Research Article | Open Access
Xiong You, "Limit-Cycle-Preserving Simulation of Gene Regulatory Oscillators", Discrete Dynamics in Nature and Society, vol. 2012, Article ID 673296, 22 pages, 2012. https://doi.org/10.1155/2012/673296
Limit-Cycle-Preserving Simulation of Gene Regulatory Oscillators
In order to simulate gene regulatory oscillators more effectively, Runge-Kutta (RK) integrators are adapted to the limit-cycle structure of the system. Taking into account the oscillatory feature of the gene regulatory oscillators, phase-fitted and amplification-fitted Runge-Kutta (FRK) methods are designed. New FRK methods with phase-fitted and amplification-fitted updated are also considered. The error coefficients and the error constant for each of new FRK methods are obtained. In the numerical simulation of the two-gene regulatory system, the new methods are shown to be more accurate and more efficient than their prototype RK methods in the long-term integration. It is a new discovery that the best fitting frequency not only depends on the problem to be solved, but also depends on the method.
Differential equations (DEs) have become one of the powerful tools for modeling the complex dynamics of gene regulatory systems, where the cellular concentrations of mRNAs, proteins, and other molecules are assumed to vary continuously in time (see, e.g., de Jong , Widder et al. , Polynikis et al. , Altinok et al. , Gérard and Goldbeter , and the references therein). An -gene activation-inhibition network can be modeled by a system of ordinary differential equations of the form (Polynikis et al. ) where for , is the concentration of mRNA produced by gene , is the concentration of protein produced by mRNA , is the degradation rate of , and is the degradation rate of . Function is the translation function. Function is the regulation function, typically defined sums and products of functions . If protein has no effect on gene , does not appear in . If , protein is an activator of gene . If , protein is an inhibitor of gene .
Since the functions , , and are nonlinear, the analytical solution of the system (1.1) is in general not acquirable. Therefore, in order to reveal the dynamics of such gene regulatory systems, one usually resorts to numerical simulation. To be theoretically clear, we first consider abstractly the initial value problem (IVP) of the autonomous system of ordinary differential equations where , “” represents the first derivative of with respect to time, and is a sufficiently smooth function. Based on experimental observation of biological facts about gene regulation systems, it is reasonable to make the following assumptions.(I)The system (1.2) has a stable limit cycle .(II)The function satisfies , that is, is an equilibrium point of the system (1.2). (III)The equilibrium point lies inside the limit cycle and there is no other equilibrium point inside .
With these assumptions, any solution near the limit cycle is oscillatory. Up till now, differential equations of gene regulatory systems are mostly simulated by Runge-Kutta methods, especially by the classical fourth-order Runge-Kutta method or by the Runge-Kutta-Felhberg adaptive method as in the MATLAB package (see [6–8]). Unfortunately, the general-purpose RK method has not taken into account the special structure of the system (1.1) and hence they cannot give satisfactory numerical results. There are mainly two deficiencies of the classical RK methods: (i) They cannot produce as accurate numerical results as required, even if they have a very high algebraic order; (ii) the true dynamical behavior of the system cannot be preserved as expected in long term integration.
Recently, researchers have proposed to adapt traditional integrators to problems whose solutions are oscillatory or periodic (see [9–12]). Bettis  constructed a three-stage method and a four-stage method which can solve the equation exactly. Paternoster  developed a family of implicit Runge-Kutta (RK) and Runge-Kutta-Nyström (RKN) methods by means of trigonometric fitting. For oscillatory problems which can be put in the form of a second-order equation , Franco  improved the update of the classical RKN methods and proposes a family of explicit RKN methods adapted to perturbed oscillators (ARKN) and a class of explicit adapted RK methods in . Anastassi and Simos  constructed a phase-fitted and amplification-fitted RK method of “almost” order five. Chen et al.  considered symmetric and symplectic extended Runge-Kutta-Nyström methods. You et al.  investigated trigonometrically fitted Scheifele two-step (TFSTS) methods, derived the necessary and sufficient conditions for TFSTS methods of up to order five based on the linear operator theory and constructed two practical methods of algebraic four and five, respectively. For other important work on frequency-dependent integrators for general second-order oscillatory equations, the reader is referred to [20–24]. Some authors aimed at obtaining effective integrators for specific categories of oscillatory problems. For instance, Vigo-Aguiar and Simos  constructed an exponentially fitted and trigonometrically fitted method for orbital problems. The papers [25–27] have designed highly efficient integrators for the Schrödinger equation.
