# What are some efficient algorithms for real-time calculation of eigenvalues?

I need to calculate eigenvalues of a square matrix for a real-time implementation using microcontrollers or dSPACE. Obviously, I can find the eigenvalues of a 2by2 matrix analytically. I am wondering if there are any efficient numerical methods/algorithms for calculating eigenvalues of square matrices of dimension greater than 2?

Any help, including introducing a paper, is appreciated.

• You can use power iteration method to extract the largest eigenvalue/eigenvector pair. Then repeat to get the next largest, and so on. In most applications, it is usually enough to get the first few largest eigenvalues - are you sure you need all of them? – Atul Ingle Jun 5 '18 at 18:34
• Thanks for your comment. I think I need all of them. Actually, I need to have the SVD. Suppose $A$ has the SVD of the form $A=USV$, then in $S$, I want to replace any singular value less than, say $\varepsilon$, by $\varepsilon$ and form $\bar S$, and then $\bar A=U\bar S V$. – Yasi Jun 5 '18 at 18:43

papers?

why not books

Golub, Gene H., and Charles F. Van Loan. Matrix computations. Vol. 3. JHU Press, 2012.

and if you need something more basic like understanding the difference between the Holder p-norm of a matrix and the Frobenious norm.

Stewart, Gilbert W. Introduction to matrix computations. Elsevier, 1973.

Although there are probably some exceptions, the symmetric square matrix is the most common in application in signal processing.

Typically, you calculate the eigenvectors and the eigenvalues follow. I'm not aware of any algorithms that just calculate the eigenvalues.

The eigenvalues are sometimes called the matrix spectrum.

It would actaully be hard to not find a numerical linear algebra library that didn't compute eigenvector and SVD of a square matrix.

A popular book for those who don't want to wade in deeply is

Press, William H., Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press, 2007.

• Yes, unless the OP has some issue or constraint that they haven't mentioned, this is a well-solved problem... – Peter K. Jun 5 '18 at 20:22