Unable to grasp the concept of Sparse Matrix - Kindly help me understand the following code.

% Sparse Matrix Indexes: S(i,j)=0 if dF(i)/dy(j)=0, else S(i,j)=1

diagonal = diag(ones(bins,1));
dcidci = diag(ones(bins,1),0)+diag(ones(bins-1,1),-1)+diag(ones(bins-1,1),+1);
dcidcj = diag(ones(bins,1))+diag(ones(bins-1,1),-1)+diag(ones(bins-2,1),-2);
dcdc = [dcidci dcidci dcidci; dcidci dcidci dcidci; dcidci dcidci dcidci];
dcdq = diag(ones(nspecies*bins,1));
dcdT = repmat(dcidci,[nspecies,1]);
dqdq = diag(ones(nspecies*bins,1));
dqdc = repmat(diagonal,nspecies);
dqdT = repmat(diagonal,[nspecies,1])
dTdc = repmat(dcidci,[1,nspecies]);
dTdq = repmat(diagonal,[1,nspecies]); 
dTdT = dcidci; 
S = [dcdc dcdq dcdT; dqdc dqdq dqdT; dTdc dTdq dTdT]; 

Answer 1

[Arun]

Hi Hamza,

I understand that you want to some explanation about “sparse” matrix and the implementation of “sparse” matrix in the shared code using MATLAB.

A matrix is a two-dimensional data object, that represent data in the form of rows and columns. A “sparse” matrix is one where most of its elements are zero. Sparse matrix can be stored using less memory and require less computation time. MATLAB provides “sparse” function to convert a full matrix into sparse form by squeezing out any zero element.

The shared code utilizes mainly two functions, “diag” and “repmat”. The “diag” function, creates a diagonal matrix. The “repmat” function helps to create a matrix by repeating a given matrix in a specified number of times in both the row and column directions.

Since the code shared is missing some values, let us assume “bins” = 4 and “nspecies” = 3. I have removed commas to display the output in order to provide a better understanding of the code:

bins = 4;
nspecies = 3;
diagonal = diag(ones(bins,1))

diagonal = 4×4

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

dcidci = diag(ones(bins,1),0)+diag(ones(bins-1,1),-1)+diag(ones(bins-1,1),+1)

dcidci = 4×4

1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1

dcidcj = diag(ones(bins,1))+diag(ones(bins-1,1),-1)+diag(ones(bins-2,1),-2);
dcdc = [dcidci dcidci dcidci; dcidci dcidci dcidci; dcidci dcidci dcidci];
dcdq = diag(ones(nspecies*bins,1));
dcdT = repmat(dcidci,[nspecies,1])

dcdT = 12×4

1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0

dqdq = diag(ones(nspecies*bins,1));
dqdc = repmat(diagonal,nspecies);
dqdT = repmat(diagonal,[nspecies,1]);
dTdc = repmat(dcidci,[1,nspecies]);
dTdq = repmat(diagonal,[1,nspecies]);
dTdT = dcidci;
S = [dcdc dcdq dcdT; dqdc dqdq dqdT; dTdc dTdq dTdT]

S = 28×28

1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0

% Find the number of ones
numOnes = sum(S(:) == 1)

numOnes = 244

% Find the number of zeros
numZeros = sum(S(:) == 0)

numZeros = 540

As we can see that more that 68% of entries are zero element, “S” is an example of “sparse” matrix.

For more information, please refer to the shared documentation links:

1. “sparse” function: https://www.mathworks.com/help/matlab/ref/sparse.html
2. “diag” function: https://www.mathworks.com/help/matlab/ref/diag.html
3. “repmat” function: https://in.mathworks.com/help/matlab/ref/repmat.html

I hope this helps.

HTH