Tutorial: Nonlinear solver

Tutorial: Nonlinear solver

Problem

Solve for x, in a non-linear function of

f(x)= 8-4.5*(x-sin(x))=0

image

Tutorial: MATLAB

Sample Code

clear all
F = inline('8-4.5*(x-sin(x))');
dF=inline('-4.5*(1-cos(x))');​
x=0:0.001:10;
plot(x,F(x)); grid on

Exercise

Download the tutorial source file and fill in the blanks. Run the code and validate your answer.

• MATLAB tutorial source file : TU_nonlinear_student.mlx

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Tutorial: C-Programming

1. Create a new project “ TU_Nonlinear” with Visual Studio, under the directory \NP\tutorial\

2. Download the tutorial source file from

• C-program tutorial source file : TU_nonlinear_student.cpp

1. Add the downloaded source file under the project folder.

2. Prepare your library header files in your project: myNP_tutorial.h and myNP_tutorial.c

   • This is the same header files used in the previous tutorial: Tutorial-Sine Taylor: Part 2

   • They must be located in \NP\include\ folder

   • If you do not have them, you can download from here

3. Rename your library header files as myNP_yourID.h and myNP_yourID.c

   • Example: myNP_20030011.c, myNP_20030011.h

   From now on, you will update your functions in the library header files.

Exercise 1

On the given source code template TU_nonlinear_student.cpp, fill-in the missing codes.

Bisection Method

Assuming (func(a) * func(b) <0 )

1. First, Write down a pseudocode for the bisection

   // YOUR pseudocode goes here// YOUR pseudocode goes here

2. Based on the pseudocode, fill in the blanks in the source code.

   double bisection(float _a, float _b, float _tol);

3. Move your functions from main source file to your header files

   • function definitions: myNP_yourID.h

   • function declaration: myNP_yourID.c

Exercise 2

Modify your Bisection function with considering the following conditions

• if(func(a) * func(b) > 0), No solution exists

• if(func(a) * func(b) ==0), Either (1) a=Xtrue or (2) b=Xtrue

• if(k==Nmax) Solution did not converge within given iteration

Exercise 3

Newton Raphson Method

1. First, Write down a pseudocode for the method

   // YOUR pseudocode goes here// YOUR pseudocode goes here

2. Based on the pseudocode, fill in the blanks in the source code.

   double newtonRaphson(float _a, float _b, float _tol);

3. You need to define the function f(x) and the derivative function dfdx(x) as

   double func(float _x);
   double dfunc(float _x);

   For dfdx(x), get the derivative formula analytically.

4. After you have compiled them successfully, move your functions to your header files

Exercise 4

Modify Newton Raphson Method with function callback

Read how to pass a function as an input argument

Modify the newton-raphson function that calls the problem functions f(x) and dfdx(x) as input arguments.

double newtonRaphson(double func(double _x), double dfunc(double _x), double _x0, double _tol);

Now, the problem function f(x) and the derivative function dfdx(x) should be located in the main source code TU_nonlinear_student.cpp

NOT in myNP_yourID.h

// Move these declaration and definitions of problem functions to
// TU_nonlinear_student.cpp
double func(float _x);
double dfunc(float _x);