Quadrature Formulas and Taylor Series of Secant and Tangent

QUADRATURE FORMULAS AND TAYLOR SERIES OF SECANT AND TANGENT

Yuri DIMITROV1, Radan MIRYANOV2, Venelin TODOROV3 1 University of Forestry, Sofia, Bulgaria [email protected] 2 University of Economics, Varna, Bulgaria [email protected] 3 Bulgarian Academy of Sciences, IICT, Department of Parallel Algorithms, Sofia, Bulgaria [email protected]

Abstract. Second-order quadrature formulas and their fourth-order expansions are derived from the Taylor series of the secant and tangent functions. The errors of the approximations are compared to the error of the midpoint approximation. Third and fourth order quadrature formulas are constructed as linear combinations of the second- order approximations and the trapezoidal approximation. Key words: Quadrature formula, Fourier transform, generating function, Euler- Mclaurin formula.

1. Introduction

Numerical quadrature is a term used for numerical integration in one dimension. Numerical integration in more than one dimension is usually called cubature. There is a broad class of computational algorithms for numerical integration. The Newton-Cotes quadrature formulas (1.1) are a group of formulas for numerical integration where the integrand is evaluated at equally spaced points. Let .

The two most popular approximations for the definite integral are the trapezoidal approximation (1.2) and the midpoint approximation (1.3).

The midpoint approximation and the trapezoidal approximation for the definite integral are Newton-Cotes quadrature formulas with accuracy . The trapezoidal and the midpoint approximations are constructed by dividing the interval to subintervals of length and interpolating the integrand function by Lagrange polynomials of degree zero and one. Another important Newton-Cotes quadrat -order accuracy and is obtained by interpolating the function by a second degree Lagrange polynomial. Denote by and the errors of the trapezoidal and the midpoint approximations.

From the Euler-Mclaurin formula the error of the trapezoidal approximation on the interval is expressed as

where is the Bernoulli 1-periodic function and is the Bernoulli polynomial of degree . The error of the midpoint approximation satisfies (

The terms and satisfy , when the function . From (4), (5) and we obtain the fourth-order expansion formulas (2.1), (2.14) of the trapezoidal and midpoint rules. The midpoint approximation is around two times more accurate than the trapezoidal approximation. The approximation and the trapezoidal approximation with weights and . With this choice of the weights the second-order terms of expansion formulas (1.4) and (1.5) are . The Gaussian quadrature formulas are approximations for the definite integral where the integrand is evaluated at unequally spaced points (Gil et al. 2007). The Gaussian quadrature formulas are constructed to yield an exact result for the polynomials of degree by using suitable values of the nodes and weights . The accuracy of the Newton-Cotes and the Gaussian quadrature formulas is lower when the function has singularities. The Monte Carlo methods for numerical integration are computational algorithms where the nodes are chosen as pseudo-random or quasi-random numbers in the area of integration, the unit hypercube (Dimov 2008). The Monte-Carlo and quasi-Monte Carlo methods are suitable for approximation of multidimensional integrals, since the convergence of the methods is independent of the dimension (Atanassov & Dimov 1999; Georgieva 2009; Todorov & Dimov 2016). The generating function of quadrature formula (1.1) is defined as

The Riemann sum approximation for the definite integral is related to the trapezoidal approximation and has a generating function The Riemann sum approximation for the fractional integral of order has a generating function where is the polylogarithm function of order . The midpoint approximation is a shifted approximation with shift parameter . The midpoint approximation has a generating function . Methods for approximation of fractional integrals and derivatives based on the Fourier transform and the generating function of the approximation have been studied by (Chen & Deng 2016; Ding & Li 2016; Lubich 1986; Tuan & Gorenflo 1995). In (Dimitrov 2016) we use the series expansion formula of the polylogarithm function , to derive the expansion formula of the trapezoidal approximation for the fractional integral of order . In the present paper we construct second, third and fourth order approximations for the definite integral with generating functions and . The two generating functions and are positive, increasing functions and have a vertical asymptote at the point . The graphs of the functions and are given in Figure 1.

