This software is for educational purposes. Mulptile Integration is an easy to use, intuitive application to numerically perform definite multiple integration. You can predefine constants and specify the error bound for the results. Handles a wide variety of functions, including trigonometric and hyperbolic functions. Results can be saved or printed. Includes a help file with instructions, example and methodology. Numerical Solutions Library : The Ordinary Differential Equations Collection solves linear or nonlinear differential equations and systems of such, subject to boundary conditions or initial conditions which may also be linear or nonlinear and may involve not only the unknown function but also its derivatives. The solution will be a finite power or trigonometric series, depending on the program.The Regression Collection fits user-chosen functionals to a given set of data points in one or more independent variables, performing linear or nonlinear regressions.The Approximation and Interpolation Collection approximates multivariable continuous or tabulated functions by finite power, trigonometric or mixed series. Continuous functions may be defined explicitly or by a linear/nonlinear equation. Depending on the program and the user setups, the function may be fitted exactly at user-specified grid points, or it may be fitted using Least Squares methods/Fast Fourier Transform methods.The Linear Algebra Collection performs computations associated with real matrices, including solution of linear systems of equations (even least squares solution of over-determined or inconsistent systems and solution by LU factors), matrix operations (add, subtract, multiply), finding the determinant, inverse, adjoint, QR or LU factors, eigenvalues and eigenvectors, establish the definiteness of a symmetric matrix, perform scalar multiplication, transposition, shift, create matrices of zeroes or ones, identity, symmetric or general matrices.The Support Collection provides ways to study the solutions found by programs of other collections. For instance, the Power and Trigonometric Series program may be used to study the solutions found by the programs of the Differential Equations collection to find integrals, roots, maxima & minima, derivatives. Even the multivariable series produced by the Approximation and Interpolation collection can be studied by support programs to compute multiple integrals, maxima & minima, partial derivatives.The Math Miscellaneous Collection is an useful and interesting addition to the library: Prime factors, base conversion, complex numbers, sorting, formulas for triangles and circles.The Stereographer Collection produces stereoscopic graphs, which most people are able to view without any paraphernalia. True depth perception is due to binocular vision, and the usual single view of a surface, space curve or scatter diagram is unable to provide it, but the STEREOGRAPHER does provide this interesting 3-D sensation. You can rotate and translate surfaces, graph partial derivatives, zoom in or out, etc. Aside from functions you enter directly, you can also stereograph the results of other collections such as mixed series, regression surfaces and scatter diagrams. The techniques used to achieve a solution have, in general, been selected in part because of their applicability in a wide variety of situations. For instance, the Nonlinear Differential Equations programs use the method of Undetermined Coefficients in conjunction with the generalized Newton method to find a finite power or trigonometric series approximation to the unknown function. This permits them to solve true Boundary Value Problems involving General Nonlinear Differential Equations with General Nonlinear Boundary Conditions.
