数值积分方法的比较教学研究与实验刘鹏飞

Comparative teaching research and experiment ofnumerical integration methodLiuPengfei, XUNainanAbstract:Numerical integration is a very important teaching content in the course of calculation method or numerical analysis. The numerical integration method is also an important method to solve practical problems.In this paper, we discuss the basic ideas of numerical integration method and some in the actual teaching comparative research and experiment teaching. Keywords:Numerical integrationCalculation methodNumerical analysisNumerical experimentsText:The invention of calculus is a great achievement in the history of human science. However, in practical problems, there are still many limitations in solving integral. If the product function is often very complex, can not find the corresponding original function, or the product function is not a function expression, butin the form of a table, its original function is not meaningful.Therefore, it is significant to investigate the numericalintegration method of approximate calculation.1.Basic ideas of numerical integration methodBy the integral mean value theorem, it is known that (x), b] (Qbaf) dx= (B-A) f (N) is set up in [a (x), which is a continuous function f (N). Problem is difficult to accurately calculate the average height of F (n) values, the different algorithms of F (n), you can get different numerical quadrature method, such as left (right) formula of rectangular, trapezoidal, Simpson formula. Generally, [a, b] some node XK at the height of F (XK) weighted average to approximate the average height of F (n), f (XK) linear combination to approximate the definite integral, mechanical quadrature formula:Qbaf(x)dxUEnk=0Akf(xk)Directly using the function computed values in the discrete node function integral, avoids the difficulty in seeking the original function, only need to be determined with the product function andproductAkquadrature coefficients are independent between partitions.In order to ensure the accuracy of the mechanical quadrature formula, it is natural to want accurate simple function as much as possible. Namely, it is required that all the m polynomials are exact, and the m+1 time polynomials are not necessarily accurate. was about coefficient of n + 1 order linear equations. 101n 220011101A A +.....A =b-a A +A x +....+A x =2A X +A X +....+A X =1n n n n n n n n n n b a x b a n -+⎧⎪++⎪-⎪⎨⎪⎪-⎪+⎩Because the coefficient determinant Vandermondedeterminant, and it is not zero,solutionAk isunique.2. Teaching analysis of several commonnumerical integration method2.1 The interpolation type quadrature formulaIn [a, b], to node n Lagrange interpolationpolynomial (x) as f(x) as the approximation function,can be obtained by interpolation quadrature formula:bb n a an bk k a k=0f x dx L x dx =f x l x dx ≈⎰⎰∑⎰（）（）（）（）即n bk a k=001110111f x dx A f (x)(x x )(x x )(x x )(x x )(x x )(x x )(x x )(x x )(x x )(x x )k bk k a bk k n a k k k k k k k n l dxdx -+-+≈=-----=-----∑⎰⎰⎰（）（x ）A The interpolation type integral formula has the accuracy of N sub algebra, and the formula of Ak>0 is stable.2.2 Newton - Cotes formulasThe[a b] is divided into N parts, interpolation quadrature formula for:k 02(n)1(n)(x)dx (x A (b a)(1)(t 1)...(t n)(k!)(n k)!nbk a k n k i k i n k b k a f A f X C t C dt n t k==-≈=----=--∑⎰∑⎰）Called Newton- Cotes formula, the Cotes coefficient is independent of the product function and the integral interval, as long as the interval and other fractions n, can be obtained. Obviously, when n= 1 is a trapezoidal formula; when n= 2 is Simpson formula.2.3 Complex formulaBy the knowledge of definite integral, the definiteintegral is only related to the integral interval.The higher the number of polynomials is, the higherthe degree of smoothness of the function is, and theRunge phenomenon occurs. Such as n> 7, the Newton-Cotes formula is not stable. Therefore, people turnto the integral interval, similar to the piecewiseinterpolation, the integral interval is divided intoa number of small areas, the use of the Newton- Cotesformula between each cell number, and then theresults of each cell is added as a function of thewhole range of the approximation, which is the basicidea of complex. Commonly used compound formula, complex trapezoid formula:1110(x x )(x)((x )(x ))2n bk k n k k a k I f dx T f f --==-=≈=+∑⎰ Complex Simpson formula:212121(x)(f ()4(x )(x ))3mbn k k k a k h f dx S f x f f --=≈==+∑⎰ 2.4 Successive bisection technology and Rombergformula How to determine the appropriate n so that the difference between the approximate value and the truevalue of the allowable range, generally speaking, it is more difficult. And successive half technology is in quadrature process according to the requirements of precision and automatically determine n choose whether or not to meet the requirements of precision, the difference of two minutes before and after twice to estimate the error, such not only reduce the step size, and can keep the original calculating results, reduce the amount of computation. For be obtained:2222221(T T )31(S )151(C )63n n n n n n n n n I T I S S I C C ≈+-≈+-≈+- 222413316115156416363n n n n n n n n n S T T C S S R C C =-=-=- Called Romberg formula. The acceleration effect of Romberg