Backslash inversion vs inv-function

[Quentin Spencer]

Rolf Fabian wrote:
> Because of its underlying algorithm, I expected that 
> inversion of a (nonsingular) square matrix should
> be faster using backslash operator  'y2 = x\eye(n)'
> instead of Octave's  builtin 'y1 = inv(x)'  function.
>
> Thus I conducted  dimension n - depending tests
> using random matrices input in order to sched
> some light on this issue.
>
> octave-3.0.0.exe:48> n=5; x=randn(n); tic; y1 = inv(x); toc,
>                                       tic; y2 = x\eye(n); toc, 
> max(max(abs(y1-y2)))
>
>              Elapsed time is sec.
>    n         inv(x)          x\eye(n)   Speed Gain Fac   max(max(abs(result
> difference)))
>    5         0.05531     0.000258        214              1.1102e-016
>   10         0.06063     0.000313       194              4.4409e-016
>   25         0.05705     0.000705         81              6.9944e-015
>   50         0.05889     0.002197         27              1.2212e-015
>  100        0.07195     0.01153          6.2              1.8952e-015
>  150        0.08721     0.03425          2.6              2.2649e-014
>  200        0.1271       0.06117         2.1               3.9413e-015
>  300        0.2892       0.1991           1.5               8.5820e-014
>  400        0.6255       0.609             1.03             8.9484e-014
>  500        1.176         1.263             0.93            2.2163e-014
>  600        2.05           2.197            0.93             7.0499e-015
>  700        3.228         3.458            0.93             2.4092e-014
>  800        4.855         5.137            0.95             2.2243e-014
>  900        6.873         7.272            0.95             3.0781e-014
> 1000       9.506         9.988            0.95             2.8283e-014
>
> I've done this on a relative old laptop 
> (Windows XP, SP2, Intel III Mobile CPU 933 MHz 512 MB RAM)
> so do not wonder about absolute performance times.
>
> >From the the table above it can be seen that the results confirm my
> expectations.
> Up to about 400x400 sized matrices the speed gain (column 4) can be
> enormous.
> Consequently replacing 'inv(x)' in existing code by 'x\eye(n)' might be an
> option and should be considered.
>
> Unfortunately the maximal absolute difference between the two results
> also increases with dimension n. Does anybody know which algorithm
> is considered to be the more precise ?
>   

Generally, the "\" algorithm is considered more numerically stable. In 
fact, I was always told in school that you should avoid inverting a 
matrix directly for this reason. In my experience with signal processing 
applications, I don't think I've ever run into an application of a 
matrix inverse that couldn't be avoided using a linear solver by posing 
the problem the right way. However, it could be justifiable in some 
situations if an inverse matrix is needed repeatedly for multiple 
subsequent calculations.


Quentin