Hello,

While reading through chapter 19.0, Introduction to Partial Differential Equations, in NR for C++ I noticed what seems to be a contradiction.

On the pages 832 to 935 the author details how to discretize a Poisson equation (a type of elliptic PDE) and then solve it using the matrix form Ax = b.

Supposedly, the matrix for the discretized form of the Poisson equation is given in figure 19.0.3 on page 834. (PS. I believe that this matrix uses Dirichlet boundary condition of 0.) I have checked this matrix and it's determinant is negative and that all of its Eigenvalues are negative as well. This means that this matrix is "negative definite."

While the fact that the matrix is "negative definite" is interesting it bothers me that this fact seems to contradict this passage from the following page (pg 835). This passage seems to suggest that the matrix should be "possible definite":

"As we discussed at the end of 2.7, if A is symmetric and positive definite (as it is usually is in elliptic problems), the conjugate-gradient algorithm can be used. [...] For the positive definite case that arises in PDEs, an examlpe of a successful implementation is the incomplete Cholskey conjugate gradient algorithm."

To me this contradiction indicates that something is wrong or misleading. Why does NR say that matrices for elliptic PDEs are positive definite but then their example matrix for the Poisson elliptic PDE isn't positive definite?

Can anyone clarify this situation? I must be missing something.

Cheers...
-ben houston
ben@exocortex.org
http://www.exocortex.org/ben

Hi Ben,

Whether A in Ax = b is positive definite or negative definite depends on the sign convention for b. In other words, if A is negative definite you can consider a problem with -b as the source and that will be a positive definite problem. The text in NR is a little careless about this point. Basically negative definite is as good as positive definite for things like the conjugate gradient method.

Hope this helps.

Saul Teukolsky

Hi Saul,

Thank you for such a prompt reply. That clarifies things considerably.

All the best!
-ben houston
ben@exocortex.org
http://www.exocortex.org/ben