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

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Hamza Shafiq 2021-9-21

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https://ww2.mathworks.cn/matlabcentral/answers/1457564-unable-to-grasp-the-concept-of-sparse-matrix-kindly-help-me-understand-the-following-code

回答： Arun 2024-2-23

% 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]; 

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Answer 1

Arun 2024-2-23

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https://ww2.mathworks.cn/matlabcentral/answers/1457564-unable-to-grasp-the-concept-of-sparse-matrix-kindly-help-me-understand-the-following-code#answer_1415168

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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