A modified regula falsi method for computing the root of an equation

Dowell, M. H.; Jarratt, P.

doi:10.1007/bf01934364

Cited by 137 publications

(59 citation statements)

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“…Such methods are standard in root finding for non-linear equations (e.g. Dowell and Jarrett, 1971;Anderson and Björk, 1973;Bus and Dekker, 1975). They are particularly suitable for our problem with its advantageous mathematical characteristics, the more so since that problem also has intrinsic a priori root bracketing, as we will show in the next section.…”

Section: Development Of a Universal And Robust Algorithmmentioning

confidence: 99%

Mathematics of the total alkalinity–pH equation – pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

Munhoven¹

2013

Geosci. Model Dev.

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Abstract. The total alkalinity-pH equation, which relates total alkalinity and pH for a given set of total concentrations of the acid-base systems that contribute to total alkalinity in a given water sample, is reviewed and its mathematical properties established. We prove that the equation function is strictly monotone and always has exactly one positive root. Different commonly used approximations are discussed and compared. An original method to derive appropriate initial values for the iterative solution of the cubic polynomial equation based upon carbonate-borate-alkalinity is presented. We then review different methods that have been used to solve the total alkalinity-pH equation, with a main focus on biogeochemical models. The shortcomings and limitations of these methods are made out and discussed. We then present two variants of a new, robust and universally convergent algorithm to solve the total alkalinity-pH equation. This algorithm does not require any a priori knowledge of the solution. SolveSAPHE (Solver Suite for Alkalinity-PH Equations) provides reference implementations of several variants of the new algorithm in Fortran 90, together with new implementations of other, previously published solvers. The new iterative procedure is shown to converge from any starting value to the physical solution. The extra computational cost for the convergence security is only 10-15 % compared to the fastest algorithm in our test series.

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Section: Development Of a Universal And Robust Algorithmmentioning

confidence: 99%

Mathematics of the total alkalinity–pH equation – pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

Munhoven¹

2013

Geosci. Model Dev.

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“…To find tJ.a n + 1, we adopted a root-finding method that has guaranteed convergence at a relatively fast rate [39,40], It is a Regula Falsi method (false position) and requires two initial guesses for the crack tip increment that bracket the root of the equation we wish to solve. We denote the first initial guess with an asterisk (*), which should yield K > K;.…”

Section: Crack Tip Advancementmentioning

confidence: 99%

“…The volume is obtained by integrating the opening profile over the fluid domain rp as follows: (37) We also introduce similarity variables for the crack length y, pressure II, and crack opening Q by scaling with the appropriate powers of r as dictated by the similarity solution: (38) The previous equations can be used to describe both similarity solutions, if we use (T~ = 0 for the zero far-field stress solution, and (To = 1 for the late-time solution. We note that the similarity solutions are governed solely by the dimensionless toughness K;, which is defined through a combination of the physical pararueters of the problem: (39) So for each value of K;, there is a corresponding value of y, and corresponding functions II(O and Q( 0, which have no dependence on r.…”

Section: Numerical Examplesmentioning

confidence: 99%

A finite element approach to the simulation of hydraulic fractures with lag

Hunsweck

Shen

Lew

2012

Num Anal Meth Geomechanics

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SUMMARYWe presented a finite-element-based algorithm to simulate plane-strain, straight hydraulic fractures in an impermeable elastic medium. The algorithm acCOllllts for the nonlinear coupling between the fluid pressure and the crack opening and separately tracks the evolution of the crack tip and the fluid front. It therefore allows the existence of a fluid lag. The fluid front is advanced explicitly in time, but an implicit strategy is needed for the crack tip to guarantee the satisfaction of Griffith's criterion at each time step. We enforced the coupling between the fluid and the rock by simultaneously solving for the pressure field in the fluid and the crack opening at each time step. We provided verification of our algorithm by performing sample simulations and comparing them with two known similarity solutions.

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“…A modification of the secant method, called the Illinois method (see Dowell and Jarratt (1971) and Kennedy and Gentle (1980)), is used to find the root.…”

Section: Methodsmentioning

confidence: 99%

Algorithm AS 261: Quantiles of The Distribution of The Square of The Sample Multiple-Correlation Coefficient

Ding¹,

Bargmann²

1991

Applied Statistics

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A modified regula falsi method for computing the root of an equation

Abstract: Abstract.The Illinois method is briefly described and the asymptotic convergence of the method investigated. Numerical examples are also given including comparisons with other similar robust methods.

Cited by 137 publications

References 1 publication

Mathematics of the total alkalinity–pH equation – pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

Mathematics of the total alkalinity–pH equation – pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

A finite element approach to the simulation of hydraulic fractures with lag

Algorithm AS 261: Quantiles of The Distribution of The Square of The Sample Multiple-Correlation Coefficient

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