined functions (input unit: [rad])
double sinTaylor2(double _x)
{
	int N_max = 10;
	double S_N = 0;

	for (int k = 0; k < N_max; k++)
		S_N = S_N + power(-1, k) * power(_x, 2 * k + 1) / factorial(2 * k + 1);

	return S_N;
}