Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Nakatsukasa, Yuji; Freund, Robert M.

doi:10.1137/140990334

Cited by 52 publications

(99 citation statements)

References 48 publications

(81 reference statements)

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“…Last but not least, we would like to investigate other QDWH algorithmic variants, which may require more FLOPs but entails an even higher level of concurrency [50]. NVIDIA 4xP100…”

Section: Discussionmentioning

confidence: 99%

Asynchronous Task-Based Polar Decomposition on Single Node Manycore Architectures

Sukkari

Ltaief²,

Faverge³

et al. 2018

IEEE Trans. Parallel Distrib. Syst.

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Abstract-This paper introduces the first asynchronous, task-based formulation of the polar decomposition and its corresponding implementation on manycore architectures. Based on a new formulation of the iterative QR dynamically-weighted Halley algorithm (QDWH) for the calculation of the polar decomposition, the proposed implementation replaces the original and hostile LU factorization for the condition number estimator by the more adequate QR factorization to enable software portability across various architectures. Relying on fine-grained computations, the novel task-based implementation is also capable of taking advantage of the identity structure of the matrix involved during the QDWH iterations, which decreases the overall algorithmic complexity. Furthermore, the artifactual synchronization points have been weakened compared to previous implementations, unveiling look-ahead opportunities for better hardware occupancy. The overall QDWH-based polar decomposition can then be represented as a directed acyclic graph (DAG), where nodes represent computational tasks and edges define the inter-task data dependencies. The StarPU dynamic runtime system is employed to traverse the DAG, to track the various data dependencies and to asynchronously schedule the computational tasks on the underlying hardware resources, resulting in an out-of-order task scheduling. Benchmarking experiments show significant improvements against existing state-of-the-art high performance implementations (i.e., Intel MKL and Elemental) for the polar decomposition on latest shared-memory vendors' systems (i.e., Intel Haswell/Broadwell/Knights Landing, NVIDIA K80/P100 GPUs and IBM Power8), while maintaining high numerical accuracy.

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“…Last but not least, we would like to investigate other QDWH algorithmic variants, which may require more FLOPs but entails an even higher level of concurrency [50]. NVIDIA 4xP100…”

Section: Discussionmentioning

confidence: 99%

Asynchronous Task-Based Polar Decomposition on Single Node Manycore Architectures

Sukkari

Ltaief²,

Faverge³

et al. 2018

IEEE Trans. Parallel Distrib. Syst.

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“…The polar decomposition is an important problem in numerical linear algebra area, and the well-known PD algorithms include the scaled Newton (SN) method [20], QDWH-PD [29], and Zolo-PD [30], etc.…”

Section: Qdwh-pd and Zolo-pdmentioning

confidence: 99%

“…subject to the constraint f pxq " x a`bx 2 1`cx 2 ď 1 on r0, 1s. The parameters in the QDWH iteration can also be obtained by finding the best type-p3, 2q rational approximation to the sign function in the infinity norm, see [30].…”

Section: Zolo-pdmentioning

confidence: 99%

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A high performance implementation of Zolo-SVD algorithm on distributed memory systems

Liu

2019

Parallel Computing

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“…For a Schur-free algorithm, the matrix sign function can be computed using a Newton algorithm or some other rational iteration [21,Chap. 5], [31]. Equation (5.8) is applicable only when A has no eigenvalue in the interval (0, 1].…”

Section: Schur-padé Algorithmmentioning

confidence: 99%

Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms

Aprahamian¹,

Higham

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. Theoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and the principal values acos, asin, acosh, and asinh are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a "round trip" formula that relates acos(cos A) to A and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Padé approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas.Key words. matrix function, inverse trigonometric functions, inverse hyperbolic functions, matrix inverse sine, matrix inverse cosine, matrix inverse hyperbolic sine, matrix inverse hyperbolic cosine, matrix exponential, matrix logarithm, matrix sign function, rational approximation, Padé approximation, MATLAB, GNU Octave, Fréchet derivative, condition number AMS subject classifications. 15A24, 65F30DOI. 10.1137/16M10575771. Introduction. Trigonometric functions of matrices play an important role in the solution of second order differential equations; see, for example, [5], [37], and the references therein. The inverses of such functions, and of their hyperbolic counterparts, also have practical applications, but have been less well studied. An early appearance of the matrix inverse cosine was in a 1954 paper on the energy equation of a free-electron model [36]. The matrix inverse hyperbolic sine arises in a model of the motion of rigid bodies, expressed via Moser-Veselov matrix equations [12]. The matrix inverse sine and inverse cosine were used by Al-Mohy, Higham, and Relton [5] to define the backward error in approximating the matrix sine and cosine. Matrix inverse trigonometric and inverse hyperbolic functions are also useful for studying argument reduction in the computation of the matrix sine, cosine, and hyperbolic sine and cosine [7].This work has two aims. The first is to develop the theory of matrix inverse trigonometric functions and inverse hyperbolic functions. Most importantly, we define the principal values acos, asin, acosh, and asinh, prove their existence and uniqueness, and develop various useful identities involving them. In particular, we determine the precise relationship between acos(cosA) and A, and similarly for the other functions. The second aim is to develop algorithms and software for computing acos, asin, acosh, and asinh of a matrix, for which we employ variable-degree Padé approximation to-

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Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Cited by 52 publications

References 48 publications

Asynchronous Task-Based Polar Decomposition on Single Node Manycore Architectures

Asynchronous Task-Based Polar Decomposition on Single Node Manycore Architectures

A high performance implementation of Zolo-SVD algorithm on distributed memory systems

Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms

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