The Microguide to Process Modeling in Bpmn 2.0 by MR Tom Debevoise and Rick Geneva_15 doc

with undefined points at the roots ofThere will be 1, 2, or 3 roots, depending on the particular values of the parameters.. Explicit solutions for the roots of a cubic polynomial are com

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4.8.1.2.8 Linear / Cubic Rational Function

http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8128.htm (5 of 5) [5/1/2006 10:23:11 AM]

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with undefined points at the roots of

There will be 1, 2, or 3 roots, depending on the particular values of the parameters Explicit solutions for the roots of a cubic polynomial are complicated and are not given here Many mathematical and statistical software programs can determine the roots of a polynomial equation numerically, and it is recommended that you use one

of these programs if you need to know where these roots occur.

Range:

with the possible exception that zero may be excluded.

Special

Features:

Horizontal asymptote at:

and vertical asymptotes at the roots of

Additional

Examples:

4.8.1.2.9 Quadratic / Cubic Rational Function

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with undefined points at the roots of

Range:

with the exception that y = may be excluded.

Special

Features:

Horizontal asymptote at:

and vertical asymptotes at the roots of

Additional

Examples:

4.8.1.2.10 Cubic / Cubic Rational Function

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A general question for rational function models is:

I have data to which I wish to fit a rational function to What degrees n and m should I use

for the numerator and denominator, respectively?

Four

Questions

To answer the above broad question, the following four specific questions need to be answered

What value should the function have at x = ? Specifically, is the value zero, a constant,

or plus or minus infinity?

Conversely, if the fitted function f(x) is such that

4.8.1.2.11 Determining m and n for Rational Function Models

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For fintite x, R(x) = 0 only when the numerator polynomial, P n, equals zero.

The numerator polynomial, and thus R(x) as well, can have between zero and n real roots Thus, for a given n, the number of real roots of R(x) is less than or equal to n.

Conversely, if the fitted function f(x) is such that, for finite x, the number of times f(x) = 0 is k3,

then n is greater than or equal to k3

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The derivative function, R'(x), of the rational function will equal zero when the numerator

polynomial equals zero The number of real roots of a polynomial is between zero and the degree

of the polynomial

For n not equal to m, the numerator polynomial of R'(x) has order n+m-1 For n equal to m, the numerator polynomial of R'(x) has order n+m-2.

From this it follows that

if n m, the number of real roots of R'(x), k4, n+m-1.

●

if n = m, the number of real roots of R'(x), k4, is n+m-2.

●

Conversely, if the fitted function f(x) is such that, for finite x and n m, the number of times f'(x)

= 0 is k4, then n+m-1 is k4 Similarly, if the fitted function f(x) is such that, for finite x and n =

m, the number of times f'(x) = 0 is k4, then n+m-2 k4

degrees of the intial rational function model

0constant

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values and the simplest case for n and m We typically start with the simplest case If the model

validation indicates an inadequate model, we then try other rational functions in the admissibleregion

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