TRIN (finance): Difference between revisions
en>Duttywine |
|||
| Line 1: | Line 1: | ||
In [[numerical analysis]], '''Ridders' method''' is a [[root-finding algorithm]] based on the [[false position method]] and the use of an [[exponential function]] to successively approximate a [[Root of a function|root]] of a [[Function (mathematics)|function]] ''f''. The method is due to C. Ridders.<ref>{{cite doi|10.1109/TCS.1979.1084580}}</ref><ref>{{cite book|title=Numerical Methods in Engineering with Python|first=Jaan |last=Kiusalaas| publisher=Cambridge University Press| year=2010| isbn=978-0-521-19132-6 | edition=2nd| pages=146–150| url=http://books.google.co.uk/books?id=9SG1r8EJawIC&pg=PT156}}</ref> | |||
Ridders' method is simpler than [[Muller's method]] or [[Brent's method]] but with similar performance.<ref>{{cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=[[Numerical Recipes]]: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 9.2.1. Ridders' Method | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=452}}</ref> The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall [[order of convergence]] of the method is √2. If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval at least halves on each iteration, so convergence is guaranteed. The algorithm also makes use of square roots, which are slower than basic floating point operations. | |||
==Method== | |||
Press et al. (2007) describe the method as follows. Given two values of the independent variable, ''x''<sub>1</sub> and ''x''<sub>2</sub>, which are on two different sides of the root being sought, the method begins by evaluating the function at the midpoint ''x''<sub>3</sub> between the two points. One then finds the unique exponential function of the form ''e''<sup>''ax''</sup> which, when multiplied by ''f'', transforms the function at the three points into a straight line. The false position method is then applied to the transformed values, leading to a new value ''x''<sub>4</sub>, between ''x''<sub>1</sub> and ''x''<sub>2</sub>, which can be used as one of the two bracketing values in the next step of the iteration. The other bracketing value is taken to be ''x''<sub>3</sub> if f(''x''<sub>3</sub>) has the opposite sign to f(''x''<sub>4</sub>), or otherwise whichever of ''x''<sub>1</sub> and ''x''<sub>2</sub> has f(x) of opposite sign to f(''x''<sub>4</sub>). | |||
The method can be summarized by the formula (equation 9.2.4 from Press et al.) | |||
:<math>x_4 = x_3 + (x_3 - x_1)\frac{\operatorname{sign}[f(x_1) - f(x_2)]f(x_3)}{\sqrt{f(x_3)^2 - f(x_1)f(x_2)}} .</math> | |||
==References== | |||
{{reflist}} | |||
[[Category:Root-finding algorithms]] | |||
{{mathapplied-stub}} | |||
Revision as of 13:11, 27 November 2013
In numerical analysis, Ridders' method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a function f. The method is due to C. Ridders.[1][2]
Ridders' method is simpler than Muller's method or Brent's method but with similar performance.[3] The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall order of convergence of the method is √2. If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval at least halves on each iteration, so convergence is guaranteed. The algorithm also makes use of square roots, which are slower than basic floating point operations.
Method
Press et al. (2007) describe the method as follows. Given two values of the independent variable, x1 and x2, which are on two different sides of the root being sought, the method begins by evaluating the function at the midpoint x3 between the two points. One then finds the unique exponential function of the form eax which, when multiplied by f, transforms the function at the three points into a straight line. The false position method is then applied to the transformed values, leading to a new value x4, between x1 and x2, which can be used as one of the two bracketing values in the next step of the iteration. The other bracketing value is taken to be x3 if f(x3) has the opposite sign to f(x4), or otherwise whichever of x1 and x2 has f(x) of opposite sign to f(x4).
The method can be summarized by the formula (equation 9.2.4 from Press et al.)
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ Template:Cite doi
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534