The objective of this paper is to develop effective integrators for simulating the gene regulation system (1.1) near its limit cycle. Motivated by the work of Van de Vyver  on the two-step hybrid methods (FTSH), this paper develops a novel type of phase-amplification-fitted methods of Runge-Kutta type. These new numerical integrators respect the limit cycle structure of the system and are expected to be more accurate than the traditional RK methods in the long-time integration of gene regulatory systems. In Section 2, we present the order conditions for the modified Runge-Kutta methods for solving initial value problems of autonomous ordinary differential equations with oscillatory solutions. Section 3 gives the conditions for a modified RK method and its update to be phase-fitted and amplification-fitted. In Section 4, we construct six phase-fitted and amplification-fitted RK (FRK) methods. For each of these methods, the error coefficients and the error constant are presented. In Section 5, the two-gene regulation system is solved by the new FRK methods as well as their prototype RK methods. Section 6 is devoted to the conclusions.
2. Modified Runge-Kutta Methods and Order Conditions
Assume that the principal frequency of the problem (1.2) is known or can be accurately estimated in advance. This estimated frequency matrix is denoted by , which is also called the fitting frequency. We consider the following -stage modified Runge-Kutta method: where is the step size, , , are real numbers and , are even functions of . The scheme (2.1) can be represented by the Butcher tableau or simply by . With the rooted tree theory of Butcher , the exact solution to the problem (1.2) and the numerical solution produced by the modified RK type method (2.1) have the -series expressions as follows: where the trees , the functions (order), (the number of monotonic labelings of tree ), (density), (the vector of elementary weights) and the elementary differentials are defined in . Then the local truncation error has the following series expansion: If for any th differentiable function , when the scheme (2.1) is applied to the problem (1.2), , then the method (2.1) is said to have (algebraic) order .
If the method (2.1) is of order , then we have where is called the error coefficient associated with the tree of order which is involved in the leading term of the local truncation error. Denote The positive number is called the error constant of the method. According to Theorem 2.1, we list the conditions for modified RK type methods to be of up to order 5 as follows: where , with , the dot “” between two vectors indicates componentwise product, indicates the componentwise square of the vector and so on. We will follow this convention in the sequel.
It is obvious that if a method satisfies the conditions for some order , then it satisfies all the conditions for orders lower than .
In Section 4, when we use the above order conditions as equations to solve for the coefficients of a method, the higher order terms will be neglected.
3. Phase-Fitting and Amplification-Fitting Conditions
For oscillatory problems, Paternoster  and Van der Houwen and Sommeijer  propose to analyze the phase properties (dispersion and dissipation) of numerical integrators. To this end, we consider the linear equation The exact solution of this equation with the initial value satisfies where . This means that after a period of time , the exact solution experiences a phase advance and the amplification remains constant.
An application of the modified RK method (2.1) to (3.1) yields where Thus, after a time step , the numerical solution obtains a phase advance and the amplification factor . is called the stability function of the method (2.1). Denoting the real and imaginary part of by and , respectively, we have Thus, and . The above analysis leads to the following definition.
Definition 3.1 (see ). The quantities are called the phase lag (or dispersion) and the error of amplification factor (or dissipation) of the method, respectively. If then the method is called dispersive of order and dissipative of order , respectively. If the method is called phase-fitted (or zero-dispersive) and amplification-fitted (or zero-dissipative), respectively. By a phase-amplification-fitted RK method we mean a modified RK method that is both phase-fitted and amplification-fitted, which we refer to as an FRK method.
