The increasing complexity of new parallel architectures has widened the gap between adaptability and efficiency of the codes. As high performance numerical libraries tend to focus more on performance, we wish to address this issue using a C++ library called NT2. By analyzing the properties of the linear algebra domain that can be extracted from numerical libraries and combining them with architectural features, we developed a generic approach to solve dense linear systems on various architectures including CPU and GPU. We have then extended our work with an example of a least squares solver based on semi-normal equations in mixed precision that cannot be found in current libraries. For the automatically generated solvers, we report performance comparisons with state-of-the-art codes, and show that it is possible to obtain a generic code with a high-level interface (similar to MATLAB) which runs either on CPU or GPU without generating a significant overhead.