Figure 1. Graphs of the generating functions (left, blue) and (right, blue) and the function (red).

The tangent and secant functions have Mclaurin series expansions

The radius of convergence of power series (1.6) and (1.7) is . The first eight elements of the sequences of Euler and Bernoulli numbers are and The Euler and Bernoulli numbers with an odd index are equal to zero. The sequence of Euler numbers is an alternating sequence of integers, and is an alternating sequence of rational numbers. The Bernoulli and Euler numbers satisfy the asymptotic relation . Efficient methods for computation of the Euler numbers, Bernoulli numbers and the series expansions of the secant and tangent functions are discussed by (Atkinson 1986; Brent & Harvey 2013; Knuth & Buckholtz 1967). The values of the Euler numbers, Bernoulli numbers and the zeta function are included in the modern software packages for scientific computing. Denote by and the coefficients of the Mclaurin series expansions of the generating functions and .

In section 2 and section 3 we obtain the fourth-order expansion formulas of the approximations with generating functions and and the second-order quadrature formulas

The substitution yields the second-order quadrature formulas (2.12) and (3.6). In Theorem 2 and Theorem 4 we derive the conditions for the integrand function, such that the errors of second-order approximations are smaller than the error of the midpoint approximation. In section 4 we obtain third and fourth order approximations for the definite integral as linear combinations of approximations (2.12), (2.12), (3.6) and (3.7) and the trapezoidal approximation

2. Second-order quadrature formulas with generating function

In this section we derive the fourth-order expansion formula (2.9) of the approximation for the definite integral with generating function , and the second-order quadrature formulas (2.12) and (2.13). The generating function of an approximation is directly related to the result of the Fourier transform of the approximation. The Fourier transform of the function f(x) is defined as

In Table 1 we compute the error and the order of approximations (2.12) and (2.13) for the definite integral of the functions and . In Theorem 2 we derive the condition for the integrand function such that the errors of second-order approximations (2.12) and (2.13) are smaller than the error of the midpoint approximation. The midpoint approximation has a fourth-order expansion (1.5)

The first and second derivatives of the function satisfy

From (2.9) and (2.15) we obtain the third-order expansions of (2.12) and (2.13)

Theorem 2 Let Then (i) The error of approximation (2.12) is smaller than the error of the midpoint approximation when the integrand function satisfies

(ii) The error of approximation (2.13) is smaller than the error of the midpoint approximation when the integrand function satisfies

Proof. From (2.14) and (2.16) the error of approximation (2.12) is smaller than the error of the midpoint approximation when

Table 1. Error and order of second-order approximation (2.12) for the definite integral of the functions on the interval , on and approximation (2.13) for on .

Error Order Error Order Error Order 0.025 0.000026603 2.0083 0.00002628 1.9973 0.00007206 1.9995 0.0125 -6 2.0044 -6 1.9987 0.00001802 1.9998 0.00625 -6 2.0022 -6 1.9994 -6 1.9999 0.003125 -6 2.0011 -7 1.9997 -6 2.0000 Source: Own calculations

Table 2. Error and order of approximation (2.12)[left] and the midpoint approximation (1.3)[right] for the definite integral of on .

Error Order Error Order 0.025 -6 2.0233 0.025 -6 1.9999 0.0125 -7 2.0120 0.0125 -7 2.0000 0.00625 -8 2.0060 0.00625 -7 2.0000 0.003125 -8 2.0030 0.003125 -8 2.0000 Source: Own calculations Table 3. Error and order of approximation (2.13)[left] and the midpoint approximation (1.3)[right] for the definite integral of on .

Error Order Error Order 0.025 -6 1.9643 0.025 -6 1.9999 0.0125 -6 1.9827 0.0125 -6 2.0000 0.00625 -7 1.9915 0.00625 -7 2.0000 0.003125 -8 1.9958 0.003125 -7 2.0000 Source: Own calculations

3. Second-order quadrature formulas with generating function

Theorem 4 Let Then (i) The error of approximation (3.6) is smaller than the error of the midpoint approximation when the integrand function satisfies

(ii) The error of approximation (3.7) is smaller than the error of the midpoint approximation when the integrand function satisfies

Table 4. Error and order of second-order approximation (3.6) for the definite integral of the functions on the interval , on and approximation (2.13) for on .

