[VA] SRC #015 - HP-15C & clones: Big NxN Matrix Inverse & Determinant

[J-F Garnier]

(09-07-2023 07:15 PM)Valentin Albillo Wrote:  

(09-05-2023 11:19 AM)J-F Garnier Wrote:  I would like very much to have a way to compute the determinant of a complex matrix in partitioned form. But I don't see any way to do it easily.

[...] computing the absolute value (modulus) of the determinant of a complex matrix is fairly trivial using partitioned real matrices [...].

Let's have an NxN complex square matrix M = A + iB, where A, B are real square matrices holding the real/imaginary parts, respectively, of its elements. If we then construct this familiar 4-block partitioned real matrix

  M' = | A  -B |
       | B   A |

then SQR(DET(M')) is the absolute value of DET(M).

That's really interesting !
Actually, I had a idea in mind when I asked for a way to compute the determinant of a partitioned complex matrix.
The HP-71B Math ROM is internally using the same partitioned scheme to invert complex matrices or solve complex systems, and for the same reason as the 15C, doesn't have a function to compute the determinant of complex matrices.
Clearly, the 71B algorithm originated from the 15C. The algorithm then changed with the 28C/S that directly computes a LU decomposition of the complex matrix, and thus was able to compute the complex value of the determinant.

So it could have been a nice feature of the 71B Math ROM to return the absolute value of the determinant with the DETL function, following a complex matrix inversion operation or system solving. It would have cost almost nothing (computing the determinant from the LU decomposition is immediate), and may have been useful as you noted to estimate the condition number.

Thanks Valentin for all !

J-F