Applied Numerical Methods Using MATLAB, 2nd Edition

Appendix HSparse Matrices

A matrix is said to be sparse if it has a large portion of zero elements. MATLAB has some built‐in functions/routines, which enables us to exploit the sparsity of a matrix for computational efficiency.

• The MATLAB function ‘ sparse()’ can be used to convert a (regular) matrix into a sparse form by squeezing out any zero elements and to generate a sparse matrix having the elements of a vector given together with the row/column index vectors. On the other hand, the MATLAB function ‘ full()’ can be used to convert a matrix of sparse form into a regular one.

 >row_index=[1 1 2 3 4]; col_index=[1 2 2 3 4]; elements=[1 2 3 4 5];  
 >m=4; n=4; As=sparse(row_index,col_index,elements,m,n) 
 As= (1,1) 1 
 (1,2) 2 
 (2,2) 3 
 (3,3) 4 
 (4,4) 5 
 >Af=full(As) 
 Af= 1 2 0 0 
 0 3 0 0 
 0 0 4 0 
 0 0 0 5  

We can use the MATLAB function ‘ sprandn(m,n,nzd)’ to generate an m× n sparse matrix having the given nonzero density nzd. Let us see how efficient the operations can be on the matrices in sparse forms.

 >As=sprandn(10,10,0.2); % A sparse matrix and  
 >Af=full(As); % Its full version 
 >AsA=As*As; % in sparse forms (50 flops) 
 >AfA=Af*Af; % in full(regular) forms (2000 flops) 
 >b=ones(10,1);  
 >x=As\b; % in sparse forms (160 flops) 
 >x=Af\b; % in full(regular) forms (592 flops) 
 >inv(As); % in sparse forms (207 flops) 
 >inv(Af); % in full(regular) forms (592 flops) 
 >[L,U,P]=lu(As); % in sparse forms (53 flops) 
 >f[L,U,P]=lu(Af); % in full(regular) forms (92 flops)