Best Rational Starting Approximations and Improved Newton Iteration for the Square Root

Ninomiya, Ichizo

doi:10.2307/2004485

Cited by 8 publications

(10 citation statements)

References 7 publications

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“…Regarding the result "composed Zolotarev is high-degree Zolotarev", a related observation has been made for the square root function originally by Rutishauser [54] (see also [13, Ch. V.5.D]), and also mentioned by Ninomiya [49] and Braess [12], in the context of accelerating Heron's iteration. They observe that by appropriately scaling Heron's iteration, which composes type (2, 1) approximants in each iteration, one can obtain the best rational approximant to the square root function, in terms of minimizing the relative error…”

Section: 7mentioning

confidence: 99%

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

Nakatsukasa¹,

Freund²

2016

SIAM Rev.

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Abstract. The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with arithmetic costs. Spectral divideand-conquer algorithms, which recursively decouple the problem into two smaller subproblems, can achieve both requirements. Such algorithms can be constructed with the polar decomposition playing two key roles: it forms a bridge between the symmetric eigendecomposition and the SVD, and its connection to the matrix sign function naturally leads to spectral-decoupling. For computing the polar decomposition, the scaled Newton and QDWH iterations are two of the most popular algorithms, as they are backward stable and converge in at most nine and six iterations, respectively. Following this framework, we develop a higher-order variant of the QDWH iteration for the polar decomposition. The key idea of this algorithm comes from approximation theory: we use the best rational approximant for the scalar sign function due to Zolotarev in 1877. The algorithm exploits the extraordinary property enjoyed by the sign function that a high-degree Zolotarev function (best rational approximant) can be obtained by appropriately composing low-degree Zolotarev functions. This lets the algorithm converge in just two iterations in double-precision arithmetic, with the whopping rate of convergence seventeen. The resulting algorithms for the symmetric eigendecompositions and the SVD have higher arithmetic costs than the QDWH-based algorithms, but are better-suited for parallel computing and exhibit excellent numerical backward stability.

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Section: 7mentioning

confidence: 99%

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

Nakatsukasa¹,

Freund²

2016

SIAM Rev.

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“…(A best Moursund fit is one whose Newton iterate has a relative error with least maximum magnitude.) This nice property (see [5]) does not appear to hold for p ^ 3, simply because /zth-root Newton iterates-even of the improved variety-do not have enough oscillations in their relative error curves. That is, it does not follow that an improved Newton approximant at a given stage becomes a best one-sided Chebyshev fit upon iteration.…”

Section: Improvement Of Iterated Approximationsmentioning

confidence: 99%

“…Such procedures correspond to exploiting the usual logarithmic error for the square root case [2], [3], [4]. In another paper [5], Ninomiya has shown how Newton iterates of Moursund fits for the square root can be improved from one-sided Chebyshev-like fits to Moursund two-sided fits for the next iteration ; each improved fit has a maximum relative error that is only about half that of the one-sided fit. By using the generalized logarithmic error, we will derive similar improved two-sided fits for the pth root, with p being any positive integer; these fits in turn will form the basis for an improved Newton iteration method to compute integral roots.…”

Section: Introductionmentioning

confidence: 99%

Improved Newton Iteration for Integral Roots

King¹

1971

Mathematics of Computation

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An improved Newton iteration procedure for computing pth roots from best Chebyshev or Moursund initial approximations is developed. It differs from the usual Newton method by a multiplicative factor at each step. This multiplier halves the relative error by translating the usual one-sided error curve into a two-sided one, and then adjusting to make a Moursund-Iike fit. The generalized logarithmic error is used in determining this set of factors.

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“…There is no reason to expect-and it is generally not true-that r k (• • • r 2 (r 1 (x))) can express the minimax rational approximant of a given type, say (m k , m k ), to a given function. However, building upon Rutishauser [15] and Ninomiya [13], Nakatsukasa and Freund [12] show a remarkable property of the best rational approximants to the function sign(x) = x/|x| on [−1, −δ] ∪ [δ, 1] for 0 < δ < 1 (called Zolotarev functions): appropriately composing Zolotarev functions gives another Zolotarev function of higher degree. In other words, the class of composite rational functions r(x) = r k (• • • r 2 (r 1 (x))), with each r i of type (m, m), contains the type-(m k , m k ) minimax approximant to the sign function.…”

Section: Introductionmentioning

confidence: 99%

Approximating the pth Root by Composite Rational Functions

Gawlik

Nakatsukasa

2019

Preprint

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A landmark result from rational approximation theory states that x 1/p on [0, 1] can be approximated by a type-(n, n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x 1/p by composite rational functions of the form. While this class of rational functions ceases to contain the minimax (best) approximant for p ≥ 3, we show that it achieves approximately pth-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate x 1/p and related functions (such as |x| and the sector function) with exceptional efficiency.

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Best Rational Starting Approximations and Improved Newton Iteration for the Square Root

Abstract: Abstract. The most important class of the best rational approximations to the square root is obtained analytically by means of elliptic function theory. An improvement of the Newton iteration procedure is proposed.

Cited by 8 publications

References 7 publications

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

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

Improved Newton Iteration for Integral Roots

Approximating the pth Root by Composite Rational Functions

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