Error Order Error Order Error Order 0.025 0.000038246 2.0065 0.00002628 1.9973 0.00003556 2.0006 0.0125 -6 2.0032 -6 1.9987 -6 2.0002 0.00625 -6 2.0016 -6 1.9994 -6 2.0001 0.003125 -7 2.0008 -7 1.9997 -7 2.0000 Source: Own calculations Table 5. Error and order of approximation (3.6)[left] and the midpoint approximation (1.3)[right] for the definite integral of on .

Error Order Error Order 0.025 -6 1.9643 0.025 -6 1.9999 0.0125 -6 1.9827 0.0125 -6 2.0000 0.00625 -7 1.9915 0.00625 -7 2.0000 0.003125 -7 1.9958 0.003125 -7 2.0000 Source: Own calculations Table 6. Error and order of approximation (3.7)[left] and the midpoint approximation (1.3)[right] for the definite integral of on .

Error Order Error Order 0.025 -6 2.0233 0.025 -6 1.9999 0.0125 -7 2.0120 0.0125 -7 2.0000 0.00625 -8 2.0060 0.00625 -7 2.0000 0.003125 -8 2.0030 0.003125 -8 2.0000 Source: Own calculations

4. Higher-order quadrature formulas

By substituting in (2.12) and (3.6) we obtain

4.1 Third-order quadrature formulas

fourth order approximations (4.2), (4.3) and (4.4) for the definite integral as linear combinations of the trapezoidal approximation (1.2) and approximations (2.12), (2.13), (3.6) and (3.7). Let

where, the numbers satisfy the system of equations

The condition for the coefficients ensures that is an approximation for the definite integral. The second and third equations of (4.1) are chose such that the second order terms of at the endpoints and are equal to zero. The system of equations (4.1) has the solution

With this choice of the coefficients , approximation for the definite integral has third-order accuracy

The weights of approximation satisfy

where, the numbers satisfy

The system of equations has the solution

We obtain the third-order approximation for the definite integral

and .

4.2 Fourth-order quadrature formula

We construct a fourth-order quadrature formula as a linear combination of the trapezoidal approximation and approximations (2.12), (2.13), (3.6), (3.7). Let

Approximation has a fourth-order accuracy when the coefficients satisfy

Let . The system of equations has the solution

With this choice of the coefficients we obtain the fourth-order approximation for the definite integral

where and

and .

Table 7. Error and order of third-order approximation (4.2) for the definite integral of the functions on the interval .

Error Order Error Order Error Order 0.025 0.00060862 3.0494 -6 2.9722 -6 2.9498 0.0125 0.00007386 3.0427 -7 2.9857 -7 2.9740 0.00625 -6 3.0261 -7 2.9928 -8 2.9868 0.003125 -6 3.0143 -8 2.9969 -9 2.9945 Source: Own calculations

Table 8. Error and order of third-order approximation (4.3) for the definite integral of the functions on the interval .

Error Order Error Order Error Order 0.025 0.00058322 3.0509 -6 2.9735 -6 2.9520 0.0125 0.00007083 3.0416 -6 2.9864 -7 2.9752 0.00625 -6 3.0251 -6 2.9930 -8 2.9872 0.003125 -6 3.0137 -7 2.9953 -9 2.9909 Source: Own calculations Table 9. Error and order of third-order approximation (4.4) for the definite integral of the functions on the interval .

Error Order Error Order Error Order 0.025 0.00016129 4.1502 0.00014554 3.3684 0.00017815 2.7198 0.0125 -6 4.1317 0.00001189 3.6129 0.00001474 3.5951 0.00625 -7 4.0828 -7 3.7781 -7 3.8829 0.003125 -8 4.0469 -8 3.8583 -8 3.9805 Source: Own calculations

5. Conclusion

In the present paper we obtained second, third and fourth order quadrature formulas based on the expansion formulas of the approximations for the definite integral with generating functions and . In Theorem 2 and Theorem 4 we derive the conditions for the integrand function which ensure that the error of second-order quadrature formulas (2.12), (2.13), (3.6) and (3.7) is smaller than the error of the midpoint rule. In future work we are going to extend the results of this paper to multidimensional and fractional integrals.