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The Mulptile Integration of a function over a region is defined as the integral of f(x) over the region D. Each region, D, is first split into a set of sub-regions, each of which has a polygonal boundary, r, and area, A. The interval, [a, b], is subdivided into a set of closed subintervals, [x, x+h], and the function is approximated by one or more power or trigonometric series of order n, defined over the subintervals. The power or trigonometric series is truncated to give the user an initial estimate, a, and a polynomial or trigonometric series with degree n has n + 1 nodes. The n terms of the polynomial or series are then multiplied by the series coefficients which have been previously calculated over the closed subintervals. This gives a polynomial approximation, denoted pn, which is estimated at the nodes. The formula for the error bound, E, is: ErrorBound = Max(f(b)-pn(b), 0.0) ; where f(b) is the exact solution at the node, and pn(b) is the approximation at the node The areas of the sub-regions are calculated by using the trapezoidal rule. Each term, pt, of the sum that computes the error bound is a function of the areas of the sub-regions of the current region. The error of the total area is computed by using the estimate of the error at the first sub-region. The value of the total error for the current region is the product of the error in the total area, times the total area. The value of the total error is the maximum of the error bounds for the sub-regions. The error bounds for the sub-regions, for the n-polynomial, must also be computed and summed before the total error is computed. If the user requests, the error bounds for the first sub-region are calculated in advance. The latter are computed by using the sum of the areas of the sub-regions. Here is an example of a common integral (integral of the Heaviside function over [0, 1]): Input example : a = log(x)^2 b = -x Output example : The error bound for the n = 10,000
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Multiple Integration is the mathematical method of evaluation of definite integrals; it was first developed by the Indian mathematician Brahmagupta in the 6th Century CE. Features: Many types of integration, including Complex, Trigonometric, Logarithmic, Common Logarithm, Exponential, Factorial, Logarithmic, Polylogarithmic, and Polynomial. Functions include logarithmic and transcendental. It's possible to work with 3-D surfaces and scatter plots. The program will find the exact numeric value of the definite integral of the selected function over the interval of integration. The integral is not evaluated exactly, but is evaluated using a method which minimizes the error in the approximation and thereby preserves the numeric value of the integral. This is done through the use of a program which is capable of transforming integrals into certain types of series. The user may also establish a specific type of series to be used to evaluate the definite integral. For example, trigonometric, infinite series or finite power series. Each type of series has a limit or error bound which controls the degree of accuracy for the definite integral. The program has command line parameters to help automate the user. The user may set several different methods for performing multiple integrals and may also specify the types of series to be used to evaluate the definite integrals. The accuracy of the results for each type of series may be controlled with command line parameters. The Logarithmic Function can be used to evaluate integrals involving factorial, exponential and other power functions. The results may be displayed as tables, graphs, histograms, or stored in the library as text. Expected Error (E) controls the degree of accuracy (magnitude) of the results for each method. The larger the error bound, the more accurate the results. The program will attempt to find an exact numeric value for the integral and the user may specify a numeric value for the error bound. Error-bound controls the degree of precision of the results and is used to control the accuracy of the numeric results. The larger the error bound, the more precise the results will be. The program will attempt to find an exact numeric value for the integral and the user may specify a numeric value for the error bound. User-defined accuracy and precision may be set with command line parameters. The program can be run interactively with command line parameters to help automate the user. User-defined functions may be added to the library.
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Multiply Integrals is an application to handle multiple integral computations. It can be used to calculate definite or indefinite integration of one or multiple integrals. The method of operation is to calculate the definite integrals directly. The indefinite integrals are then calculated by the related definite integrals. You can predefine constants and specify the error bound for the results. You can also change the method of operation.Many Integrals : ManyIntegrals is a highly useful collection of functions. It provides comprehensive programs to calculate definite and indefinite integrals for various functions, and for fixed and variable integration limits. By using of These integrals, one can calculate definite integrals of trigonometric, exponential, logarithmic, inverse trigonometric, hyperbolic, logarithmic, inverse hyperbolic, power, and inverse power series.You can add new functions to ManyIntegrals from user-defined integrals. All integrals are added to the ManyIntegrals collection in the ManyIntegrals source code file. ... Concurrent Collections : Concurrent Collections is the science of concurrent programming. Programs that use the same data from different memory locations (sometimes known as noncoherent) may yield unpredictable results. In particular, two or more independent processes using the same variable value may be unable to synchronize with each other or their results may not always be consistent. A common example of a noncoherent program is a web browser. When two browsers accessing the same URL open the same web page, they may display different contents. If one browser accesses and the other accesses then the browsers may display different contents. This phenomenon is known as thread safety. Concurrent Collections is an easy to use, intuitive application to numerically perform definite multiple integration. You can predefine constants and specify the error bound for the results.Handles a wide variety of functions, including trigonometric and hyperbolic functions. Results can be saved or printed. Includes a help file with instructions, example and methodology.Numerical Solutions Library : The Ordinary Differential Equations Collection solves linear or nonlinear differential equations and systems of such, subject to boundary conditions or initial conditions which may also be linear or nonlinear and may involve not only the unknown function but also its derivatives. The solution will be a finite power or trigonometric series, depending on the program.The Regression Collection fits user-ch
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