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How Does mldivide Work for Full, Square, Full Rank, Triangular Matrix?

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How does mldivide work when the input matrix A is full, square, full rank, and triangular? The doc page just say it uses a triangular solver, and I'd like more insight into what that triangular solver is.
An iterative approach can be used, e.g., for a lower triangular A
A = [1 0 0;2 3 0;4 5 6];
b = [1; 2; 3];
% A*x = b
% solve for x(i) iteratively, could be easily done with a for loop
x(1) = b(1)/A(1,1);
x(2) = (b(2) - A(2,1)*x(1)) / A(2,2);
x(3) = (b(3) - A(3,1)*x(1) - A(3,2)*x(2)) / A(3,3);
A*x(:) - b
ans = 3×1
0 0 0
Does mldivide use a solver that's more efficient and/or more numerically robust?
  13 Comments
Bruno Luong
Bruno Luong on 12 Mar 2024
Edited: Bruno Luong on 12 Mar 2024
My guess is that the warning during solving x=A\b is not alway triggered based on matrix conditioning number or matrix norm. Those are not cheap to compute.
Rather it's based first on eventual intermediate quantity computed while the linear solver algorithm does the work: smallest and largest diagonal elements of A if A is triangular form, and diagonal of R if QR decomposition is used (generic full matrix).
This code seems to indicate that
  1. substitution is used directly on T (small is compared to 1 for singularity detection) and
  2. QR is used on Tf (small is compared to norm([1; 1]) = sqrt(2) for singularity detection)
Christine Tobler
Christine Tobler on 13 Mar 2024
Yes, rcond is not computed in x = A\b if A is triangular. This is because computing the approximate rcond value would be several times more expensive than solving the linear system itself. Instead, a heuristic based on the diagonal values of A is used.

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