Dear Krishna, Allow me to clarify my previous statement. From Strang (intro to linear algebra, 5th edition, page 352):
When a symmetric matrix S has one of these five properties, it has them all: 1. all n pivots of S are positive. 2. all n upper left determinants are positive. 3. all n eigenvalues of S are positive. 4. x^T S x is positive except at x = 0. This is the energy-based definition. 5. S equals A^T A for a matrix A with independent columns. i.e., the statement on eigenvalue positivity and the statement on pivots are *equivalent*, i.e., containing a zero block causes the same difficulties as the issues with the spectra since one implies the other. Thanks On Sat, Feb 1, 2020 at 2:45 PM Krishnakumar Gopalakrishnan <krish...@vt.edu> wrote: > > Dear Dr David Wells, > > Thank you for the explanation. However, this only satisfies me partially > because the very next statement in that tutorial says: > > "We would have to resort to other iterative solvers instead, such as MinRes, > SymmLQ, or GMRES, that can deal with indefinite systems. However, then the > next problem immediately surfaces: due to the zero block, there are zeros on > the diagonal and none of the usual, "simple" preconditioners (Jacobi, SSOR) > will work as they require division by diagonal elements." > > > If the cause of the indefiniteness is not due to the eigen-value test, but > rather due to the simple fact the pivot elements in the lower-right block is > zero, then we have only ONE root cause of all difficulties. I am rephrasing > the issues as follows (please correct me if my understanding is incorrect) > > > "Two difficulties arise due to the zero pivots in the bottom-right block of > the matrix. > > Firstly, following a classical result from linear algebra, such matrices are > indefinite and the conjugate gradient solver cannot be applied. We would have > to resort to other iterative solvers instead, such as MinRes, SymmLQ, or > GMRES, that can deal with indefinite systems. > Secondly, due to the zero block, there are zeros on the diagonal and none of > the usual, "simple" preconditioners (Jacobi, SSOR) will work as they require > division by diagonal elements" > > > Regards, > Krishna > > On Saturday, February 1, 2020 at 7:13:18 PM UTC, David Wells wrote: >> >> Hi Krishna, >> >> This is a classic linear algebra result - a symmetric matrix is >> positive definite if and only if it has positive pivots. Since this >> matrix has a zero block it does not even have a full set of pivots so >> it cannot be positive definite. >> >> Thanks, >> David Wells >> >> On Sat, Feb 1, 2020 at 9:49 AM Krishnakumar Gopalakrishnan >> <kris...@vt.edu> wrote: >> > >> > I realize that this question is not exactly about the code/concepts behind >> > deal.II library itself, but rather about a mathematical statement from >> > step-20. >> > >> > "After assembling the linear system we are faced with the task of solving >> > it. The problem here is: the matrix has a zero block at the bottom right >> > (there is no term in the bilinear form that couples the pressure p with >> > the pressure test function q), and it is indefinite". >> > >> > My question is about the indefiniteness of the matrix. In my >> > understanding, a matrix is indefinite if it has both positive and negative >> > eigen values. However, there has not been any discussion thus far in >> > step-20 that comment about the eigen-values. How was the statement about >> > the indefiniteness then made? Can someone here explain why that matrix >> > is indefinite? >> > >> > Regards, >> > Krishna >> > >> > -- >> > The deal.II project is located at http://www.dealii.org/ >> > For mailing list/forum options, see >> > https://groups.google.com/d/forum/dealii?hl=en >> > --- >> > You received this message because you are subscribed to the Google Groups >> > "deal.II User Group" group. >> > To unsubscribe from this group and stop receiving emails from it, send an >> > email to dea...@googlegroups.com. >> > To view this discussion on the web visit >> > https://groups.google.com/d/msgid/dealii/ec88d69a-4b26-43cb-915b-b3ae1fdfa3aa%40googlegroups.com. > > -- > The deal.II project is located at http://www.dealii.org/ > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > --- > You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to dealii+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/2e309994-0d67-4809-bb28-0c07a142184e%40googlegroups.com. -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. 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