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Results tagged with jacobian
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Questions related to the calculation or use of the Jacobian matrix or its determinant. Not to be confused with the Jacobi iterative method for solving systems of linear equations. For those, use [iterative-method] instead.
7
votes
Accepted
Sanity checking jacobians for Finite Element code
For example:
As @BlaisB suggestes, if you rotate the reference cell, the determinant of the Jacobian needs to be one. You don't need to know the exact Jacobian matrix. … If you scale the cell by a factor, the Jacobian needs to be a multiple of the identity matrix. The multiplication factor is simply the scaling factor. …
- 56.8k
3
votes
How to calculate/derive analytic FEM Newton Jacobian
You need to understand how to actually compute derivatives when you want to take the derivative with respect to a function. I've recorded a lengthy example in lecture 31.55 here: http://www.math.colos …
- 56.8k
6
votes
Accepted
Why can bad jacobians sometimes works better for implicit ODE method?
Finite difference approximations of the Jacobian are really only good if the step lengths are chosen appropriately for each coordinate. … You are almost always better off if you provide an approximation of the Jacobian based on expert insight, rather than black box choices. …
- 56.8k
3
votes
How to deal with complexity in numerical code, for example, when dealing with large Jacobian...
Here is an example of where we have used automatic differentiation using Sacado in one code:
http://www.dealii.org/developer/doxygen/deal.II/step_33.html
- 56.8k
0
votes
Poisson-Nernst-Planck equations with ill-conditioned sparse matrix
If you write your system as a single matrix, you may need to scale the two equations so that they have roughly the same order of magnitude. Rather than describing in great detail what to do and why, l …
- 56.8k
5
votes
Spot redundant equations within nonlinear system of equations
Each of the two equations describes a set of lines in the 2d plane, and the lines happen to be tangential at a specific point -- which is expressed as the Jacobian computed at that point being singular …
- 56.8k