MATLAB jacobi itteration Exercise 1: Poisson\'s Equation Consider the linear sys
ID: 3600714 • Letter: M
Question
MATLAB jacobi itteration
Exercise 1: Poisson's Equation Consider the linear system A, = p, where An is an n × n matrix with 2's on the main diagonal,-1's directly above and below the main diagonal and 0's everywhere else. For instance, As is 2-100 0 1 2-1 0 0 As=10-1 2-1 0 0 1 2-1 0 0 01 2 This is a discretized version of Poisson's equation: This equation appears very often in physics. We'll learn about discretizations of differential equations later this quarter Construct the matrix Aso in Matlab. Try reading the documentation for diag() to find a simple way to do this. Make the vector such that the jth entry in p, pj, is defined according to the formula P 2 (1-cos (23/51)) sin (23j/51) (a) Write down the matrix form of the Jacobi iteration 1 = MPa + c. Con- catenate the nnatrix M and the vector c and save the resulting 50 × 51 matrix as Al.dat. (b) Use Jacobi iteration to solve for given an initial guess of a column of ones. Continue to iterate the Jacobi method until every term in the vector is within 10 of the previous iteration. L.e., norm(phic,k+1) . phi(nk), Inf)Explanation / Answer
Solving Basic Algebraic Equations in MATLAB
The solve function is used for solving algebraic equations. In its simplest form, the solve function takes the equation enclosed in quotes as an argument.
For example, let us solve for x in the equation x-5 = 0
solve('x-5=0')
MATLAB will execute the above statement and return the following result
ans =
5
You can also call the solve function as
y = solve('x-5 = 0')
MATLAB will execute the above statement and return the following result
y =
5
You may even not include the right hand side of the equation
solve('x-5')
MATLAB will execute the above statement and return the following result
ans =
5
If the equation involves multiple symbols, then MATLAB by default assumes that you are solving for x, however, the solve function has another form
solve(equation, variable)
where, you can also mention the variable.
For example, let us solve the equation v – u – 3t2 = 0, for v. In this case, we should write
solve('v-u-3*t^2=0', 'v')
MATLAB will execute the above statement and return the following result
ans =
3*t^2 + u
Solving Basic Algebraic Equations in Octave
The roots function is used for solving algebraic equations in Octave and you can write above examples as follows:
For example, let us solve for x in the equation x-5 = 0
roots([1, -5])
Octave will execute the above statement and return the following result
ans = 5
You can also call the solve function as
y = roots([1, -5])
Octave will execute the above statement and return the following result
y = 5
Solving Quadratic Equations in MATLAB
The solve function can also solve higher order equations. It is often used to solve quadratic equations. The function returns the roots of the equation in an array.
The following example solves the quadratic equation x2 -7x +12 = 0. Create a script file and type the following code
eq = 'x^2 -7*x + 12 = 0';
s = solve(eq);
disp('The first root is: '), disp(s(1));
disp('The second root is: '), disp(s(2));
When you run the file, it displays the following result
The first root is:
3
The second root is:
4
Solving Quadratic Equations in Octave
The following example solves the quadratic equation x2 -7x +12 = 0 in Octave. Create a script file and type the following code
s = roots([1, -7, 12]);
disp('The first root is: '), disp(s(1));
disp('The second root is: '), disp(s(2));
When you run the file, it displays the following result
The first root is:
4
The second root is:
3
Solving Higher Order Equations in MATLAB
The solve function can also solve higher order equations. For example, let us solve a cubic equation as (x-3)2(x-7) = 0
solve('(x-3)^2*(x-7)=0')
MATLAB will execute the above statement and return the following result
ans =
3
3
7
In case of higher order equations, roots are long containing many terms. You can get the numerical value of such roots by converting them to double. The following example solves the fourth order equation x4 7x3 + 3x2 5x + 9 = 0.
Create a script file and type the following code
eq = 'x^4 - 7*x^3 + 3*x^2 - 5*x + 9 = 0';
s = solve(eq);
disp('The first root is: '), disp(s(1));
disp('The second root is: '), disp(s(2));
disp('The third root is: '), disp(s(3));
disp('The fourth root is: '), disp(s(4));
% converting the roots to double type
disp('Numeric value of first root'), disp(double(s(1)));
disp('Numeric value of second root'), disp(double(s(2)));
disp('Numeric value of third root'), disp(double(s(3)));
disp('Numeric value of fourth root'), disp(double(s(4)));
When you run the file, it returns the following result
The first root is:
6.630396332390718431485053218985
The second root is:
1.0597804633025896291682772499885
The third root is:
- 0.34508839784665403032666523448675 - 1.0778362954630176596831109269793*i
The fourth root is:
- 0.34508839784665403032666523448675 + 1.0778362954630176596831109269793*i
Numeric value of first root
6.6304
Numeric value of second root
1.0598
Numeric value of third root
-0.3451 - 1.0778i
Numeric value of fourth root
-0.3451 + 1.0778i
Please note that the last two roots are complex numbers.
