Speeding up 2D Finite Difference Matrix
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I have two matrices B and D, where B is a 
matrix and D is a 
matrix defined as follows:
matrix and D is a matrix defined as follows:
with 
,where 
is the identity matrix and 
denotes the Kronecker product of matrices P and Q. I am currently computing B as a sparse matrix as follows:
is the identity matrix and denotes the Kronecker product of matrices P and Q. I am currently computing B as a sparse matrix as follows:o = ones(n,1);
z = zeros(n,1);
D = spdiags([-o o z],-1:1,n,n);
D(1,1) = 0;
I = speye(n);
B1 = kron(I, D);
B2 = kron(D, I);
B = [B1;B2];
I know that 
can be quickly computed using
can be quickly computed using [0; diff(x)]
I need to find a quick way of computing 
and 
in a similar manor. Currently my code is spending a signifigant amount of its time to compute and so I was wondering if anyone had some incite into computing this efficently.
and in a similar manor. Currently my code is spending a signifigant amount of its time to compute and so I was wondering if anyone had some incite into computing this efficently.采纳的回答
Chris J
2020-3-9
I solved it here is my solution:
U = reshape(x, [n,n]);
u1 = [zeros([1,n]); diff(U, 1, 1)];
u2 = [zeros(n, 1), diff(U, 1, 2)];
y = [u1(:); u2(:)];
To solved for 
we solve the equavalent formulation 
:
we solve the equavalent formulation :x1vec = x(1:end/2);
x2vec = x(end/2+1:end);
x1mat = reshape(x1vec, [n,n]);
x2mat = reshape(x2vec, [n,n]);
p1 = -diff(x1mat, 1, 1);
p1 = [-x1mat(2, :); p1(2:end,:); x1mat(end,:)];
p1 = p1(:);
p2 = -diff(x2mat, 1, 2);
p2 = [-x2mat(:,2), p2(:, 2:end), x2mat(:,end)];
p2 = p2(:);
y = p1+p2;
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