It is interesting to consider the phase properties of the update of the scheme (2.1). Suppose that the internal stages have been exact for the linear equation (3.1), that is, , then the update gives where Denote the real and imaginary part of by and , respectively. Then, for small ,
Definition 3.2. The quantities are called the dispersion (or phase lag) and the dissipation (or error of amplification factor) of the update of the method, respectively. If then the update is called dispersive of order and dissipative of order , respectively. If the update is called phase-fitted (or zero-dispersive) and amplification-fitted (or zero-dissipative), respectively.
In general, a classical RK method with constant coefficients carries a nonzero dispersion and a nonzero dissipation when applied to the linear oscillatory equation (3.1). Therefore, they are neither phase-fitted nor amplification-fitted. So does the update. For example, the classical RK method of order four with constant coefficients (denoted as RK4, see ) given by has a phase lag and an error of amplification factor For the update, Therefore, both the method and its update are dispersive of order four and dissipative of order five.
The following theorem gives the necessary and sufficient conditions for a modified RK method and its update to be phase-fitted and amplification-fitted, respectively. The proof of this theorem is immediate.
4. Construction of FRK Methods
In this section, we construct modified RK type methods that are phase-amplification-fitted based on the internal coefficients of familiar classical RK methods. For convenience we restrict ourselves to explicit methods.
4.1. FRK Methods of Type I
The coefficients of Type I methods are obtained by solving the phase-amplification-fitting conditions and some low-order conditions.
4.1.1. Three-Stage Methods of Order Three
We begin by considering the following three-stage method The phase-fitting and amplification-fitting conditions (3.18) for the scheme (4.1) have the form On the other hand, the first-order condition is given by Solving (4.2) and (4.3), we obtain As , have the following Taylor expansions
The error coefficients for FRK3s3aI are given as follows: The error constant is
Next we consider the following three-stage method based on Heun's method of order three Solving the phase-fitting and amplification-fitting conditions and the first-order condition, we obtain As , , have the following Taylor expansions: The method given by (4.8) and (4.9) is also of order three and we denote it by FRK3s3bI.
The error coefficients for FRK3s3bI are given as follows: The error constant is
4.1.2. Four-Stage Methods of Order Four
Next we consider the following four-stage method based on the so-called 3/8 rule (see the right method in Table 1.2 of Hairer et al. )
For , the phase-fitting and amplification-fitting conditions (3.18) have the form On the other hand, the first-order and second-order conditions become Solving (4.29) and (4.15), we obtain As , have the following Taylor expansions: The method given by (4.13) and (4.16) is also of order four and we denote it by FRK4s4aI.
Corresponding to the nine fifth-order rooted trees , , the error coefficients are given by The error constant is
Now we consider the following four-stage method based on the classical Runge-Kutta method (see Hairer et al. [8, Page 138], the left method in Table 1.2)
The phase-fitting and amplification-fitting conditions (3.18) and the first-order and second-order conditions give As , have the following Taylor expansions:
Corresponding to the nine fifth-order rooted trees , , the error coefficients of FRK4s4bI are given by Then the error constant is
4.2. FRK Methods of Type II
The coefficients of Type II methods are determined by solving the phase-amplification-fitting conditions and update phase-amplification-fitting conditions. Due to the limitation of the number of parameters, we can only consider methods of at least four stages.
For the method given by (4.13), the conditions in Theorem 3.3 assume Solving this system, we can obtain , . Their analytic expressions are complicated. Here we present their Taylor expansions as follows: It can be verified that the method given by (4.13) and (4.26) is of order four. We denote the method by FRK4s4bII.
The error coefficients of FRK4s4aII are given by Then the error constant is
For the method given by (4.13), the conditions in Theorem 3.3 assume Solving this system, we obtain where It can be verified that the method given by (4.20) and (4.30) is of order four. We denote the method by FRK4s4bII.
The error coefficients of FRK4s4bII are given by where Then the error constant is
5. Numerical Simulation
In this section, we proceed to solve numerically a two-gene regulatory system without self-regulation (see Widder et al.  and Polynikis et al. ) where for , is the concentration of mRNA produced by gene , is the concentration of protein , is the maximal transcription rate of , is the translation rate of , is the degradation rate of and is the degradation rate of . The two functions are the Hill functions of activation and repression, respectively. The parameters , are the Hill coefficients, , are the expression thresholds.
