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My Matlab code has a section of code which repeatedly performs matrix multiplication and division (through the backslash operator =A\b). When testing large datasets, a certain function named solvebvp_colloc.m is called millions of times. As this function contains multiple matrix multiplications and divisions, it tends to take up a lot of runtime. I'm curious if I could rewrite solvebvp_collect.m (or the code before solvebvp_collect.m) in order to improve performance. I'm not sure if matrix decomposition will help or if I could remove one of the for loops referencing j=1:Nx completely.

My original implementation is shown below. The function solvebvp_colloc.m is harmful to the overall performance of my Matlab code.

% Test Data

N = 3;

M = 3;

% In real test scenarios, I would have larger N,M such as N=M=30

Nx = 32; % Ny=Nx=Nz

Nz = 32;

Ny = 32;

% My production data doesn't have rand(). I am only creating random

% matrices to test peformance.

Fnmhat = rand(Nx,Nz+1);

Jnmhat = rand(Nx,1);

xi_n_m_hat = rand(Nx,N+1,M+1);

Uhat = zeros(Nx,Nz+1);

Uhat_new = zeros(Nx,Nz+1);

identy = eye(Ny+1,Ny+1);

p = rand(Nx,1);

gammap = rand(Nx,1);

D = rand(Nx+1,Ny+1);

D2 = rand(Nx+1,Ny+1);

gamma = 1.5;

alpha = 0; % this could be non-zero

ntests = 150; % I'm creating a ntests variable to test running the code below multiple times. The

% ntests variable is not part of my production code (and is only used for testing performance).

tic

% Original Implementation

for ii=1:ntests

for n=0:N

for m=0:M

for j=1:Nx

Fnmhat_p = Fnmhat(j,:).';

alphaalpha = 1.0;

betabeta = 0.0; % this could be non-zero

gammagamma = gamma*gamma - p(j)^2 - 2*alpha*p(j);

d_min = 1.0;

n_min = 0.0; % this could be non-zero

r_min = xi_n_m_hat(j,n+1,m+1);

d_max = -1i*gammap(j);

n_max = 1.0;

r_max = Jnmhat(j);

% This subroutine is expensive for large N,M. The problem is that

% solvebvp_collec.m needs to run against three for loops from

% 1:N,1:M,1:Nx and can be called millions of times.

uhat_p = solvebvp_colloc(Fnmhat_p,alphaalpha,betabeta,gammagamma,...

d_min,n_min,r_min,d_max,n_max,r_max,identy,D,D2);

Uhat(j,:) = uhat_p.';

end

end

end

end

toc

where the function solvebvp_colloc is of the form

function [utilde] = solvebvp_colloc(ftilde,alpha,beta,gamma,d_min,n_min,...

r_min,d_max,n_max,r_max,identy,D,D2)

A = alpha*D2 + beta*D + gamma*identy;

b = ftilde;

A(end,:) = n_min*D(end,:);

A(end) = A(end) + d_min;

b(end) = r_min;

A(1,:) = n_max*D(1,:);

A(1,1) = A(1,1) + d_max;

b(1) = r_max;

utilde = A\b;

return;

This code is slow for large N,M (large being anything over N=M=10. I would consider N=M=30 as a resonable test case for testing large datasets.) I think that my code design has three major performance flaws including:

- The function solvebvp_colloc.m is processed a large number of times (as it needs to run for all values of the for loops 1:N, 1:M, and 1:Nx). When N and M are large (say N=M=20), the profiler shows that this function is called 20,829,312 times and takes a total runtime of 1042 seconds or 17.3 minutes. The operation utilde = A\b is expensive and calling this millions of times takes up a lot of runtime. Calling the matrix multiplication operations in solvebvp_colloc.m is also expensive.
- Running an extra for loop with j=1:Nx creates larger runtime.
- I don't use matrix decomposition. Since the matrix A changes in every iteration of j (because of the value of gammagamma is different for different values of j), I'm not sure if I can use matrix decomposition to speed up this calculation.

My initial thought was to avoid writing the additional for loop from j=1:Nx. I implemented this in a second impementation below. I wasn't able to figure out how to move the matrix A out of the for loop because the size of gammagamma is (Nx,1) and the size of the identity matrix (Nx+1,Nx+1) are different.

