I simply edited the previous patch file and removed all the sections doing more than #pragma insertions. I'll do better investigation in future patches.

(file #27620)

Your patch does more than add #pragmas.  For example, I see changes like the following:

@@ -2021,15 +2027,13 @@
                             {
                               Complex tmp = work[k] / data (cidx (k+1)-1);
                               work[k] = tmp;
-                              for (octave_idx_type i = cidx (k);
-                                   i < cidx (k+1)-1; i++)
-                                {
-                                  octave_idx_type iidx = ridx (i);
-                                  work[iidx] = work[iidx] - tmp * data (i);
-                                }
+                              #pragma omp parallel for
+                              for (octave_idx_type i = cidx (k); i < cidx (k+1)-1; i++)
+                                work[ridx (i)] -= tmp * data (i);
                             }

Could you please undo these kinds of changes and make the patch add #pragmas only?  That will be much easier to evaluate.

I'm not opposed to the other kinds of changes, but let's make them separately.