Difference between revisions of "Linear Systems in Matlab"

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[[Category:Matlab]]
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== Solving Linear Systems of Equations in MATLAB ==
 
== Solving Linear Systems of Equations in MATLAB ==
  
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<math>[A](x)=(b)</math>
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This section discusses how to solve a set of linear equations <math>[A](x)=(b)</math> in MATLAB.
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See the discussion of [[Linear_Algebra|linear algebra]] for help on writing a linear system of equations in matrix-vector format.  There is also help on [[Matlab_Arrays#Creating Arrays in Matlab|creating matrices and vectors]] in MATLAB.
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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:
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<source lang="matlab">
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  x = A\b;  % Solve the linear system Ax=b for x.
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</source>
  
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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 creating [[Matlab_Arrays#Creating Matrices|matrices]] and [[Matlab_Arrays#Creating Vectors|vectors]] in MATLAB.
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=== Example ===
  
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  Insert discussion of the "\" operator.
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Consider the following set of equations: [[image:linsys_example.png|right|350px]]
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  Also discuss "inv"
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<center><math>
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\begin{cases}
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  6y = -5x \\
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  y = -3x -5
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\end{cases}.
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</math></center>
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These can be easily solved by hand to obtain
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<math>(x,y) = \left( -\tfrac{30}{13}, \tfrac{25}{13} \right) </math>. These equations and their solution (intersection) are [[:Matlab_Plotting#X-Y_Line_Plots|plotted]] in the figure to the right.
  
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=== Example - Dense System ===
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To solve the system of equations using MATLAB, first [[Linear_Algebra#Linear_Systems_of_Equations|rewrite these in a matrix-vector form]] as
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<center><math>
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\begin{align}
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  5x + 6y &= 0 \\
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  3x + 1y &= -5 \\
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\end{align}
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\quad \Leftrightarrow \quad
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\left[ \begin{array}{cc} 5 & 6 \\ 3 & 1 \end{array} \right]
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\left( \begin{array}{c} x \\ y \end{array} \right) =
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\left( \begin{array}{c} 0 \\ -5 \end{array} \right) .
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</math></center>
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Once in matrix-vector form, the solution is obtained in MATLAB by using the following commands (see [[Matlab_Arrays#Creating_Arrays_in_Matlab|here]] for help on creating matrices and vectors):
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<source lang="matlab">
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  A = [ 5 6; 3 1 ];  % define the matrix
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  b = [ 0; -5 ];      % define the vector
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  solution = A\b;    % solve the system of equations.
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</source>
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In this example, <tt>solution</tt> is a column vector whose elements are <tt>x</tt> and <tt>y</tt>:
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<center><math>
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\mathrm{solution} = \left[ \begin{smallmatrix}-2.3077 \\1.9231 \end{smallmatrix} \right],
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</math></center>
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which is consistent with the answer we obtained by hand above.
  
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=== Example - Sparse System ===
 
  
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==== Matrix Inverse ====
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Note that we can also form the inverse of a matrix,
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<center><math>
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  [A] (x)=(b) \quad \Leftrightarrow \quad (x)=[A]^{-1}(b).
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</math></center>
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This can be done in MATLAB as illustrated by the following:
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<source lang="matlab">
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  A = [ 5 6; 3 1 ];
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  b = [ 0; -5 ];
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  Ainv = A^-1;      % calculate the inverse of A
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  solution = Ainv*b; % calculate the solution
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</source>
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We could also calculate A<sup>-1</sup> by
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<source lang="matlab">
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  Ainv = inv(A);  % entirely equivalent to A^-1.
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</source>
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=== Sparse Systems ===
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{{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.
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 +
{{Stub|section}}

Latest revision as of 08:14, 26 August 2009


Solving Linear Systems of Equations in MATLAB

This section discusses how to solve a set of linear equations [A](x)=(b) in MATLAB. See the discussion of linear algebra for help on writing a linear system of equations in matrix-vector format. 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;  % Solve the linear system Ax=b for x.

Example

Consider the following set of equations:
Linsys example.png
\begin{cases} 6y = -5x \\ y = -3x -5 \end{cases}.

These can be easily solved by hand to obtain (x,y) = \left( -\tfrac{30}{13}, \tfrac{25}{13} \right) . These equations and their solution (intersection) are plotted in the figure to the right.

To solve the system of equations using MATLAB, 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) .

Once in matrix-vector form, the solution is obtained in MATLAB by using the following commands (see here for help on creating matrices and vectors): 

  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:

\mathrm{solution} = \left[ \begin{smallmatrix}-2.3077 \\1.9231 \end{smallmatrix} \right],

which is consistent with the answer we obtained by hand above.


Matrix Inverse

Note that we can also form the inverse of a matrix,

[A] (x)=(b) \quad \Leftrightarrow \quad (x)=[A]^{-1}(b).

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

Warn.jpg
This section is a stub and needs to be expanded.
If you can provide information or finish this section you're welcome to do so and then remove this message afterwards.

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.

Warn.jpg
This section is a stub and needs to be expanded.
If you can provide information or finish this section you're welcome to do so and then remove this message afterwards.
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