% Test Data

N = 3;

M = 3;

% In real test scenarios, I would have larger N,M such as N=M=30

Nx = 32; % Ny=Nx=Nz

Nz = 32;

Ny = 32;

% My production data doesn't have rand(). I am only creating random

% matrices to test peformance.

Fnmhat = rand(Nx,Nz+1);

Jnmhat = rand(Nx,1);

xi_n_m_hat = rand(Nx,N+1,M+1);

Uhat = zeros(Nx,Nz+1);

Uhat_new = zeros(Nx,Nz+1);

identy = eye(Ny+1,Ny+1);

p = rand(Nx,1);

gammap = rand(Nx,1);

D = rand(Nx+1,Ny+1);

D2 = rand(Nx+1,Ny+1);

gamma = 1.5;

alpha = 0; % this could be non-zero

ntests = 150;

% New Implementation

tic

for ii=1:ntests

for n=0:N

for m=0:M

% Moved outside of the for loop

Fnmhat_p = Fnmhat.';

alphaalpha = 1.0;

betabeta = 0.0; % this could be non-zero

gammagamma = gamma*gamma - p.^2 - 2*alpha.*p; % size (Nx,1)

d_min = 1.0;

n_min = 0.0; % this could be non-zero

r_min = xi_n_m_hat(:,n+1,m+1);

d_max = -1i.*gammap;

n_max = 1.0;

r_max = Jnmhat;

% Moved b outside of the for loop

b = Fnmhat_p;

% I don't know how to move A outside of the for loop as gammagamma

% is of size(Nx,1) and identy is of size(Ny+1,Ny+1) so writing

% A = alphaalpha*D2 + betabeta*D + gammagamma.*identy

% wouldn't work since the two matrices are of different sizes.

for j=1:Nx

uhat_p_new = solvebvp_colloc_new(b,alphaalpha,betabeta,gammagamma,...

d_min,n_min,r_min,d_max,n_max,r_max,identy,D,D2,j);

Uhat_new(j,:) = uhat_p_new.';

end

end

end

end

toc

diff = norm(Uhat - Uhat_new,inf); % is zero

where the new function solvebvp_colloc_new.m is written as

function [utilde] = solvebvp_colloc_new(b,alpha,beta,gamma,d_min,n_min,...

r_min,d_max,n_max,r_max,identy,D,D2,j)

A = alpha*D2 + beta*D + gamma(j)*identy;

A(end,:) = n_min*D(end,:);

A(end) = A(end) + d_min;

b(end,j) = r_min(j);

A(1,:) = n_max*D(1,:);

A(1,1) = A(1,1) + d_max(j);

b(1,j) = r_max(j);

utilde = A\b(:,j);

return;

A local test shows that the second implementation is "slightly faster."

Elapsed time of original implementation is 4.087457 seconds.

Elapsed time of new implementation 3.934619 seconds.

For large M,N (say N=M=20 or N=M=30) this difference would be larger. However, I wasn't able to significantly speed up the calculation because I still do a large amount of matrix multiplications and divisions inside the new function solvebvp_colloc_new.m. I'm interested in analyzing how I can speed up this calculation (I will need to test N=M=20 and N=M=30) and the profiler shows that this calculation takes up about 1/3 of the total runtime. I'm interested in analyzing the following performance improvements:

- Can Matlab matrix decomposition help here even though A changes for every value of j (since gammagamma is different for every value of j)?
- Is it possible to avoid writing the for loop from j=1:Nx in order to perform utilde = A\b less and matrix multiplication less?
- Are there any other performance improvements that would speed up this calculation (the function solvebvp_colloc.m is called millions of times for large N,M)? I think that bsxfun may help where the backslash operator becomes utilde = bsxfun(@rdivide, A, b); although this doesn't match the dimensions of the inner for loop through j.

Chunru
on 25 Jul 2021

Edited: Chunru
on 25 Jul 2021

Analyse your program to see if the matrix A is independent of which loop variables (it seems A is independent to some loop variables at the first look of the program). Compute A outside the loops of the independent variables. In inner-loop contactate the right-hand b into a matrix B. Then solve multiple linear systems with the same A in one-go: X = A \ B. This way you will only do one matrix "division" for all the independent loops (if any) and hopefully the performance can be improved.

If A is changing over all loop variable, then you may not be able to improve the performance drastically.

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