It is the first time I deal with column-compress storage (CCS) format to store matrices. After googling a bit, if I am right, in a matrix having n nonzero elements the CCS is as follows:
-we define a vector A_v of dimensions n x 1 storing the n non-zero elements
of the matrix
- we define a second vector A_ir of dimensions n x 1 storing the rows of the
non-zero elements of the matrix
-we finally define a third vector A_jc whose elements are the indices of the
elements of A_v which corresponds to the beginning of new column, plus a
final value which is by convention equal t0 n+1, and identifies the end of
the matrix (pointing theoretically to a virtual extra-column).
So for instance, if
M = [1 0 4 0 0;
0 3 5 2 0;
2 0 0 4 6;
0 0 7 0 8]
we get
A_v = [1 2 3 4 5 7 2 4 6 8];
A_ir = [1 3 2 1 2 4 2 3 3 4];
A_jc = [1 3 4 7 9 11];
my questions are
I) is what I wrote correct, or I misunderstood anything?
II) what if I want to represent a matri with some columns which are zeroes, e.g.,
M2 = [0 1 0 0 4 0 0;
0 0 3 0 5 2 0;
0 2 0 0 0 4 6;
0 0 0 0 7 0 8]
wouldn't the representation of M2 in CCS be identical to the one of M?
Thanks for the help!
I) is what I wrote correct, or I misunderstood anything?
You are perfectly correct. However, you have to take care that if you use a C or C++ library offsets and indices should start at 0. Here, I guess you read some Fortran doc for which indices are starting at 1. To be clear, here is below the C version, which simply translates the indices of your Fortran-style correct answer:
A_v = unmodified
A_ir = [0 2 1 0 1 3 1 2 2 4] (in short [1 3 2 1 2 4 2 3 3 4] - 1)
A_jc = [0 2 3 6 8 10] (in short [1 3 4 7 9 11] - 1)
II) what if I want to represent a matri with some columns which are zeroes, e.g., M2 = [0 1 0 0 4 0 0; 0 0 3 0 5 2 0; 0 2 0 0 0 4 6; 0 0 0 0 7 0 8]
wouldn't the representation of M2 in CCS be identical to the one of M?
I you have an empty column, simply add a new entry in the offset table A_jc. As this column contains no element this new entry value is simply the value of the previous entry. For instance for M2 (with index starting at 0) you have:
A_v = unmodified
A_ir = unmodified
A_jc = [0 0 2 3 6 8 10] (to be compared to [0 2 3 6 8 10])
Hence the two representations are differents.
If you just start learning about sparse matrices there is an excelllent free book here: http://www-users.cs.umn.edu/~saad/IterMethBook_2ndEd.pdf