A method of determination for the interval of convolution integral by use of the graphic partition
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A method for determining the interval of convolution integrating by use of the graphic partition is given in this paper,and the application and promotion of this method are exemplified.By use of this method,the interval of independent variable t and the mute variable τin the convolution integral can be determined quickly,only if in the two convoluted functions which value of the end point of the interval is unequal to zero is known,and the computational result is in closed form.This method can avoid the repetition and pretermission of integrating interval(or summarizing interval),especially in the convolution computation of multi-partition function.Keywords:
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By utilizing the relationship between the knot intervals,a method for calculating the transform matrix between different B-spline basis functions is presented.Furthermore,an interval skipping algorithm is given to calculate B-spline functions multiplication.This algorithm is so-called just as only part of knot intervals need to be considered to calculate rapidly B-spline functions multiplication.This algorithm partially solves the multiplication of piecewise polynomial and B-spline curve;and it can be applied into smooth B-spline surfaces blending and elevation of B-spline.Some examples are given to illustrate the efficiency of the presented algorithm.
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Integrals of a function of a single variable can be expressed as the sum of a numerical quadrature rule and a remainder term. The quadrature rule is a linear combination of function values and weights, or the integral of a Taylor polynomial, while the remainder term depends on some derivative of the integrand evaluated at an unknown point in the interval of integration. Numerical quadrature is made self validating by using interval computations to capture both the roundoff and truncation errors made when using a given formula. Necessary derivatives can be generated automatically by using well-known recurrence relations for Taylor coefficients. In order for quadrature methods of this type to be accurate (in the sense that small intervals are produced) and efficient (to obtain results of given accuracy in a reasonably short time), an accurate scalar product and an adaptive strategy are required. The necessary scalar product and support for interval arithmetic are provided in Pascal-SC (for microcomputers) and ACRITH (for IBM 370 computers). The adaptive strategy chooses the subintervals of integration and the order of the quadrature formula in each subinterval on the basis of guaranteed, rather than estimated, information about the error of numerical integration in each subinterval. The program described here implements standard Newton-Cotes, Gaussian and Taylor series formulas for numerical integration. The handling of singularities is discussed, and comparisons are given with a standard numerical integration method.
Adaptive quadrature
Tanh-sinh quadrature
Clenshaw–Curtis quadrature
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Gauss–Hermite quadrature
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This paper presents a hybrid method for finding real solution of nonlinear equations with arbitrary precision. In contrast with symbolic computation, the system of nonlinear equations don't need to be triangularized. In the procedure of computation, we combine the methods, including contraction of the initial interval by analysis, factorization and squarefree decomposition, to preprocess the nonlinear systems firstly. Then, interval dichotomy is used to bisect the designed interval vector. After this,it is examined whether there is zero point in each sub-interval vector. If these is no solution in sub-interval vector,the sub-interval vector is abandoned, or else we use multivariate Newton Gauss-Seidel method with symbolic preconditioner to refine this sub-interval vector. It is one solution if each interval in interval vector is not greater than the tolerance. Or else, the above procedure is repeated till the error is less than the tolerance. In the algorithm, as interval dichotomy and extended interval division are used, it is certain that all sub-interval boxes can be examined to guarantee all real roots of system of nonlinear equation can be attained. Its performance is shown in solving examples from various applications. Finally, it is pointed out that there is some related works to be researched further. This method can solve some complex problem in practice effectively.
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In linear control theory the frequency domain approach and the state-space approach are equivalent. But for the implementation of numerical algorithms the use of state-space equations is preferred. To avoid the numerical problems caused by a polynomial arithmetic, the use of an interval arithmetic with variable length of the mantissa is suggested. Some basic algorithms for polynomial and transfer function matrices implemented in interval arithmetic will be stated. A control example shows the feasibility of this method.
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In this paper, we present a new approach which is based on the technique of parametric identification to estimate one‐dimensional definite integrals. So we suppose a continuous function on the closed interval. This function can be represented at the points of continuity by Fourier series. A basic description of this method is that you take a number N of samples. For each sample, we estimate the function value for the considered function. We sum all of these values, and divide by N to get the mean value from our samples. We then multiply this value by the interval to get the integral.The method is implemented and tested on MATLAB for several examples. Simulations results confirmed the analytical methods, and the error of estimation can be more accurate. Furthermore, we give evidence that the method proposed here is much simpler and use less arithmetic operations, this method is especially suitable for digital signal processing.
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