Acknowledgements This work was supported by the Bulgarian Academy of Sciences through the Program for Career Development of Young Scientists, Grant DFNP-17-88/2017, Project "Efficient Numerical Methods with an Improved Rate of Convergence for Applied Computational Problems'' and by the Bulgarian National Science Fund under Grant DFNI I02-20/2014, Project "Efficient Parallel Algorithms for Large-Scale Computational Problems''.

Literature Abramowitz, M. & Stegun I. A., 1964. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York. Atanassov, E. & Dimov, I., 1999. A new optimal Monte Carlo method for calculating integrals of smooth functions. Monte Carlo Methods and Applications 5(2), pp. 149- 167. Atkinson, M. D., 1986. How to compute the series expansions of secx and tanx. Amer. Math. Monthly 93, pp. 387-388. Borwein, J. M., Borwein, P. B. & Dilcher, D., 1989. Pi, Euler numbers and asymptotic expansions. The American Mathematical Monthly 96(8), pp. 681-687. Brent, R. P. & Harvey, D., 2013. Fast computation of Bernoulli, tangent and secant numbers. Proceedings of amWorkshop on Computational and Analytical Mathematics in honour of Jonathan Borwein Statistics, 50, pp. 127-142. Chen, M. & Deng, W., 2016. A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation. arXiv:1610.02661. Davis, P. J. & Rabinowitz, P., 1984. Methods of numerical integration: Second edition, Academic Press. Dimitrov, Y., 2014. Numerical approximations for fractional differential equations. Journal of Fractional Calculus and Applications, 5(3S), No. 22, pp. 1-45. Dimitrov, Y., 2016. Second-order approximation for the Caputo derivative. Journal of Fractional Calculus and Applications, 7(2), pp. 175-195. Dimitrov, Y., 2016. Higher-order numerical solutions of the fractional relaxation-oscillation equation using fractional integration. arXiv:1603.08733. Dimov, I., 2008. Monte Carlo methods for applied scientists. World Scientific. Ding, H. & Li, C., 2016. High-order numerical algorithms for Riesz derivatives via constructing new generating functions. Journal of Scientific Computing. Ding, H. & Li, C., 2016. High-order algorithms for Riesz derivative and their applications (III). Fract. Calc. Appl. Anal. 19(1), 19 55 (2016) Georgieva, R., 2009. Computational complexity of Monte Carlo algorithms for multidimensional integrals and integral equations. Ph.D. thesis, BAS. Gil, A., Segura, J. & Temme, N. M., 2007. Numerical methods for special functions. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Knuth, D. E. & Buckholtz, T. J., 1967. Computation of tangent, Euler and Bernoulli numbers. Math. Comput. 21, pp. 663 688. Kouba O., 2013. Bernoulli polynomials and applications. Lecture Notes. Lubich, C., 1986. Discretized fractional calculus. SIAM J. Math. Anal. 17, pp. 704-719. Sahakyan, K., 2014. New quadrature and cubature formulas of Hermite type. Universal Journal of Applied Mathematics 2(9), pp. 299-304. Tadjeran, C., Meerschaert, M. M. & Scheffer, H. P., 2006. A second-order accurate numerical approximation for the fractional diffusion equation. Journal of Computational Physics 213, pp. 205-213 (2006) Tian, W. Y., Zhou, H. & Deng, W.H., 2015. A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703-1727. Todorov, V. & Dimov, I., 2016. Monte Carlo methods for multidimensional integration for European option pricing. AIP Conf. Proc. 1773, 100009Atkinson, M. D., 1986. Tuan, V. K. & Gorenflo, R., 1995. Extrapolation to the limit for numerical fractional differentiation. Z. Agnew. Math. Mech. 75, pp. 646-648. doi:10.1002/19950750826.