The documentation page "Catalogue of dense decompositions" ("http://eigen.tuxfamily.org/dox-devel/group__TopicLinearAlgebraDecompositions.html
") tells that the JacobiSVD decomposition can give you least squares solutions to overdetermined linear systems. However, that table doesn't tell you that the FullPivHouseholderQR decomposition can also do that. The documentation for the FullPivHouseholderQR::solve function ("http://eigen.tuxfamily.org/dox-devel/classEigen_1_1FullPivHouseholderQR.html#a18fa0031972cf8575deff77d65f75453") claims that it can do that.
Make sure that the FullPivHouseholderQR::solve function indeed gives linear least squares solutions. If so, then please do the following changes to the documentation. Firstly, in the Catalogue page, in row FullPivHouseholderQR, change the "Allows to compute" column value to "Orthogonalization, least squares for overdetermined systems", and change the "Linear solver provided by Eigen" column value to "Yes (and does least squares)". Note that the "overdetermined systems" part is important, because sometimes people talk about least squares solutions for systems with more variables than equations, in which case the least squares solution means one where the solution has the least 2-norm. Then, in the documentation for FullPivHouseholderQR::solve, explain that the least squares solution means a solution where the 2-norm of (Ax-b) is minimal. This is already explained for JacobiSVD in the docs for JacobiSVD::solve ("http://eigen.tuxfamily.org/dox-devel/classEigen_1_1SVDBase.html#a34c762f9b7eeb5ffc593ac20a5efe4a4").
Yes, all QR decompositions compute the least square solution for overdetermined system. Feel free to submit a patch.
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