The inverse of a banded matrix

  1. Hello all,

    I have say 512-by-512 matrix, but based on the structure of this matrix most elements not on the diagonals between -5 to +5 (- stand for diagonal below the main diagonal, and + for diagonal above the main diagonal) are small relative to the elements of the mentioned diagonals. So, I create a 512-by-512 banded matrix, where I null all other elements not on the mentioned diagonals.

    Now the question is: will there be a huge complexity saving if I want the inverse of the matrix by inverting its banded version instead of the original matrix?

    Thanks
     
  2. jcsd
  3. AlephZero

    AlephZero 7,298
    Science Advisor
    Homework Helper

    In computer calculations, "inverting a matrix" is almost always the wrong thing to do, even if you have a nice looking math equation with an inverse matrix in it.

    In this case there will be a huge "complexity" increase, because the inverse matrix will be fully populated, not banded.

    What you really want to do is probably solve a set of equations or something similar. If you decompose your banded matrix as A = LDU or something similar, where L and U are lower and upper triangular and have the same bandwidth as A, you preserve the efficiency by not needing to process all the zero terms in L and U.
     
  4. Could you please tell me more about this process?
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?

0
Draft saved Draft deleted