Are the algorithms used in mldivide direct solvers?

In numerics, it is often distinguished whether a solver is direct or iterative. We are most of the time using "mldivide" (backslash) in Matlab. Are the algorithms used by "mldivide" for any input direct solvers (not matter if (not) sparse, (not) symmetric, (not) banded etc.)?

Thank you for your help!

Answer 1

[John D'Errico]

The algorithms in mldivide are not iterative, in the sense that an iterative algorithm converges to a result, improving upon it at every iteration. Instead, MLDIVIDE uses tools for matrix factorizations. For example, depending on the specific matrix, you might find a Cholesky, a QR, an LU, etc. Even in the case of sparse matrices, the algorithms in mldivide are still not iterative.

Some other tools, such as LSQR, are indeed iterative. But LSQR is not part of mldivide.

Answer 2

[Torsten]

Are the algorithms used by "mldivide" for any input direct solvers (not matter if (not) sparse, (not) symmetric, (not) banded etc.)?

Yes.

For the algorithms used in the different cases, see

https://de.mathworks.com/help/matlab/ref/mldivide.html