Difference between revisions of "Linear Systems in Matlab"

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This section discusses how to solve a set of linear equations <math>[A](x)=(b)</math> in MATLAB.
 
This section discusses how to solve a set of linear equations <math>[A](x)=(b)</math> in MATLAB.
See the discussion of [[LinearAlgebra|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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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.
  
 
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:
 
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">
 
<source lang="matlab">
   x = A\b;
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   x = A\b; % Solve the linear system Ax=b for x.
 
</source>
 
</source>
  
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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.
 
<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.
  
To solve the system of equations using MATLAB, first [[LinearAlgebra#Linear_Systems_of_Equations|rewrite these in a matrix-vector form]] as
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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
 
<center><math>
 
<center><math>
 
  \begin{align}
 
  \begin{align}
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In this example, <tt>solution</tt> is a column vector whose elements are <tt>x</tt> and <tt>y</tt>:
 
In this example, <tt>solution</tt> is a column vector whose elements are <tt>x</tt> and <tt>y</tt>:
 
<center><math>
 
<center><math>
  \mathrm{solution} = \left[ \begin{array}{c} -2.3077 \\1.9231 \end{array} \right],
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  \mathrm{solution} = \left[ \begin{smallmatrix}-2.3077 \\1.9231 \end{smallmatrix} \right],
 
</math></center>
 
</math></center>
 
which is consistent with the answer we obtained by hand above.
 
which is consistent with the answer we obtained by hand above.
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   Ainv = inv(A);  % entirely equivalent to A^-1.
 
   Ainv = inv(A);  % entirely equivalent to A^-1.
 
</source>
 
</source>
 
  
 
=== Sparse Systems ===
 
=== Sparse Systems ===
 
{{Stub|section}}
 
{{Stub|section}}
 
 
  
 
== Linear Systems using the Symbolic Toolbox ==
 
== Linear Systems using the Symbolic Toolbox ==

Latest revision as of 09:14, 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

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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.

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