The solution of the system is an equilibrium point of the system (5.1). Now we take the values of parameters as follows: For any initial point near the equilibrium point, the system (5.1) has a stable limit cycle. Figure 1 presents three phase-orbits projected in the protein plane starting at , , and , respectively.
In order to test the effectiveness of the new FRK methods proposed in this paper, we apply them to the above two-gene regulatory system. For comparison, we also employ several highly efficient integrators from the literature. The numerical methods we choose for experiments are as follows: (i)FRK3s3aI: the three-stage RK method of order three given by (4.1) and (4.4) in Section 4 of this paper; (ii)FRK3s3bI: the three-stage RK method of order three given by (4.8) and (4.9) in Section 4 of this paper; (iii)FRK4s4aI: the four-stage RK method of order four given by (4.13) and (4.16) in Section 4 of this paper; (iv)FRK4s4bI: the four-stage RK method of order four given by (4.20) and (4.21) in Section 4 of this paper; (v)FRK4s4aII: the four-stage RK method of order four given by (4.13) and (4.26) in Section 4 of this paper; (vi)FRK4s4bII: the four-stage RK method of order four given by (4.20) and (4.30) in Section 4 of this paper; (vii)ARK4: the second four-stage adapted RK method of order four given in Subsection 3.2 of Franco ; (viii)EFRK4: the four-stage exponentially fitted RK method of order four given in ; (ix)RK4: the classical RK method of order four presented in  (the prototype method of FRK4s4bI and FRK4s4bII);(x)RK38: the so-called 3/8 rule, the right method in Table 1.2 of Hairer et al.  (the prototype method of FRK4s4aI and FRK4s4aII).
5.1. Experiment 1: Protein Error Growth
We first integrate the problem (5.1)–(5.3) on the interval with the fixed step size and the initial values In Figure 2, we plot the absolute global errors of Protein 1 produced by the methods FRK3s3aI and FRK3s3bI compared with their prototype RK methods. In Figure 3, we plot the absolute global errors of Protein 1 produced by the methods FRK4s4aI, FRK4s4aII, FRK4s4bI and FRK4s4bII compared with their prototype RK methods.
It can be seen from Figures 2 and 3 that the FRK methods are more accurate than their prototype RK methods. Moreover, for appropriate fitting frequency (here ), the FRK methods of type II can be more accurate than the corresponding FRK methods of type I and their prototype RK methods.
5.2. Experiment 2: Efficiency
Next we compare the computational efficiency of the methods under consideration. In this experiment, we also take the same initial data as Experiment I. The problem is integrated on the interval with step sizes for the four-stage methods and with for the three-stage methods. The numerical results are presented in Figure 4, where the horizontal axis stands for the computational effort measured by the number of function evaluations required and the vertical axis stands for the maximal global error (MGE) of Protein 1, both in the digital logarithm scale.
It can be seen from Figure 4 that the FRK methods are more efficient than their prototype RK methods and are more efficient than the other frequency depending methods of the same algebraic order.
6. Conclusions and Discussions
In this paper, classical Runge-Kutta methods are adapted to stable limit cycles of first-order dynamical systems of genetic regulation. The newly developed phase-fitted and amplification-fitted Runge-Kutta methods (FRK) adopt functions of the product of the fitting frequency and the step size as weight coefficients in the update. FRK methods have zero dispersion and zero dissipation when applied to the standard linear oscillator . That is to say, they can preserve initial phase and amplification as time extends. Therefore, FRK integrators are a kind of integrators which preserve the oscillation structure of the problem.
We note that as the fitting frequency tends to zero, FRK methods reduce to their classical prototype methods. Furthermore, an FRK method has the same algebraic order and the same error constant with its prototypes method. Numerical experiments illustrate the high efficiency of FRK methods compared with their prototype methods and some other frequency depending methods like exponentially fitted RK methods.
Theorem 3.3 gives a pair of sufficient conditions for modified RK type methods to be phase-fitted and amplification-fitted. The coefficients of the FRK methods of type I (FRK3s3aI, FRK3s3bI, FRK4s4aI, and FRK4s4b) are obtained with these conditions combined with an appropriate number of low order conditions. Since the FRK methods of type II (FRK4s4aII and FRK4s4bII) have updates that are also phase-amplification-fitted, they can be more accurate than those methods whose updates do not have this property.
