Envelope Matrix

Methods to generate and work with envelopes of matrices.

Notice Use envelope-matrix methods for educational purposes only. The methods are NOT suitable for professional application as it is not meant to be the most efficient, optimized, correct and secure implementation.

The envelope of a symmetric or upper-triangular matrix A is a set of ordered pairs (i, j), i < j, representing element locations in the upper triangle of A, defined as follows: (i, j) is in the envelope of A if and only if a_kj != 0 for some k <= i. Thus, if the first nonzero entry of the j th column is a_mj and m < j, then (m, j), (m + 1, j), …, (j - 1, j) are the members of the envelope of A from the j th column. [1]

About

A basic data structure of four arrays is one way to store the envelope a matrix A.

Array	Description
DIAG	Stores the elements from the diagonal of A.
ENV	Stores the envelope of A.
ENVcol	Points to elements of ENV. ENVcol(j) points to position in ENV that stores the first non-zero element of the j th column. If the j th column is “empty”, then ENVcol(j) points to ENVcol(j + 1).
ENVlin	Stores the line index of each element from ENV.

Example

A = [
  11  12  0   14  0   0;
  0   22  23  0   0   0;
  0   0   33  0   0   0;
  0   0   0   44  0   46;
  0   0   0   0   55  0;
  0   0   0   0   0   66
]

DIAG = [11   22   33   44   55   66];
ENV = [12   23   14    0    0   46    0    null];
ENVcol = [1   1   2   3   6   6   8];
ENVlin = [1   2   1   2   3   4   5   null];

To retrieve an element a_ij from an envelope:

1. If i == j, return DIAG(i).
2. Get pos = ENVcol(j).
3. If pos == ENVcol(j + 1), then the j th column is empty: return 0.
4. If ENVlin(pos) > i, then the column has 0’s up to the ENVlin(pos) th line.
5. Get pos = pos + i - ENVlin(pos).
6. Return ENV(pos).

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Reference

1. Watkins, David S. (2010), Fundamentals of matrix computations (Third ed.), New York: John Wiley & Sons, Inc., ISBN 0-470-52833-4