Computing the singular values of a bidiagonal matrix is the fin al phase of the standard algow rithm for the singular value decomposition of a general matrix. We present a new algorithm hich computes all the singular values of a bidiagonal matrix to high relative accuracy independent of their magnitudes. In contrast, the standard algorithm for bidiagonal matrices may compute small singular values with no relative accuracy at all. Numerical experiments show that the new algorithm is comparable in speed to the standard algorithm , and frequently faster.

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 pro-portional 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.

this paper we explore the use of this paradigm in the design of numerical algorithms. We exploit the fact that there are numerical algorithms that run quickly and usually give the right answer as well as other, slower, algorithms that are always right. By "right answer" we mean that the algorithm is stable, or that it computes the exact answer for a problem that is a slight perturbation of its input [9]; this is all we can reasonably ask of most algorithms. To take advantage of the faster but occasionally unstable algorithms, we will use the following paradigm: (1) Use the fast algorithm to compute an answer; this will usually be done stably. (2) uickly and reliably assess the accuracy of the computed answer. (3) In the unlikely event the answer is not accurate enough, recompute it slowly but accurately.

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We present a Jacobi-like 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 off-diagonal 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.

We consider computing the singular value decomposition of a bidiagonal matrixB. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive de nite tridiagonal matrix. We show that if the entries of B are known with high relative accuracy, the singular values and singular vectors ofB will be determined to much higher accuracy than the standard perturbation theory suggests. We also show that the algorithm in [Demmel and Kahan] computes the singular vectors as well as the singular values to this accuracy. We also give a Hamiltonian interpretation of the algorithm and use di erential equation methods to prove many of the basic facts. The Hamiltonian approach suggests a way to use ows to predict the accumulation of error in other eigenvalue algorithms as well.

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The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on invariant subspace variations that are proportional to the reciprocals of absolute gaps between subsets of spectra or subsets of singular values. These bounds may be bad news for invariant subspaces corresponding to clustered eigenvalues or clustered singular values of much smaller magnitudes than the norms of matrices under considerations when some of these clustered eigenvalues or clustered singular values are perfectly relatively distinguishable from the rest. In this paper, we consider how eigenspaces of a Hermitian matrix A change when it is perturbed toe A = D AD and how singular values of a (nonsquare) matrix B change when it is perturbed toe B = D1 BD2, where D, D1 and D2 are assumed to be close to identity matrices of suitable dimensions, or either D1 or D2 close to some unitary matrix. It is proved that under these kinds of perturbations, the change of invariant subspaces are proportional to the reciprocals of relative gaps between subsets of spectra or subsets of singular values. We have been able to extend well-known Davis-Kahan

We consider the solution of the homogeneous equation (J \Gamma I)x = 0 where J is a tridiagonal matrix, is a known eigenvalue, and x is the unknown eigenvector corresponding to . Since the system is under-determined, x could be obtained by setting x k = 1 and solving for the rest of the elements of x. This method is not entirely new and it can be traced back to the times of Cauchy (1829). In 1958, Wilkinson demonstrated that, in finite-precision arithmetic, the computed x is highly sensitive to the choice of k; the traditional practice of setting k = 1 or k = n can lead to disastrous results. We develop algorithms to find optimal k which require a LDU and a UDL factorisation of J \Gamma I and are based on the theory developed by Fernando for general matrices. We have also discovered new formulae (valid also for more general Hessenberg matrices) for the determinant of J \Gamma øI, which give better numerical results when the shifted matrix is nearly singular. These formulae could be ...

. In this paper we present an algorithm for the eigenvalue problem of symmetric tridiagonal matrices. Our algorithm employs the determinant evaluation, split-and-merge strategy and Laguerre's iteration. The method directly evaluates eigenvalues and uses inverse iteration as an option when eigenvectors are needed. This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection and Cuppen's divide-and-conquer method. It is fully parallel, and competitive in speed with the most efficient QR algorithm in serial mode. On the other hand, our algorithm is as accurate as any standard algorithm for the symmetric tridiagonal eigenproblem and enjoys the flexibility in evaluating partial spectrum. Key words. eigenvalue, Laguerre's iteration, symmetric tridiagonal matrix 1. Introduction. For a symmetric tridiagonal matrix T with nonzero subdiagonal entries, the eigenvalues of T or the zeros of its characteristic polynomial f() = det[T \Gamma I ] (1.1) are all rea...