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

3 visualizzazioni (ultimi 30 giorni)
Paul il 9 Mar 2024
Commentato: Christine Tobler il 13 Mar 2024
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 CommentiMostra 11 commenti meno recentiNascondi 11 commenti meno recenti
Bruno Luong il 12 Mar 2024
Modificato: Bruno Luong il 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 il 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.

Accedi per commentare.

### Categorie

Scopri di più su Linear Algebra in Help Center e File Exchange

R2023b

### Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by