# Solve large linear equations with backslash operator. Why is sparse matrix solved slower than the full matrix?

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Hello Community and Staff,

maybe somebody could explain to me why sparse matrix operation for a not so well condidioned matrix is maybe 2-5 times slower than the backslash operator.

And maybe somebody have an idea how to improve the speed.

% This script measure the time for solving large number of equations

% N is the number of N-1 chebyshev collocations points

% s is the size of the matrix and vector

% indices with an s denotes that tha variable is sparse

% rf is the ratio of numbers that are zero and the size s for vector f

% rA is the ratio of numbers that are zero and the size s for matrix A

% rA is approx.: 0.0255 for N=32

% 0.0175 for N=48

% 0.0133 for N=64

N = 48;

s = 2*(N+1)^2+(N-1)^2;

rf = (2*((N+1)^2-(N-1)^2)/s);

rA = 0.0175; %this variabel has to be set

f = rand(s,1);

f(f<(1-rf))=0;

fs = sparse(f);

A = rand(s,s);

A(A<1-rA) = 0;

As = sparse(A);

whos A* f*

disp([' solving q1 = A\f take ',num2str(timeit(@() A\f)),' seconds'])

disp([' solving q2 = A\fs take ',num2str(timeit(@() A\fs)),' seconds'])

disp([' solving q3 = As\f take ',num2str(timeit(@() As\f)),' seconds'])

disp([' solving q4 = As\fs take ',num2str(timeit(@() As\fs)),' seconds'])

I get these results:

Name Size Bytes Class Attributes

A 7011x7011 393232968 double

As 7011x7011 13821760 double sparse

f 7011x1 56088 double

fs 7011x1 6176 double sparse

solving q1 = A\f take 4.7096 seconds

solving q2 = A\fs take 4.7589 seconds

solving q3 = As\f take 12.0466 seconds

solving q4 = As\fs take 11.1677 seconds

something is also curious:

how much memory each "disp..." line need for solving:

q1 need approx. 0.37GB

q2 need approx. 0.35GB

q3 need approx. 2.79GB

q4 need approx. 1.43GB

------------------------------------------------------

Why does it take so much memory and time for solving A sparse matrix instead of solving a non-sparse matrix.

Maybe somebody could reproduce these results.

Any help would be appreciated.

### Answers (1)

Christine Tobler
on 22 Feb 2021

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