Twisted Factorizations and qd-Type Transformations for the $\mbox{MR}^{3}$ Algorithm---New Representations and Analysis

Willems, Paul R.; Lang, Bruno

doi:10.1137/110834044

Cited by 5 publications

(15 citation statements)

References 12 publications

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“…Starting with the corresponding canonical vector on the right-hand side, one step of inverse iteration will deliver a residual norm satisfying (3.5), which is at most a factor √ n away from being optimal [19]. For details about twisted factorizations and new variants of the dqds transformations relating them in the MR 3 algorithm we refer the reader to [9,29,39].…”

Section: The Ideamentioning

confidence: 99%

“…This is fulfilled, for example, when using twisted factorizations combined with qd-transformations, as described in [9]. Improved variants of these techniques and a completely new approach based on block decompositions are presented in [38,39].…”

Section: Requirement 4: Shiftrel (Shift Relation)mentioning

confidence: 99%

“…The requirements declare a small set of constants to control the error bounds globally; an overview of them is given in Table 4.1, together with reasonable assumptions about their sizes for an average test case. Bounds for α ↓ and α ↑ can be obtained from mixed error analyses of the transformations [38,39]; the others arise from experience.…”

Section: Requirement 5: Getvec (Computation Of Eigenvectors)mentioning

confidence: 99%

“…• Shiftrel is fulfilled if the routine to compute the child representation in line 12 is componentwise mixed relatively stable. This goal can be achieved, e.g., with twisted factorizations or block factorizations [9,38,39]. • Getvec is not always easy to achieve, but given a faithful residual norm for the vector, fulfillment of the requirement can be monitored during runtime.…”

Section: How To Achieve the Requirementsmentioning

confidence: 99%

“…This was done by deflecting small pivots from zero using a suitable threshold pivmin, or alternatively by exploiting IEEE 754 arithmetic; cf. [9,39]. The new cause for loss of relative accuracy does not concern the pivots, but rather is due to underflow during a sequence of multiplications and divisions.…”

Section: Outline Of Our Implementation Xmrmentioning

confidence: 99%

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A Framework for the $MR^3$ Algorithm: Theory and Implementation

Willems¹,

Lang²

2013

SIAM J. Sci. Comput.

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This paper provides a streamlined and modular presentation of the MR 3 algorithm for computing selected eigenpairs of symmetric tridiagonal matrices, thus disentangling the principles driving MR 3 and the (recursive) "core" algorithm from the specific (e.g., twisted) decompositions used to represent the matrices at different recursion depths and from the (dqds) transformations converting between them. Our approach allows a modular full proof for the correctness of the MR 3 algorithm. This proof is based on five requirements concerning the interplay between the core algorithm and its subcomponents. These requirements can also guide in implementing the algorithm, because they expose quantities that can and should be monitored at runtime. Our new implementation XMR, which is based on the above analysis, is described and compared to xSTEMR from Lapack 3.2.2. Numerical experiments comparing the robustness and performance of both implementations are given.1. Introduction. The established strategy for solving a dense real symmetric or complex Hermitian eigenproblem numerically is to reduce the matrix to real tridiagonal form by an orthogonal similarity transformation [1,16] and then solve the tridiagonal problem. Before there was MR 3 , usually one of the following three standard algorithms was employed for the second part.QR iteration has been the workhorse since the 1960's [14,15]. The current implementation xSTEQR in Lapack [1] is rock solid. At the time of this writing, the best method for computing all eigenvalues is the dqds algorithm [13,31], which can be considered to be a specially tailored QR iteration. But for computing eigenvectors, QR exhibits a true O(n 3 ) complexity, making it very slow in practice. In fact, for the dense symmetric problem, the solution of the tridiagonal problem using QR can outweigh the reduction phase by far [9]. On the other hand, QR is the only method that allows accumulating the eigenvectors of the original dense problem in place, thus saving a final matrix-matrix multiplication. Together with recently proposed high-performance realizations of sequences of rotations [21,33], this can make QR (almost) competitive with the approaches discussed below if most or all eigenvectors are required.Divide and conquer (DC) has been known quite a long time [3], but it took nearly 15 years until a stable implementation was found [18]. Complexity is O(n 3 ) in theory, but in practice DC behaves more like O(n 2.5 ) on average [4, 6], due to effective use of BLAS3-operations and a large amount of work that can be saved via deflation.

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Section: The Ideamentioning

confidence: 99%

Section: Requirement 4: Shiftrel (Shift Relation)mentioning

confidence: 99%

Section: Requirement 5: Getvec (Computation Of Eigenvectors)mentioning

confidence: 99%

Section: How To Achieve the Requirementsmentioning

confidence: 99%

Section: Outline Of Our Implementation Xmrmentioning

confidence: 99%

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A Framework for the $MR^3$ Algorithm: Theory and Implementation

Willems¹,

Lang²

2013

SIAM J. Sci. Comput.

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On Computing Eigenvectors of Symmetric Tridiagonal Matrices

Mastronardi

Taeter

Dooren

2019

Springer INdAM Series

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A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

Ferreira

Parlett

2022

Numer. Math.

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The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fast real arithmetic algorithm and compare it with alternative published approaches.

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Twisted Factorizations and qd-Type Transformations for the $\mbox{MR}^{3}$ Algorithm---New Representations and Analysis

Cited by 5 publications

References 12 publications

A Framework for the $MR^3$ Algorithm: Theory and Implementation

A Framework for the $MR^3$ Algorithm: Theory and Implementation

On Computing Eigenvectors of Symmetric Tridiagonal Matrices

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

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