formula is very significant, and thecalculation amount is very small under the same precision.2.5Gauss type quadrature formulaTo further improve the algebraic precision of quadrature formula, through the proper selection of interpolation nodes and quadrature coefficients, the algebraic accuracy up to a maximum of (2n + 1). The quadrature nodes and quadrature coefficients as parameters. The equations can be obtained by using the orthogonal polynomials.You can use orthogonal polynomials with different weighting functions of Gauss type quadrature formulas of different types areparison of several classes of numerical integration methods in TeachingDue to practical problems should be taken into accountwhen the calculation precision and calculation, thus different quadrature formulas have different characteristics:Trapezoidal quadrature formula and Simpson quadrature formula although the calculation is simple, easy to use, but poor accuracy, but for less smooth integrand function sometimes ratio, high precision method is more effective. The effect of the periodic function is more prominent, especially the trapezoidal formula. Newton-Cotes, the n>7 formula is unstable, while the complex trapezoid formula and Simpson formula in the retention of the advantages of low order formula, and can get higher accuracy, so the most widely used in the actual calculation.Romberg method using two points method is simple and easy to program. When the node encryption improve integral approximation degree, previous calculation results can be used for the back, very beneficial to reduce the calculation amount, and the estimation error is relatively simple, can get some integral sequence, if in the convergence of the control, and to check the rows and columns, to function in different state canwith the fastest convergence sequence to approximate the integral.High precision Gauss type quadrature formulas, especially other methods cannot match the calculation of infinite integral and defect integral. But because of the node is irregular, when the node is increased, the function value of the previous calculation can not be used, and the N value of the node value and the coefficient of the different values are also stored in the computer.4.Numerical integration method in the teaching of numerical experimentsIn the teaching process of numerical integration method, the numerical experiments are carried out to improve the students' understanding of the teaching contents and to master the basic method of numerical integration. The Matlab program is written in each method, and the advantages and disadvantages of various numerical integration methods are grasped and understood by comparison.4.1 Comparison of the accuracy of different algorithms with the same number of nodesIn the course of teaching, the accuracy of various methods can be determined, but it is not conducive to students' understanding. Using the powerful function of the popular software, the students can be used to operate the machine. It can be used to realize the accuracy of different numerical integration methods, and it can also improve the students' experimental ability and practical ability.For example the precision parison with the exact solutions of the trapezoidal formula, Simpson formula, Gauss formula. Use Matlab to write programs to run after the results show[Precision Trapezoidal Simpson Gauss][0.7853981635 0.7849814972 0.7853981535 0.7853981635]Obviously, the trapezoidal formula with low accuracy, low Simpson formula and the Gauss quadrature formula is better, the highest precision in a given precision and exact solutions of (Precision) agreement. And then describes the method of various quadrature Matlab commands by calling and format, compared with their ownprogramming, and make the students master the calculation method of numerical integration, in the calculation of actual problems skillfully.4.2 Comparison of the different points of the same algorithm.integrate Using trapezoid formula considering the same algorithm sync long more or different integral interval of scores in the comparison of calculation accuracy and computation time, let the students fully understand in numerical calculation of integral interval the importance of correct treatment. Matlab programming operation result shows that:Interval number:[10 20 40 80]The calculation results:[0.63050.6228 0.6209 0.6204](精确解为: 0.6203)Computation time:[0 0 0 0.0100]Graphical presentation:In short, along with the rise of the mathematics experiment, it has played a positive role in promoting the whole mathematics curriculum teaching reform, but also put forward a severe test to the reform of teaching methods, methods and contents of mathematics curriculum. In the teaching process of the traditional mathematics course, the theory is strong, the more boring, the full use of modern teaching methods, to cultivate students' ability of practical experiments and application of mathematical theory and computer technology to solve practical problems, to cultivate more adapted to the times and social development needs, has a strong practical ability of high qualitypersonnel with very important significance.[References][1] Xu Cui Wei. 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