Difference between revisions of "Linear Systems in Matlab"
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See the discussion of [[LinearAlgebra|linear algebra]] for help on setting up a linear system of equations. There is also help on [[Matlab_Arrays#Creating Arrays in Matlab|creating matrices and vectors]] in MATLAB. | See the discussion of [[LinearAlgebra|linear algebra]] for help on setting up a linear system of equations. There is also help on [[Matlab_Arrays#Creating Arrays in Matlab|creating matrices and vectors]] in MATLAB. | ||
| − | + | The simplest way of solving a system of equations in MATLAB is by using the '''\''' operator. Given a matrix '''A''' and a vector '''b''', we may solve the system using the following MATLAB commands | |
| − | + | <source lang="matlab"> | |
| + | x = A\b; | ||
| + | </source> | ||
| + | For example, if we wanted to solve the system of equations | ||
| + | <center><math> | ||
| + | \begin{cases} | ||
| + | 6y = -5x \\ | ||
| + | y = -3x -5 | ||
| + | \end{cases} | ||
| + | </math></center> | ||
| + | we would first rewrite these in a matrix-vector form as | ||
| + | <center><math> | ||
| + | \begin{align} | ||
| + | 5x + 6y = 0 \\ | ||
| + | 3x + 1y &= -5 \\ | ||
| + | \end{align} | ||
| + | \quad \Leftrightarrow \quad | ||
| + | \left[ \begin{array}{cc} 5 & 6 \\ 3 & 1 \end{array} \right] | ||
| + | \left( \begin{array}{c} x \\ y \end{array} \right) = | ||
| + | \left( \begin{array}{c} 0 \\ -5 \end{array} \right) | ||
| + | </math></center> | ||
| + | This is implemented in MATLAB as | ||
| + | <source lang="matlab"> | ||
| + | A = [ 5 6; 3 1 ]; % define the matrix | ||
| + | b = [ 0; -5 ]; % define the vector | ||
| + | solution = A\b; % solve the system of equations. | ||
| + | </source> | ||
| + | In this example, <tt>solution</tt> is a column vector whose elements are <tt>x</tt> and <tt>y</tt>. | ||
| − | == | + | Note that we can also form the inverse of a matrix, |
| + | <center><math> | ||
| + | [A] (x)=(b) \quad \Leftrightarrow \quad (x)=[A]^{-1}(b). | ||
| + | </math></center> | ||
| − | === | + | This can be done in MATLAB as illustrated by the following: |
| + | <source lang="matlab"> | ||
| + | A = [ 5 6; 3 1 ]; | ||
| + | b = [ 0; -5 ]; | ||
| + | Ainv = A^-1; % calculate the inverse of A | ||
| + | solution = Ainv*b; % calculate the solution | ||
| + | </source> | ||
| + | We could also calculate A<sup>-1</sup> by | ||
| + | <source lang="matlab"> | ||
| + | Ainv = inv(A); % entirely equivalent to A^-1. | ||
| + | </source> | ||
| + | |||
| + | |||
| + | === Sparse Systems === | ||
| + | {{Stub|section}} | ||
== Linear Systems using the Symbolic Toolbox == | == Linear Systems using the Symbolic Toolbox == | ||
Occasionally we may want to find the symbolic (general) solution to a system of equations rather than a specific numerical solution. The symbolic toolbox provides a way to do this. | Occasionally we may want to find the symbolic (general) solution to a system of equations rather than a specific numerical solution. The symbolic toolbox provides a way to do this. | ||
| + | |||
| + | {{Stub|section}} | ||
Revision as of 15:09, 18 August 2008
Solving Linear Systems of Equations in MATLAB
=(b)](/wiki/images/math/b/3/a/b3ae30ff4dcdb54b97d86d814ca503d6.png)
See the discussion of linear algebra for help on setting up a linear system of equations. There is also help on creating matrices and vectors in MATLAB.
The simplest way of solving a system of equations in MATLAB is by using the \ operator. Given a matrix A and a vector b, we may solve the system using the following MATLAB commands
x = A\b;
For example, if we wanted to solve the system of equations
we would first rewrite these in a matrix-vector form as
![\begin{align}
5x + 6y = 0 \\
3x + 1y &= -5 \\
\end{align}
\quad \Leftrightarrow \quad
\left[ \begin{array}{cc} 5 & 6 \\ 3 & 1 \end{array} \right]
\left( \begin{array}{c} x \\ y \end{array} \right) =
\left( \begin{array}{c} 0 \\ -5 \end{array} \right)](/wiki/images/math/4/9/8/4988602ede2c5f89c04678734acaff3b.png)
This is implemented in MATLAB as
A = [ 5 6; 3 1 ]; % define the matrix
b = [ 0; -5 ]; % define the vector
solution = A\b; % solve the system of equations.
In this example, solution is a column vector whose elements are x and y.
Note that we can also form the inverse of a matrix,
![[A] (x)=(b) \quad \Leftrightarrow \quad (x)=[A]^{-1}(b).](/wiki/images/math/9/6/0/96008bab47a724a093e8ed6374cec8f5.png)
This can be done in MATLAB as illustrated by the following:
A = [ 5 6; 3 1 ];
b = [ 0; -5 ];
Ainv = A^-1; % calculate the inverse of A
solution = Ainv*b; % calculate the solution
We could also calculate A-1 by
Ainv = inv(A); % entirely equivalent to A^-1.
Sparse Systems
 |
|
Linear Systems using the Symbolic Toolbox
Occasionally we may want to find the symbolic (general) solution to a system of equations rather than a specific numerical solution. The symbolic toolbox provides a way to do this.
|
|
