Results 1 
4 of
4
Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices
, 1980
"... When computing eigenvalues of sym metric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound proportional to the product of machine precision and the norm of the matrix. In particular, we do not expect to comp ..."
Abstract

Cited by 83 (14 self)
 Add to MetaCart
When computing eigenvalues of sym metric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound proportional to the product of machine precision and the norm of the matrix. In particular, we do not expect to compute tiny eigenvalues and singular values to high relative accuracy. There are some important classes of matrices where we can do much better, including bidiagonal matrices, scaled diagonally dominant matrices, and scaled diagonally dominant definite pencils. These classes include many graded matrices, and all sym metric positive definite matrices which can be consistently ordered (and thus all symmetric positive definite tridiagonal matrices). In particular, the singular values and eigenvalues are determined to high relative precision independent of their magnitudes, and there are algorithms to compute them this accurately. The eigenvectors are also determined more accurately than for general matrices, and may be computed more accurately as well. This work extends results of Kahan and Demmel for bidiagonal and tridiagonal matrices.
Numerical Methods for Simultaneous Diagonalization
 SIAM J. Matrix Anal. Applicat
, 1993
"... We present a Jacobilike algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations. The algorithm uses a sequence of similarity transformations by elementary complex rotations to drive the offdiagonal entries to zero. We show th ..."
Abstract

Cited by 42 (0 self)
 Add to MetaCart
We present a Jacobilike algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations. The algorithm uses a sequence of similarity transformations by elementary complex rotations to drive the offdiagonal entries to zero. We show that its asymptotic convergence rate is quadratic and that it is numerically stable. It preserves the special structure of real matrices, quaternion matrices and real symmetric matrices.
P.J.: Singular value decomposition on gpu using cuda
 In: IPDPS ’09: Proceedings of the 2009 IEEE International Symposium on Parallel & Distributed Processing
, 2009
"... Linear algebra algorithms are fundamental to many computing applications. Modern GPUs are suited for many general purpose processing tasks and have emerged as inexpensive high performance coprocessors due to their tremendous computing power. In this paper, we present the implementation of singular ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Linear algebra algorithms are fundamental to many computing applications. Modern GPUs are suited for many general purpose processing tasks and have emerged as inexpensive high performance coprocessors due to their tremendous computing power. In this paper, we present the implementation of singular value decomposition (SVD) of a dense matrix on GPU using the CUDA programming model. SVD is implemented using the twin steps of bidiagonalization followed by diagonalization. It has not been implemented on the GPU before. Bidiagonalization is implemented using a series of Householder transformations which map well to BLAS operations. Diagonalization is performed by applying the implicitly shifted QR algorithm. Our complete SVD implementation outperforms the MATLAB and Intel R○Math Kernel Library (MKL) LAPACK implementation significantly on the CPU. We show a speedup of upto 60 over the MATLAB implementation and upto 8 over the Intel MKL implementation on a Intel Dual Core 2.66GHz PC on NVIDIA GTX 280 for large matrices. We also give results for very large matrices on NVIDIA Tesla S1070. 1.