Henrik Sönnerlind
COMSOL Employee
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Posted:
6 years ago
06.08.2018, 07:47 GMT-4
Hi Sarah,
This is used if your eigenvalue problem is nonlinear in the eigenvalue itself. Think about a simple standared eigenvalue problem like
![](data:image/png;base64,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)
If the matrix A itself depends on the eigenvalue, that is
![](data:image/png;base64,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)
then the problem is linearized using
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKIAAAAXCAQAAAAi/FxWAAAAAmJLR0QA/4ePzL8AAAAJcEhZcwAAAHgAAAB4AJ31WmAAAAAHdElNRQfoBhgKHRNaaxGLAAADfklEQVRYw+2Z3ZGjOhCFv97aBEiBFDQhaELAIcghsCEwIeAQ7BBwCCaEIQQTQt8H8yOBMGY841u1ta0XAzo6rUOrpcai/LNn7df/7cDfYIGI4sRsg29DSC7Jz0/pNSy+/fbILYnWM5cMFW/aRN0NEGJwnPR8h+1Awf5JiSYsYrHe40YP38Gy6kMbKKJdI+GozBslShF9MkPgUNJY36GHjY+1pYUspFhyFKXEYr6LZZE9o8CSUXo+eGLZqFBXlM8FeWcIruQrTlT3ZX5oIhMWLBq+6u9giTKnvRYYLv3dLidKgo0uREcLpJLNQjqOOJFx3w7kT6+n17DELOd0+6E19Kr0G0vWP5yYY0cLEafjiDNG0nte6LoA6/Yalpj5o7Z9Nu5FtNRzhFgarTkB2czpKEJPtKvu12J5yrawiJPq1gCkGH9/yRL81ZeGIqa0Ecgt2g4AuMmzOAJOs55Ta9h4kHqGRQ/6Dlhuu+mZlL2+h13FiI22iZ+Tl1/THaX6I46Z5zdJyHQPWkuNIeNP8NgsHGbOODFas2wtz5/jtrHsqHBSc6JkFzmuLUf1PQb6YPh9p4vrYhAOlKSS6YkVE4OhIbuRS0LOGcuHtndRyUJstnGZQpZ101b2VJQ4drER9c9j48yseUTEtsseSXe9IqIYCnYwRO2RnbZSU/F2F5gQz5J1NFNPWR6RsZY9x8UUtEW0Mb7NqohiacbcISUOK2m8chkmd+RNWzmRi9FaDGgL2kp7f+lp87gcU5aHBbCcsRzlfb4qJF9KMGGMaiNgvFBqQxEbmZ76xsUMcMABzpvsBCGGindtQWtpcOyD6PL38uTRZRiVcMqyZAGLFKDvUuKiJeH54Sztr8Vxp+7O30ey+MncqwGUq3cdIDBcccNVwVWhoBqwYTVhvlgvRFjiFUvIgrsVqCRc0GdKQuxQsaRcSYKyj3wcGsMFRUeHcVR8oiiXQZg8EOYSXKUo2aKI1y9PIcbiqDp/P32eQeCiu2G6ERSlGl/FZh9yShLS4CUNDl02DhYgprGFISXzRByilozyyxOIsCz0fILlgZnnuD4KPRGXPkDcHWwFQTKE/udI6X/9+Ln2Gpa+jR9lP1ZrgKmtILTlQwpJpRjPibMvcT9ir2Hxpzq+Pbc1U6wjSLF+TFD6y+AH4/AFLB5fcJFt3Te3IchfIuFLWPwm//7te97+A5pcnSAnnrNPAAAALXRFWHRpY2M6Y29weXJpZ2h0AENvcHlyaWdodCBBcnRpZmV4IFNvZnR3YXJlIDIwMTEIusW0AAAAMXRFWHRpY2M6ZGVzY3JpcHRpb24AQXJ0aWZleCBTb2Z0d2FyZSBzUkdCIElDQyBQcm9maWxlEwwBhgAAACF0RVh0cHM6SGlSZXNCb3VuZGluZ0JveAA5N3gxNCsyNTcrNjM2JBgAnAAAAB50RVh0cHM6TGV2ZWwAUFMtQWRvYmUtMi4wIEVQU0YtMi4wQfkzEwAAAABJRU5ErkJggg==)
where
is the value you supply as Value of eigenvalue linearization point.
This means that you can expect accurate eigenvalues only in the vicinity of
, and may have to do several analyses with different values of the the linearization point.
Regards,
Henrik
-------------------
Henrik Sönnerlind
COMSOL
Hi Sarah,
This is used if your eigenvalue problem is nonlinear in the eigenvalue itself. Think about a simple standared eigenvalue problem like
( \mathbf A-\lambda \mathbf I) \mathbf x = 0
If the matrix **A** itself depends on the eigenvalue, that is
( \mathbf A(\lambda)-\lambda \mathbf I) \mathbf x = 0
then the problem is linearized using
( \mathbf A(\lambda_0)-\lambda \mathbf I) \mathbf x = 0
where \lambda_0 is the value you supply as _Value of eigenvalue linearization point_.
This means that you can expect accurate eigenvalues only in the vicinity of \lambda_0, and may have to do several analyses with different values of the the linearization point.
Regards,
Henrik