In practical computations of oscillatory problems, the true frequency is, in general, not available. The fitting frequency contained in an FRK method is just an estimate of the true frequency. For techniques of estimating principal frequencies we refer to the papers [32–37]. In this experiment, we have a new discovery that for a specific FRK method, the choice of the value of the fitting frequency affects its efficiency to some extent. For instance, in Section 5, we have discovered that is the best choice for FRK4s4aII, while is the best for FRK4s4bII. This interesting discovery constitutes a challenge to the traditional viewpoint on the choice of fitting frequency for frequency depending methods for which a “best” estimate of frequency can be obtain from the problem under consideration. The results of Experiment 2 of Section 5 show that this best frequency also depends on the method itself.
The author is grateful to the anonymous referees for their careful reading of the paper and for their invaluable comments and suggestions. This work was supported by NSFC (Grants nos. 11171155 and 11271186), the foundation of Scientific the Fundamental Research Fund for the Central Universities (Grant no. Y0201100265), and the Research Fund for the Doctoral Program of Higher Education (Grant no. 20100091110033).
- H. de Jong, “Modeling and simulation of genetic regulatory systems: a literature review,” Journal of Computatational Biology, vol. 9, no. 1, pp. 67–103, 2002.
- S. Widder, J. Schicho, and P. Schuster, “Dynamic patterns of gene regulation. I. Simple two-gene systems,” Journal of Theoretical Biology, vol. 246, no. 3, pp. 395–419, 2007.
- A. Polynikis, S. J. Hogan, and M. di Bernardo, “Comparing different ODE modelling approaches for gene regulatory networks,” Journal of Theoretical Biology, vol. 261, no. 4, pp. 511–530, 2009.
- A. Altinok, D. Gonze, F. Lévi, and A. Goldbeter, “An automaton model for the cell cycle,” Interface Focus, vol. 1, no. 1, pp. 36–47, 2011.
- C. Gérard and A. Goldbeter, “A skeleton model for the network of cyclin-dependent kinases driving the mammalian cell cycle,” Interface Focus, vol. 1, no. 1, pp. 24–35, 2011.
- J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2nd edition, 2008.
- J. C. Butcher and G. Wanner, “Runge-Kutta methods: some historical notes,” Applied Numerical Mathematics, vol. 22, no. 1–3, pp. 113–151, 1996.
- E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer, Berlin, Germany, 1993.
- D. G. Bettis, “Numerical integration of products of Fourier and ordinary polynomials,” Numerische Mathematik, vol. 14, no. 5, pp. 421–434, 1970.
- W. Gautschi, “Numerical integration of ordinary differential equations based on trigonometric polynomials,” Numerische Mathematik, vol. 3, no. 1, pp. 381–397, 1961.
- P. Martín and J. M. Ferrándiz, “Multistep numerical methods based on the Scheifele -functions with application to satellite dynamics,” SIAM Journal on Numerical Analysis, vol. 34, no. 1, pp. 359–375, 1997.
- A. D. Raptis and T. E. Simos, “A four-step phase-fitted method for the numerical integration of second order initial value problems,” BIT Numerical Mathematics, vol. 31, no. 1, pp. 160–168, 1991.
- D. G. Bettis, “Runge-Kutta algorithms for oscillatory problems,” Zeitschrift für Angewandte Mathematik und Physik, vol. 30, no. 4, pp. 699–704, 1979.
- B. Paternoster, “Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials,” Applied Numerical Mathematics, vol. 28, no. 2–4, pp. 401–412, 1998.
- J. M. Franco, “Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators,” Computer Physics Communications, vol. 147, no. 3, pp. 770–787, 2002.
- J. M. Franco, “Runge-Kutta methods adapted to the numerical integration of oscillatory problems,” Applied Numerical Mathematics, vol. 50, no. 3-4, pp. 427–443, 2004.
