Symmetric Matrix-Vector Multiplication with only lower triangular stored
21 views (last 30 days)
Show older comments
I have a very big (n>1000000) sparse symmetric positive definite matrix A and I want to multiply it efficiently by a vector x. Only the lower triangular of matrix A is stored with function "sparse". Is there any function inbuilt or external that anyone knows of that takes as input only the lower triangular and performs the complete operation Ax without having to recover whole A (adding the strict upper triangular again)?
Thank you very much,
Jan
0 Comments
Accepted Answer
Jan
on 24 Apr 2021
Edited: Jan
on 24 Apr 2021
Plese check if this is efficient in your case:
A = rand(5, 5);
A = A + A'; % Symmetric example matrix
B = tril(A); % Left triangular part
x = rand(5, 1);
y1 = A * x;
y2 = B * x + B.' * x - diag(B) .* x;
y1 - y2
I assume that Matlab's JIT can handle B.' * x without computing B.' explicitly. Alternative:
y2 = B * x + B.' * x - diag(diag(B)) * x;
More Answers (1)
Clayton Gotberg
on 24 Apr 2021
If
is a symmetric matrix and
is the lower triangular part of the matrix and
is the upper triangular part of the matrix:




where the diagonal function only finds the diagonal elements of
. This is because of a few relations:



To save time and space on MATLAB (because the upper triangular matrix will take up much more space), take advantage of the relations:


To get:

Now, the MATLAB calculation is
A_times_x = A_LT*x+(x.'*A_LT).'+ diag(A_LT).*x;
This should only perform transposes on the smaller resultant matrices.
7 Comments
Clayton Gotberg
on 24 Apr 2021
I begin to understand why engineers are specifically trained and hired to test software. It's deeply interesting to get this peek into what MATLAB does to ensure I can keep writing ill-considered code.
See Also
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!