Acceleration of Hessenberg reduction for nonsymmetric matrix.

The worth of finding a general solution for nonsymmetric eigenvalue problems is specified in many areas of engineering and science computations, such as reducing noise to have a quiet ride in automotive industrial engineering or calculating the natural frequency of a bridge in civil engineering. The main objective of this thesis is to design a hybrid algorithm (based on CPU-GPU) in order to reduce general non-symmetric matrices to Hessenberg form. A new blocks method is used to achieve great efficiency in solving eigenvalue problems and to reduce the execution time compared with the most recent related works. The GPU part of proposed algorithm is thread based with asynchrony structure (based on FFT techniques) that is able to maximize the memory usage in GPU. On a system with an Intel Core i5 CPU and NVIDA GeForce GT 635M GPU, this approach achieved 239.74 times speed up over the CPU-only case when computing the Hessenberg form of a 256 * 256 real matrix. Minimum matrix order (n), which the proposed algorithm supports, is sixteen. Therefore, supporting this matrix size is led to have the large matrix order range.

Acceleration of Hessenberg reduction for nonsymmetric matrix.

The worth of finding a general solution for nonsymmetric eigenvalue problems is specified in many areas of engineering and science computations, such as reducing noise to have a quiet ride in automotive industrial engineering or calculating the natural frequency of a bridge in civil engineering. The main objective of this thesis is to design a hybrid algorithm (based on CPU-GPU) in order to reduce general non-symmetric matrices to Hessenberg form. A new blocks method is used to achieve great efficiency in solving eigenvalue problems and to reduce the execution time compared with the most recent related works. The GPU part of proposed algorithm is thread based with asynchrony structure (based on FFT techniques) that is able to maximize the memory usage in GPU. On a system with an Intel Core i5 CPU and NVIDA GeForce GT 635M GPU, this approach achieved 239.74 times speed up over the CPU-only case when computing the Hessenberg form of a 256 * 256 real matrix. Minimum matrix order (n), which the proposed algorithm supports, is sixteen. Therefore, supporting this matrix size is led to have the large matrix order range.

An Input/Output Efficient Algorithm for Hessenberg Reduction

Reduction of an [Formula: see text] nonsymmetric matrix to Hessenberg form which takes [Formula: see text] flops and [Formula: see text] I/Os is a major performance bottleneck in the computing of its eigenvalues. Usually to improve the performance, this Hessenberg reduction is computed in two steps: the first one reduces the matrix to a banded Hessenberg form, and the second one further reduces it to Hessenberg form by incorporating more matrix-matrix operations in the computation. We analyse the two steps of the Hessenberg reduction problem on the external memory model (of Aggarwal and Vitter) for their I/O complexities. We propose and analyse a tile based algorithm for the first step of the reduction and show that it takes [Formula: see text] I/Os. For the reduction of a banded Hessenberg matrix of bandwidth [Formula: see text] to Hessenberg form, we propose an algorithm, which uses tight packing of bulges, and requires only [Formula: see text] I/Os. Combining the results of the two steps of the reduction, we show that the Hessenberg reduction can be performed in [Formula: see text] I/Os, when [Formula: see text] is sufficiently large. To the best of our knowledge, the proposed algorithm is the first I/O optimal algorithm for Hessenberg reduction; the worst case I/O complexity matches the known lower bound of the problem.

A nonsymmetrical matrix and its factorizations

We introduce a nonsymmetric matrix defined by q-integers. Explicit formulæ are derived for its LU-decomposition, the inverse matrices L−1 and U−1 and its inverse. Nonsymmetric variants of the Filbert and Lilbert matrices come out as consequences of our results for special choices of q and parameters. The approach consists of guessing the relevant quantities and proving them later by traditional means.