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Triangular linear systems are fundamental in numerical linear algebra. A triangular linear system has a straight-forward and efficient solution strategy, namely forward substitution for lower triangular systems and backward substitution for upper triangular systems. Triangular systems, or, more generally, systems of triangular type occur frequently in algorithms for more complex problems. This thesis addresses three systems that involve linear systems of triangular type. The first system concerns quasi-triangular matrices. Quasi-triangular matrices are block triangular with 1-by-1 and 2-by-2 blocks on the diagonal. Quasi-triangular systems arise in the computation of eigenvectors from the real Schur form for the non-symmetric eigenvalue problem. This thesis contributes two algorithms for the eigenvector computation, which solve shifted quasi-triangular linear systems in an efficient and scalable way. The second system addresses scaled triangular linear systems. During the solution of a triangular linear system, the entries of the solution can grow. This growth can exceed the representable range of floating-point numbers. Such an overflow can be avoided by solving a scaled triangular system. The solution is scaled prior to every operation that would otherwise result in an overflow. After scaling, the operations can be executed safely. This thesis analyzes the scalability of a recently developed tiled, robust solver for scaled triangular systems, which ensures that at no point in the computation the overflow threshold is exceeded. The third system tackles the scaled continuous-time triangular Sylvester equation, which couples two quasi-triangular matrices. The solution process is prone to overflow. This thesis contributes a robust, tiled solver and demonstrates its practicability. These three systems can be addressed with a variation of forward or backward substitution. Compared to the highly optimized and scalable implementations of standard forward and backward substitution available in HPC libraries,the existing implementations of these three systems run at a smaller fraction of the peak performance. This thesis presents techniques to improve on the performance and robustness of the implementations of the three systems.

An effective strategy in dense linear algebra is the design of algorithms as tiled algorithms. Tiled algorithms that express the bulk of the computation as matrix-matrix operations (level-3 BLAS) have proven successful in achieving high performance on cache-based architectures. At the same time, tiled algorithms interoperate with dynamic data-driven execution models such as task parallelism and promise good parallel scalability.

This thesis applies the concept of tiled algorithms and task-centric execution to algorithms related to the computation of eigenvectors for the dense, non-symmetric eigenvalue problem. First, a standard algorithm for computing eigenvectors from the Schur form is recast such that all computational steps are rich in matrix-matrix operations. Second, inverse iteration on the Hessenberg matrix as an alternative approach to computing eigenvectors is addressed. An existing algorithm is revised to express the computationally most expensive step with matrix-matrix operations. Third, a task-parallel, tiled triangular Sylvester equation solver is amended to solve a larger class of problems. All algorithms have an enhanced performance, which is demonstrated through numerical experiments.