Numerical Methods Contents - SAM

2.6.1 Sparse matrix storage formats
Special sparse matrix storage formats store only non-zero entries:
(➣ usually O(n + m) storage required for sparse n × m-matrix)
• Compressed Row Storage (CRS)
• Compressed Column Storage (CCS)
• Block Compressed Row Storage (BCRS)
• Compressed Diagonal Storage (CDS)
• Jagged Diagonal Storage (JDS)
• Skyline Storage (SKS)
→ used by MATLAB
Option:
⎛
⎞
10 0 0 0 −2 0
3 9 0 0 0 3
A =
0 7 8 7 0 0
⎜ 3 0 8 7 5 0
⎟
⎝ 0 8 0 9 9 13⎠
0 4 0 0 2 −1
val 10 -2 3 9 3 7 8 7 3 . . . 9 13 4 2 -1
col_ind 1 5 1 2 6 2 3 4 1 . . . 5 6 2 5 6
row_ptr 1 3 6 9 13 17 20
diagonal CRS format (matrix diagonal stored in separate array)
2.6.2 Sparse matrices in MATLAB
✸
mandatory for large sparse matrices.
Detailed discussion in [17]
Example 2.6.2 (Compressed row-storage (CRS) format).
Data for matrix A = (a ij ) ∈ K n,n kept in three arrays
Ôº½½ ¾º
Initialization: A = sparse(m,n); A = spalloc(m,n,nnz)
A = sparse(i,j,s,m,n);
A = spdiags(B,d,m,n); A = speye(n); A = spones(S);
Ôº½¿ ¾º
double * val size nnz(A) := #{(i,j) ∈ {1, ...,n} 2 , a ij ≠ 0}
unsigned int * col_ind size nnz(A)
unsigned int * row_ptr size n + 1 & row_ptr[n + 1] = nnz(A) + 1
Example 2.6.3 (Accessing rows and columns of sparse matrices).
0
Pattern for matrix A for n = 16
nnz(A) ˆ= (number of nonzeros) of A
2
10 0
row access
column access
O(n)
Access to matrix entry a ij ≠ 0, 1 ≤ i,j ≤ n:
4
10 −1
val
val[k] = a ij
⇔
{
col_ind[k] = j ,
row_ptr[i] ≤ k < row_ptr[i + 1] ,
a ij
1 ≤ k ≤ nnz(A) .
6
8
10
access time [s]
10 −2
10 −3
10 −4
col_ind
j
12
14
10 −5
row_ptr
beginning of i-th row
16
0 2 4 6 8
nz = 32
10 12 14 16
Fig. 11
10 1 size n of sparse quadratic matrix
10 −6
10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7
Fig. 12
i
Ôº½¾ ¾º
Code 2.6.4: timing access to rows/columns of a sparse matrix
1 figure ; spy ( spdiags ( repmat([−1 2 5 ] ,16 ,1) ,[ −8 ,0 ,8] ,16 ,16) ) ;
2 t i t l e ( ’ Pattern f o r m a trix { \ b f A} f o r n = 16 ’ , ’ f o n t s i z e ’ ,14) ;
3 p r i n t −depsc2 ’ . . / PICTURES/ spdiagsmatspy . eps ’ ;
Ôº½ ¾º
5 t = [ ] ;
4