Contents of MGS Example

MGS Example

**Figure:** Modified Gram-Schmidt Function
$\begin{figure} \begin{verbatim} // // Modified Gram-Schmidt // Given A (MxN), w... ...:m;k] * r[k;j]; } } return << q = q; r = r >>; };\end{verbatim} \end{figure}$

This example (See Figure $[*]$ ) implements a modified Gram-Schmidt algorithm to find an orthonormal basis for the input matrix. There are several important points to recognize in this function.

The function returns a list. mgs returns a list containing the matrices Q and R. The members of the list can be accessed by:

> aa = rand(4,4)
 aa =
 matrix columns 1 thru 4
      0.7       0.96      0.924      0.148  
     0.95      0.915     0.0882      0.879  
   0.0918      0.441      0.908    0.00543  
    0.902     0.0735      0.362      0.222  
> x = mgs (aa)
   q            r            
> x.q
 q =
 matrix columns 1 thru 4
     0.47      0.513      0.261     -0.669  
    0.638      0.243     -0.613      0.396  
   0.0617      0.436      0.642      0.628  
    0.606     -0.698      0.379     0.0378  
> mgs (aa).q
 q =
 matrix columns 1 thru 4
     0.47      0.513      0.261     -0.669  
    0.638      0.243     -0.613      0.396  
   0.0617      0.436      0.642      0.628  
    0.606     -0.698      0.379     0.0378

The function argument is copied. The argument A is copied to the local variable a to avoid destroying the original contents of A (or aa in the callers environment).