3
$\begingroup$

I have read in many research papers related with iteration methods to find the generalized inverses. Where to show efficiency of the methods randomly generated matrices of higher order have been used sometime matrices with elements randomly taken from certain interval are used. I want to know why we take randomly generated matrices of higher order When simple matrices are available in literature?

I want to know importance and significance of randomly generated matrices. Please help me to understand this. I would be very much thankful.

  • 0
    simple matrices? do you mean $\mathbf{I}$?2012-08-29
  • 0
    @chaohuang I mean lower order matrices. I want to know why to take randomly generated matrices? Is there any specific reasons?2012-08-29
  • 1
    $\mathbf{I}$ is just an extreme example if the matrix is not randomly generated. Also, with lower order matrices, all the methods may probably work well, but you cannot extrapolate to tell the situation in high order cases.2012-08-29
  • 0
    @chaohuang Thanks for replying me? Is there not any other way to create large size matrices?2012-08-29
  • 1
    There could be some other ways, but creating random matrices may be the simplest, thus most common used approach. And it's sufficient to tell the performance of a method, unless you're so unlucky.2012-08-29
  • 0
    @chaohuang Thanks for replying me and clearing my doubt..2012-08-29
  • 1
    A random matrix models a real-life input. You're writing a method and expecting that a random user calls it with some input. You don't know if the input is just a special case (e.g. zero matrix, identity matrix, triangular matrix, full rank matrix etc) or more general. So you assume it's just *random*. I think there is a more formal statement about the expected behaviour of any algorithm over a set of input drawn from a random distribution.2012-08-31
  • 0
    @JenniferDylan Thanks for making such a useful comment for me. I have one more query. I have read in few papers where they take random generated matrices where elements are uniformaly distributed over some specified intervals. For example take random matrix of order 20 where elements are uniformly distributed over [-1.1 1.1]. Is there any specific reasons for taking random matrices this way? Thanks a lot for helping me.2012-08-31
  • 1
    He is a much worded version of my comment above. To establish a statement about correctness/efficiency of an algorithm, we need to test with all possible inputs which is impractical/impossible. But we can test with a sample of the input space. Here comes the catch phrase: with random samples we can establish "statistically meaningful estimates" of the program bahvior over the whole input space. (quote from this [PPT](http://www.cs.cmu.edu/~agroce/CS119/l4.ppt))2012-08-31
  • 1
    For you other question. Frankly, I'm not sure. The uniformly distributed radnom elements are typically generated from a finite interval. In computational finance & machine learning, for example, I've read that they encourage to [normalize](http://en.wikipedia.org/wiki/Normalization_(statistics)) all inputs to the interval $[-1, +1]$. There was a good reason for this, but I forgot it :( The mean is zero or something?. See for example [this question](http://stackoverflow.com/questions/4674623/why-do-we-have-to-normalize-the-input-for-an-artificial-neural-network).2012-08-31
  • 0
    @JenniferDylan Better you would have written it in answer form. I got what i wanted to know. Heartily thanks to you.2012-09-01

0 Answers 0