Reducing the influence of tiny normwise relative errors on performance profiles

Dingle, Nicholas J.; Higham, Nicholas J.

doi:10.1145/2491491.2491494

Cited by 10 publications

(7 citation statements)

References 17 publications

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“…To improve the plots of forward error, we map any errors outside the displayed range onto the nearest edge (top or bottom) of the plot. We also present the results in the form of performance profiles [13], and use the technique of Dingle and Higham [12] to rescale errors smaller than u.…”

Section: Extension To the Sine And Itsmentioning

confidence: 99%

Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

Al-Mohy¹,

Higham²,

Liu³

2022

SIAM J. Matrix Anal. Appl.

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Existing algorithms for computing the matrix cosine are tightly coupled to a specific precision of floating-point arithmetic for optimal efficiency so they do not conveniently extend to an arbitrary precision environment. We develop an algorithm for computing the matrix cosine that takes the unit roundoff of the working precision as input, and so works in an arbitrary precision. The algorithm employs a Taylor approximation with scaling and recovering and it can be used with a Schur decomposition or in a decomposition-free manner. We also derive a framework for computing the Fréchet derivative, construct an efficient evaluation scheme for computing the cosine and its Fréchet derivative simultaneously in arbitrary precision, and show how this scheme can be extended to compute the matrix sine, cosine, and their Fréchet derivatives all together. Numerical experiments show that the new algorithms behave in a forward stable way over a wide range of precisions. The transformation-free version of the algorithm for computing the cosine is competitive in accuracy with the state-of-the-art algorithms in double precision and surpasses existing alternatives in both speed and accuracy in working precisions higher than double.

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Section: Extension To the Sine And Itsmentioning

confidence: 99%

Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

Al-Mohy¹,

Higham²,

Liu³

2022

SIAM J. Matrix Anal. Appl.

Self Cite

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“…When plotting the forward error, we choose the limits of the y-axis in order to show the area where most of the data points lie, and move the outliers to the closest edge of the box containing the plot. In several cases, we present our experimental results with the aid of performance profiles [10], and adopt the technique of Dingle and Higham [9] to rescale values smaller than u.…”

Section: Schur-padé Variantsmentioning

confidence: 99%

An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential

Fasi¹,

Higham²

2019

SIAM J. Matrix Anal. Appl.

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The most popular algorithms for computing the matrix exponential are those based on the scaling and squaring technique. For optimal efficiency these are usually tuned to a particular precision of floating-point arithmetic. We design a new scaling and squaring algorithm that takes the unit roundoff of the arithmetic as input and chooses the algorithmic parameters in order to keep the forward error in the underlying Padé approximation below the unit roundoff. To do so, we derive an explicit expression for all the coefficients in an error expansion for Padé approximants to the exponential and use it to obtain a new bound for the truncation error. We also derive a new technique for selecting the internal parameters used by the algorithm, which at each step decides whether to scale or to increase the degree of the approximant. The algorithm can employ diagonal Padé approximants or Taylor approximants and can be used with a Schur decomposition or in transformation-free form. Our numerical experiments show that the new algorithm performs in a forward stable way for a wide range of precisions and that the most accurate of our implementations, the Taylor-based transformation-free variant, is superior to existing alternatives.

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“…The comparison method use, perfprof (for "performance profile") [6,7,13], comparatively plots the performance of different algorithms against each other. This type of analysis is typically used when comparing the runtimes of multiple algorithms against each other, preferring low runtime over high.…”

Section: Performance Profilesmentioning

confidence: 99%

Comparative study of clustering techniques for real‐time dynamic model reduction

Purvine

Cotilla‐Sanchez

Halappanavar

et al. 2017

Statistical Analysis

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Dynamic model reduction in power systems is necessary for improving computational efficiency. Traditional model reduction using linearized models or offline analysis is not adequate to capture dynamic behaviors of the power system, especially with the new mix of intermittent generation and intelligent consumption, making the power system more dynamic and nonlinear. Real‐time dynamic model reduction has emerged to fill this important need. This paper explores using clustering techniques to analyze real‐time phasor measurements to identify groups of generators with similar behavior, as well as a representative generator from each group for dynamic model reduction. Two clustering techniques—graph clustering and k‐means—are considered. These techniques are compared with a previously developed dynamic model reduction approach using singular value decomposition. Two sample power grid datasets are used to test these different model reduction techniques. Based on the algorithms' relative performance, recommendations are provided for practical use.

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Reducing the influence of tiny normwise relative errors on performance profiles

Cited by 10 publications

References 17 publications

Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential

Comparative study of clustering techniques for real‐time dynamic model reduction

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