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] 4. Process Modeling 4.8. Some Useful Functions for Process Modeling 4.8.1. Univariate Functions 4.8.1.2. Rational Functions 4.8.1.2.9.Quadratic / Cubic Rational Function Function: 4.8.1.2.9. Quadratic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8129.htm (1 of 4) [5/1/2006 10:23:12 AM] Function Family: Rational Statistical Type: Nonlinear Domain: 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 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. Additional Examples: 4.8.1.2.9. Quadratic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8129.htm (2 of 4) [5/1/2006 10:23:12 AM] 4.8.1.2.9. Quadratic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8129.htm (3 of 4) [5/1/2006 10:23:12 AM] 4.8.1.2.9. Quadratic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8129.htm (4 of 4) [5/1/2006 10:23:12 AM] 4. Process Modeling 4.8. Some Useful Functions for Process Modeling 4.8.1. Univariate Functions 4.8.1.2. Rational Functions 4.8.1.2.10.Cubic / Cubic Rational Function Function: 4.8.1.2.10. Cubic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (1 of 4) [5/1/2006 10:23:13 AM] Function Family: Rational Statistical Type: Nonlinear Domain: 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 exception that y = may be excluded. Special Features: Horizontal asymptote at: and vertical asymptotes 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. Additional Examples: 4.8.1.2.10. Cubic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (2 of 4) [5/1/2006 10:23:13 AM] 4.8.1.2.10. Cubic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (3 of 4) [5/1/2006 10:23:13 AM] 4.8.1.2.10. Cubic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (4 of 4) [5/1/2006 10:23:13 AM] 4. Process Modeling 4.8. Some Useful Functions for Process Modeling 4.8.1. Univariate Functions 4.8.1.2. Rational Functions 4.8.1.2.11.Determining m and n for Rational Function Models General Question 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? 1. What slope should the function have at x = ? Specifically, is the derivative of the function zero, a constant, or plus or minus infinity? 2. How many times should the function equal zero (i.e., f (x) = 0) for finite x?3. How many times should the slope equal zero (i.e., f '(x) = 0) for finite x?4. These questions are answered by the analyst by inspection of the data and by theoretical considerations of the phenomenon under study. Each of these questions is addressed separately below. Question 1: What Value Should the Function Have at x = ? Given the rational function or then asymptotically From this it follows that if n < m, R( ) = 0● if n = m, R( ) = a n /b m ● if n > m, R( ) = ● Conversely, if the fitted function f(x) is such that 4.8.1.2.11. Determining m and n for Rational Function Models http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (1 of 13) [5/1/2006 10:23:15 AM] [...]... m, the number of real roots of R'(x), k4, is n+m-1 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 Tables for Determining Admissible Combinations of m and n In summary, we can determine the. .. functions can have and shows the admissible 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 admissible region Shape 1 Shape 2 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (4 of 13) [5/1/2006 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational... admissible combinations of n and m by using the following four tables to generate an n versus m graph Choose the simplest (n,m) combination for the degrees of the intial rational function model 1 Desired value of f( ) 0 constant 2 Desired value of f'( Relation of n to m nm ) Relation of n to m 0 constant nm+1 3 For finite x, desired number, k3, of times f(x) = 0 Relation of n to k3... Zero for Finite ? 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 m, the number... 4 For finite x, desired number, k4, of times f'(x) = 0 Relation of n to k4 and m k4 (n m) k4 (n = m) n n k3 (1 + k4) - m (2 + k4) - m http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (3 of 13) [5/1/2006 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Examples for Determing m and n The goal is to go from a sample data set to a specific rational function The graphs... 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 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (2 of 13) [5/1/2006 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Question 4: How Many Times Should the. .. 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 6 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (8 of 13) [5/1/2006 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 7 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (9 of 13) [5/1/2006 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models... ) = Conversely, if the fitted function f(x) is such that q f'( ) = 0, this implies n m q f'( ) = constant, this implies n = m + 1 q f'( )= , this implies n > m + 1 q Question 3: How Many Times Should the Function Equal Zero for Finite ? For fintite x, R(x) = 0 only when the numerator polynomial, Pn, equals zero The numerator polynomial, and thus R(x) as well, can have between zero and n real roots Thus,...4.8.1.2.11 Determining m and n for Rational Function Models q q q Question 2: What Slope Should the Function Have at x = ? f( f( f( ) = 0, this implies n < m ) = constant, this implies n = m )= , this implies n > m The slope is determined by the derivative of a function The derivative of a rational function is with Asymptotically From this it follows that q if n < m,... 3 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (5 of 13) [5/1/2006 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 4 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (6 of 13) [5/1/2006 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 5 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (7 . [5/1 / 20 06 10 :23 :15 AM] Shape 8 4.8.1 .2. 11. Determining m and n for Rational Function Models http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm ( 10 of 13) [5/1 / 20 06 10 :23 :15 AM] Shape. 4 4.8.1 .2. 11. Determining m and n for Rational Function Models http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (6 of 13) [5/1 / 20 06 10 :23 :15 AM] Shape 5 4.8.1 .2. 11. Determining m and. Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (2 of 4) [5/1 / 20 06 10 :23 :13 AM] 4.8.1 .2. 10. Cubic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (3 of 4) [5/1 / 20 06