- Z. A. Anastassi and T. E. Simos, “A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for the numerical solution of orbital problems,” New Astronomy, vol. 10, no. 1, pp. 31–37, 2004.
- Z. Chen, X. You, W. Shi, and Z. Liu, “Symmetric and symplectic ERKN methods for oscillatory Hamiltonian systems,” Computer Physics Communications, vol. 183, no. 1, pp. 86–98, 2012.
- X. You, Y. Zhang, and J. Zhao, “Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators,” Computer Physics Communications, vol. 182, no. 7, pp. 1481–1490, 2011.
- J. Vigo-Aguiar and H. Ramos, “Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 187–211, 2003.
- J. Vigo-Aguiar and J. M. Ferrándiz, “A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems,” SIAM Journal on Numerical Analysis, vol. 35, no. 4, pp. 1684–1708, 1998.
- T. E. Simos and J. Vigo-Aguiar, “An exponentially-fitted high order method for long-term integration of periodic initial-value problems,” Computer Physics Communications, vol. 140, no. 3, pp. 358–365, 2001.
- A. Tocino and J. Vigo-Aguiar, “Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods,” Mathematical and Computer Modelling, vol. 42, no. 7-8, pp. 873–876, 2005.
- J. Vigo-Aguiar and T. E. Simos, “An exponentially fitted and trigonometrically fitted method for the numerical solution of orbital problems,” Astronomical Journal, vol. 122, no. 3, pp. 1656–1660, 2001.
- G. Avdelas, T. E. Simos, and J. Vigo-Aguiar, “An embedded exponentially-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation and related periodic initialvalue problems,” Computer Physics Communications, vol. 131, no. 1-2, pp. 52–67, 2000.
- T. E. Simos and J. Vigo-Aguiar, “A symmetric high order method with minimal phase-lag for the numerical solution of the Schrodinger equation,” International Journal of Modern Physiscs C, vol. 12, no. 7, pp. 1035–1042, 2001.
- T. E. Simos and J. Vigo-Aguiar, “A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems,” Computer Physics Communications, vol. 152, no. 3, pp. 274–294, 2003.
- H. Van de Vyver, “Phase-fitted and amplification-fitted two-step hybrid methods for ,” Journal of Computational and Applied Mathematics, vol. 209, no. 1, pp. 33–53, 2007.
- P. J. van der Houwen and B. P. Sommeijer, “Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 595–617, 1987.
- T. E. Simos, “An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions,” Computer Physics Communications, vol. 115, no. 1, pp. 1–8, 1998.
- G. Vanden Berghe, H. De Meyer, M. Van Daele, and T. Van Hecke, “Exponentially fitted Runge-Kutta methods,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 107–115, 2000.
- G. Vanden Berghe, L. Gr. Ixaru, and M. Van Daele, “Optimal implicit exponentially fitted Runge-Kutta methods,” Computer Physics Communications, vol. 140, no. 3, pp. 346–357, 2001.
- G. Vanden Berghe, L. Gr. Ixaru, and H. De Meyer, “Frequency determination and step-length control for exponentially-fitted Runge-Kutta methods,” Journal of Computational and Applied Mathematics, vol. 132, no. 1, pp. 95–105, 2001.
- L. Gr. Ixaru, G. Vanden Berghe, and H. De Meyer, “Frequency evaluation in exponential fitting multistep algorithms for ODEs,” Journal of Computational and Applied Mathematics, vol. 140, no. 1-2, pp. 423–434, 2002.
- J. Vigo-Aguiar, T. E. Simos, and J. M. Ferrándiz, “Controlling the error growth in long-term numerical integration of perturbed oscillations in one or several frequencies,” Proceedings of The Royal Society of London A, vol. 460, no. 2042, pp. 561–567, 2004.
- H. Van de Vyver, “Frequency evaluation for exponentially fitted Runge-Kutta methods,” Journal of Computational and Applied Mathematics, vol. 184, no. 2, pp. 442–463, 2005.
- H. Ramos and J. Vigo-Aguiar, “On the frequency choice in trigonometrically fitted methods,” Applied Mathematics Letters, vol. 23, no. 11, pp. 1378–1381, 2010.
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