[vtkusers] Computing eigenvalues for asymmetric matrices &detecting complex conjugate

[Adriano Gagliardi]

Alexis,
 
I couldn't find such functionality in VTK. However, you can get the roots
for your 3x3 matrix directly. Theory here (although you may know this
already):
http://en.wikipedia.org/wiki/Eigenvalue_algorithm#Eigenvalues_of_3.C3.973_ma
trices. It is what I used in the end for l2.
 
Regards,
 
Adriano
 
===================================

Adriano Gagliardi MEng PhD
Business Sector Leader
Computational Aerodynamics
Aircraft Research Association Ltd.
Manton Lane
Bedford

Tel: 01234 32 4644
E-mail: agagliardi at ara.co.uk
Url: www.ara.co.uk 
 

  _____  

From: vtkusers-bounces at vtk.org [mailto:vtkusers-bounces at vtk.org] On Behalf
Of Alexis Chan
Sent: 06 April 2011 20:32
To: vtkusers at vtk.org; vis-contest-2011 at freelists.org
Subject: [vtkusers] Computing eigenvalues for asymmetric matrices &detecting
complex conjugate


Hi
I am trying to compute the eigenvalue of S^2 + Omega^2 where S: strain-rate
tensor and Omega:spin tensor [1]

Question 1:

Since vtkMath::Jacobi computes the eigenvalues for a symmetric matrix, does
that mean that I have compute the eigenvalues for S separately from the
eigenvalues of Omega?

Is there a way to compute eigenvalues of asymmetric matrices in VTK ?
Should I run the Matlab eig function using vtkMatlabEngineInterface instead?

Here's how I compute S^2+Omega^2 in my modification of vtkCellDerivatives:

else if(this->TensorMode == VTK_TENSOR_MODE_COMPUTE_LAMBDA2)// This is not
actually Lambda2, but the tensor used for calculating lambda2
          {
          tens->SetComponent(0,0, derivs[0]*derivs[0] +
0.25*derivs[1]*derivs[1] + 0.25*derivs[3]*derivs[3] +
0.25*derivs[2]*derivs[2] + 0.25*derivs[6]*derivs[6]);
tens->SetComponent(0,1, 0.5*derivs[0]*derivs[1] + 0.5*derivs[4]*derivs[1] +
0.25*derivs[2]*derivs[5] + 0.25*derivs[6]*derivs[7]);
          tens->SetComponent(0,2, 0.5*derivs[0]*derivs[2] +
0.25*derivs[1]*derivs[5] + 0.25*derivs[3]*derivs[7] +
0.5*derivs[8]*derivs[1]);
          tens->SetComponent(1,0, 0.5*derivs[0]*derivs[1] +
0.5*derivs[4]*derivs[1] + 0.25*derivs[5]*derivs[2] +
0.25*derivs[7]*derivs[6]);
          tens->SetComponent(1,1, 0.25*derivs[1]*derivs[1] +
0.25*derivs[3]*derivs[3] + 1*derivs[4]*derivs[4] + 0.25*derivs[5]*derivs[5]
+ 0.25*derivs[7]*derivs[7]);
          tens->SetComponent(1,2, 0.25*derivs[1]*derivs[2] +
0.25*derivs[3]*derivs[6] + 0.5*derivs[4]*derivs[5] +
0.5*derivs[8]*derivs[5]);
          tens->SetComponent(2,0, 0.5*derivs[1]*derivs[2] +
0.25*derivs[5]*derivs[1] + 0.25*derivs[7]*derivs[5] +
0.5*derivs[8]*derivs[2]);
          tens->SetComponent(2,1, 0.25*derivs[2]*derivs[1] +
0.25*derivs[6]*derivs[5] + 0.5*derivs[4]*derivs[5] +
0.5*derivs[8]*derivs[5]);
          tens->SetComponent(2,2, 0.25*derivs[2]*derivs[2] +
0.25*derivs[6]*derivs[6] + 0.25*derivs[5]*derivs[5] +
0.25*derivs[7]*derivs[7] + 1*derivs[8]*derivs[8]);
          
          outTensors->InsertTuple(cellId, tens->T);
          }


full file here:
https://github.com/alexisylchan/VTK/blob/master/Graphics/myVTKCellDerivative
s.cxx

Question2:

How do I determine if the eigenvalue returned by vtkMath::Jacobi is a real
or complex-conjugate value?

I would appreciate any help! Thanks.

[1] Jinhee Jeong and Fazle Hussain. On the Identification of a Vortex.
<http://journals.cambridge.org/download.php?file=%2FFLM%2FFLM285%2FS00221120
95000462a.pdf&code=b89c005d1fed041d4786ecfec8f757c2> Journal of Fluid
Mechanics, pages 69-94, 285 1995 
[2]  <http://citeseer.ist.psu.edu/viewdoc/summary?cid=543964> Sujudi,D., and
Haines,R., "Identification of Swirling Flow in 3-D Vector Fields", Proc.
AIAA (Am. Inst. of Aeronautics and Astronautics) Computational Fluid
Dynamics Conf., American Institute of Aeronautics and Astronautics, (Reston,
Va., June 1995, pp. 151-158. 
-- 
Regards,
Alexis



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