Algorithm 467: matrix transposition in place [F1]

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n 3) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report [Van Barel, Vandebril, Mastronardi 2003] a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method.

2013

The orthogonal qd algorithm with shifts (oqds algorithm), proposed by von Matt, is an algorithm for computing the singular values of bidiagonal matrices. This algorithm is accurate in terms of relative error, and it is also applicable to general triangular matrices. In particular, for lower tridiagonal matrices, BLAS Level 2.5 routines are available in preprocessing stage for this algorithm. BLAS Level 2.5 routines are faster than BLAS Level 2 routines widely used in preprocessing for bidiagonalization. Generally, it takes O(n 3 ) operations to reduce a full n-by-n matrix to a band matrix such as bidiagonal or lower tridiagonal matrix. On the other hand, computing the singular values of a bidiagonal or lower tridiagonal matrices takes only O(n 2 ) operations. Consequently, if we have an algorithm for computing the singular values of lower tridiagonal matrices, we can expect that the total computation time including preprocessing to obtain the singular values is reduced. In this paper, we consider the oqds algorithm for lower tridiagonal matrices. We propose a shift strategy for lower tridiagonal matrices to accelerate convergence and derive criteria for deflation or splitting.

Report TW

Applied and Computational Harmonic Analysis, 2013

We propose a new fast algorithm for computing the spectrum of an N × N symmetric tridiagonal matrix in O (N ln N) operations. Such an algorithm may be combined with any of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues. The underlying technique is a divide-andconquer approach which determines eigenvalues of a larger tridiagonal matrix from those of constituent matrices by the use of relations of their characteristic polynomials. The evaluation of characteristic polynomials is accelerated by the use of a technique known as the fast multipole method. An implementation of the algorithm has been developed in Fortran, providing for a comparison with existing techniques in terms of running time and accuracy. We present numerical results which demonstrate the effectiveness of the method.

Linear algebra and its applications, 1995

We present a new, fast, and practical parallel algorithm for computing a few eigenvalues of a symmetric tridiagonal matrix by the explicit QR method. We present a new divide and conquer parallel algorithm which is fast and numerically stable. The algorithm is work efficient and of low communication overhead, and it can be used to solve very large problems infeasible by sequential methods.

arXiv (Cornell University), 2022

In the current paper the authors linked two methods in order to evaluate general n-th order tridiagonal determinants. A breakdown free numerical algorithm is developed for computing the inverse of any n×n general nonsingular tridiagonal matrix without imposing any constrains. The algorithm is suited for implementation using any computer language such as FORTRAN, PYTHON, MATLAB, MAPLE, C, C++, MACSYMA, ALGOL, PASCAL and JAVA. Some illustrative examples are presented.

IEEE Transactions on Computers, 1977

We show that if the size of the tridiagonal matrix in any given iteration is n, then the parallel QR algorithm requires 0(log2n) steps with 0(n) processors per iteration and no square roots. This results in a speedup of 0(n/log2n) over the sequential algorithm with an efficiency of 0(1/log2n). We also give an error analysis of the parallel triangular system

Proceedings Euromicro Workshop on Parallel and Distributed Processing

I n this paper we propose a parallel organization of the Q R algorithm for computing the complete eigensystem of symmetric matrices. W e developed Occam versions of standard sequential implementations of the Q R algorithm: the procedure qrl which computes only eigenvalues and qr2 for the computation of all eigenvalues and eigenvectors. The Occam procedure parqr2 is a parallel implementation of qr2 and was tested on a pipeline of 16 transputers. Although parqr2 could be used to compute the eigenvalues and eigenvectors of a symmetric tridiagonal matrix, it is best suited to be used in conjunction with a parallel algorithm for the reduction of a dense symmetric matrix to tridiagonal f o r m where the orthogonal transformations are accumulated in an explicit way. We have proposed a parallel algorithm for this purpose in 1151. In the pratical tests parqr2 has proved to be efficient and we have carried out a simple analysis that appears to indicate that it is possible to use efficiently a number p of processors of the same order of magnitude of the size n o f t h e matrix (p 5 n/6). This is an interesting result from the point of view of the scalability of our parallel algorithm.

SIAM Journal on Scientific and Statistical Computing, 1987

A multiprocessor algorithm for finding few or all eigenvalues and the corresponding eigenvectors of a symmetric tridiagonal matrix is presented. It is a pipelined variation of EISPACK routinesmBISECT and TINVIT which consists ofthe three steps: isolation, extraction-inverse iteration, and partial orthogonalization. Multisections are performed for isolating the eigenvalues in a given interval, while bisection or the Zeroin method is used to extract these isolated eigenvalues. After the corresponding eigenvectors have been computed by inverse iteration, the modified Gram-Schmidt method is used to orthogonalize certain groups of these vectors. Experiments on the Alliant FX/8 and CRAY X-MP/48 multiprocessors show that this algorithm achieves high speed-up over BISECT and TINVIT; in fact it is much faster than TQL2 when all the eigenvalues and eigenvectors are required.

2009

In this paper two fast algorithms that use orthogonal similarity transformations to convert a symmetric rationally generated Toeplitz matrix to tridiagonal form are developed, as a means of finding the eigenvalues of the matrix efficiently. The reduction algorithms achieve cost efficiency by exploiting the rank structure of the input Toeplitz matrix. The proposed algorithms differ in the choice of the generator set for the rank structure of the input Toeplitz matrix.