Function evaluation on branch cuts

Rich, Albert D.; Jeffrey, David J.

doi:10.1145/235699.235704

Cited by 10 publications

(8 citation statements)

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“…9 First do whatever is necessary so that fractional powers use the principal branch and w is regarded as a complex indeterminate.…”

Section: As Indicated Inmentioning

confidence: 99%

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Simplifying products of fractional powers of powers

Stoutemyer

2013

ACM Commun. Comput. Algebra

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Most computer algebra systems incorrectly simplifyto 0 rather than to 0/0. The reasons for this are: 1. The default simplification does not succeed in simplifying the denominator to 0. 2. There is a rule that 0 is the result of 0 divided by anything that does not simplify to either 0 or 0/0.Many of these systems have more powerful optional transformation and general purpose simplification functions. However that is unlikely to help this example even if one of those functions can simplify the denominator to 0, because the input to those functions is the result of default simplification, which has already incorrectly simplified the overall ratio to 0. Try it on your computer algebra systems! This article describes how to simplify products of the form w α w β 1 γ 1 · · · w βn γn correctly and well, where w is any real or complex expression and the exponents are rational numbers.It might seem that correct good simplification of such a restrictive expression class must already be published and/or built into at least one widely used computer-algebra system, but apparently this issue has been overlooked. Default and relevant optional simplification was tested with 86 examples on 5 systems with n = 1. Using a spectrum from the most serious flaw being a result that is not equivalent to the input somewhere to the least serious being not rationalizing a denominator when that does not cause a more serious flaw, the overall percentage of most flaw types is alarming: flaw: ≡ 0-recognition cancelable singularity extra factor excessive |γ 1 | ¬ canonical ¬ idempotent ··· √ ···

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“…9 First do whatever is necessary so that fractional powers use the principal branch and w is regarded as a complex indeterminate.…”

Section: As Indicated Inmentioning

confidence: 99%

“…with other form 4 notations for results containing both half-integer and other nested powers. 9 If you are familiar enough with those systems, then most of them probably have a quick way to generate all of the results for test families 1 and 2 by entries analogous to the following one for Mathematica: …”

Section: Four Alternative Formsmentioning

confidence: 99%

Simplifying products of fractional powers of powers

Stoutemyer

2013

ACM Commun. Comput. Algebra

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“…It clearly makes sense to have it continuous with one side or the other, and the common choice, as in [2], is to choose the value of ln on the branch cut to be continuous with the upper half-plane, so that −π < ln z ≤ π. However, this choice is essentially arbitrary, and [16] would like to make the function twovalued on the branch cut: ln(−1) = ±πi. This has the drawback of not fitting readily with numerical evaluation.…”

Section: The "Elementary Transcendental Functions"mentioning

confidence: 99%

MKM from Book to Computer: A Case Study

Davenport

2003

Lecture Notes in Computer Science

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Abstract.[2] is one of the great mathematical knowledge repositories. Nevertheless, it was written for a different era, and for human readership. In this paper, we describe the sorts of knowledge in one chapter (elementary transcendental functions) and the difficulties in making this sort of knowledge formal. This makes us ask questions about the nature of a Mathematical Knowledge Repository, and whether a database is enough, or whether more "intelligence" is required.

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“…There is one area of research in multivalued functions which is slowly developing a body of published literature, and that is the simplification of principal-value functions, see [13,12,22,19,2,11,5,4,3]. However, the underlying assumption of the area, namely that the principal value is the best way to implement a multivalued function, is not an assumption that is uniformly accepted outside the CAS community, and perhaps not even within it.…”

Section: Introductionmentioning

confidence: 99%

Not seeing the roots for the branches

Jeffrey

Norman

2004

SIGSAM Bull.

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We discuss the multiple definitions of multivalued functions and their suitability for computer algebra systems. We focus the discussion by taking one specific problem and considering how it is solved using different definitions. Our example problem is the classical one of calculating the roots of a cubic polynomial from the Cardano formulae, which contain fractional powers. We show that some definitions of these functions result in formulae that are correct only in the sense that they give candidates for solutions; these candidates must then be tested. Formulae that are based on single-valued functions, in contrast, are efficient and direct.

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Function evaluation on branch cuts

Cited by 10 publications

References 1 publication

Simplifying products of fractional powers of powers

Simplifying products of fractional powers of powers

MKM from Book to Computer: A Case Study

Not seeing the roots for the branches

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