Solving Higher Order Equations in Octave
The following example solves the fourth order equation x4 7x3 + 3x2 5x + 9 = 0.
Create a script file and type the following code
v = [1, -7, 3, -5, 9];
s = roots(v);
% converting the roots to double type
disp('Numeric value of first root'), disp(double(s(1)));
disp('Numeric value of second root'), disp(double(s(2)));
disp('Numeric value of third root'), disp(double(s(3)));
disp('Numeric value of fourth root'), disp(double(s(4)));
When you run the file, it returns the following result
Numeric value of first root
6.6304
Numeric value of second root
-0.34509 + 1.07784i
Numeric value of third root
-0.34509 - 1.07784i
Numeric value of fourth root
1.0598
Solving System of Equations in MATLAB
The solve function can also be used to generate solutions of systems of equations involving more than one variables. Let us take up a simple example to demonstrate this use.
Let us solve the equations
5x + 9y = 5
3x – 6y = 4
Create a script file and type the following code
s = solve('5*x + 9*y = 5','3*x - 6*y = 4');
s.x
s.y
When you run the file, it displays the following result
ans =
22/19
ans =
-5/57
In same way, you can solve larger linear systems. Consider the following set of equations
x + 3y -2z = 5
3x + 5y + 6z = 7
2x + 4y + 3z = 8
Solving System of Equations in Octave
We have a little different approach to solve a system of 'n' linear equations in 'n' unknowns. Let us take up a simple example to demonstrate this use.
Let us solve the equations
5x + 9y = 5
3x – 6y = 4
Such a system of linear equations can be written as the single matrix equation Ax = b, where A is the coefficient matrix, b is the column vector containing the right-hand side of the linear equations and x is the column vector representing the solution as shown in the below program
Create a script file and type the following code
A = [5, 9; 3, -6];
b = [5;4];
A b
When you run the file, it displays the following result
ans =
1.157895
-0.087719
In same way, you can solve larger linear systems as given below
x + 3y -2z = 5
3x + 5y + 6z = 7
2x + 4y + 3z = 8
Expanding and Collecting Equations in MATLAB
The expand and the collect function expands and collects an equation respectively. The following example demonstrates the concepts
When you work with many symbolic functions, you should declare that your variables are symbolic.
Create a script file and type the following code
syms x %symbolic variable x
syms y %symbolic variable x
% expanding equations
expand((x-5)*(x+9))
expand((x+2)*(x-3)*(x-5)*(x+7))
expand(sin(2*x))
expand(cos(x+y))
% collecting equations
collect(x^3 *(x-7))
collect(x^4*(x-3)*(x-5))
When you run the file, it displays the following result
ans =
x^2 + 4*x - 45
ans =
x^4 + x^3 - 43*x^2 + 23*x + 210
ans =
2*cos(x)*sin(x)
ans =
cos(x)*cos(y) - sin(x)*sin(y)
ans =
x^4 - 7*x^3
ans =
x^6 - 8*x^5 + 15*x^4
Expanding and Collecting Equations in Octave
You need to have symbolic package, which provides expand and the collect function to expand and collect an equation, respectively. The following example demonstrates the concepts
When you work with many symbolic functions, you should declare that your variables are symbolic but Octave has different approach to define symbolic variables. Notice the use of Sin and Cos, which are also defined in symbolic package.
Create a script file and type the following code
% first of all load the package, make sure its installed.
pkg load symbolic
% make symbols module available
symbols
% define symbolic variables
x = sym ('x');
y = sym ('y');
z = sym ('z');
% expanding equations
expand((x-5)*(x+9))
expand((x+2)*(x-3)*(x-5)*(x+7))
expand(Sin(2*x))
expand(Cos(x+y))
% collecting equations
collect(x^3 *(x-7), z)
collect(x^4*(x-3)*(x-5), z)
When you run the file, it displays the following result
ans =
-45.0+x^2+(4.0)*x
ans =
210.0+x^4-(43.0)*x^2+x^3+(23.0)*x
ans =
sin((2.0)*x)
ans =
cos(y+x)
ans =
x^(3.0)*(-7.0+x)
ans =
(-3.0+x)*x^(4.0)*(-5.0+x)
Factorization and Simplification of Algebraic Expressions
The factor function factorizes an expression and the simplify function simplifies an expression. The following example demonstrates the concept
Example
Create a script file and type the following code
syms x
syms y
factor(x^3 - y^3)
factor([x^2-y^2,x^3+y^3])
simplify((x^4-16)/(x^2-4))
When you run the file, it displays the following result
ans =
(x - y)*(x^2 + x*y + y^2)
ans =
[ (x - y)*(x + y), (x + y)*(x^2 - x*y + y^2)]
ans =
x^2 + 4
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