## Linear algebra in C/C++ using Eigen

I’ve recently started using the free, open-source library Eigen for linear algebra routines in C/C++. The library is absolutely fantastic in terms of perfomance and functionality, however, as is sometimes the case with open-source software, the API documentation is somewhat lacking. If I get some time in the future I hope to contribute a few articles to the online documentation. Here is an example of what I mean.

Suppose you want to do a QR decomposition of a matrix X. There are three classes that can do this for you, one of which is ColPivHouseholderQR, and you may write something like

```1 2 // perform a QR decomposition of the [n x p] X matrix; where n > p Eigen::ColPivHouseholderQR< Eigen::MatrixXd > qr = X.colPivHouseholderQr();```

Seems fairly straightforward, right? But, how do we get Q and R? If we look at the online documentation for ColPivHouseholderQR, we find a public method called matrixQR() which seems to return a reference to a matrix related to Q and R in some fashion. The documentation doesn’t state anything about how to get Q and R from the returned matrix. It turns out, that the matrix returned by matrixQR() contains both Q and R, with R occupying the ‘top’ block and Q the ‘bottom’ block. Thus, to get, say, R we need to use something like

```1 2 3 4 // perform a QR decomposition of the [n x p] X matrix; where n > p Eigen::ColPivHouseholderQR< Eigen::MatrixXd > qr = X.colPivHouseholderQr(); Eigen::Matrix<double,p,p> R; R = qr.matrixQR().topRightCorner(p,p).triangularView<Eigen::Upper>(); // R matrix```

Here, topRightCorner() extracts the matrix of interest and triangularView() ensures that R is stored as an upper triangular dense matrix.

It didn’t take too long to figure out how QR works in Eigen. After discovering the solution I found a post on the Eigen forum that describes a similar solution to the aforementioned. I urge anyone who uses Eigen to contribute code and documentation to the project, if possible. It really is a fantastic open-source library and equals and surpasses many other commercial linear algebra packages in both performance and quality.