Numerical Analysis: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=207
en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 14 Jul 2020 20:53:32 GMTTue, 14 Jul 2020 20:53:32 GMTNew applications in the Numerical Analysis categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgNumerical Analysis: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=207
Car Loan Calculator
https://www.maplesoft.com/applications/view.aspx?SID=145174&ref=Feed
<p>This loan calculator facilitates the life of a borrower. By entering the <strong>purchase price</strong>, the <strong>down payment</strong>, the <strong>number of years</strong> it takes to repay the loan, the <strong>payment frequency</strong>, the <strong>annual interest rate</strong>, and clicking on the "<strong>calculate</strong>" buttom, the calculator will give you the <strong>amount of payment</strong> for each payment period.</p>
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<p>Want to make a loan? Try it out and see how things change with respect to each element.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Car Loan Calculator" style="max-width: 25%;" align="left"/><p>This loan calculator facilitates the life of a borrower. By entering the <strong>purchase price</strong>, the <strong>down payment</strong>, the <strong>number of years</strong> it takes to repay the loan, the <strong>payment frequency</strong>, the <strong>annual interest rate</strong>, and clicking on the "<strong>calculate</strong>" buttom, the calculator will give you the <strong>amount of payment</strong> for each payment period.</p>
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<p>Want to make a loan? Try it out and see how things change with respect to each element.</p>https://www.maplesoft.com/applications/view.aspx?SID=145174&ref=FeedWed, 27 Mar 2013 04:00:00 ZZinan WangZinan WangPhénomène de Runge - subdivision de Chebychev
https://www.maplesoft.com/applications/view.aspx?SID=35301&ref=Feed
<p>On observe d'abord la divergence du polynôme de Lagrange interpolant la fonction densité de probabilité de la loi de Cauchy lorsque la <strong>subdivision est équirépartie</strong> sur [-1;1]. C'est le <u>phénomène de Runge</u>.<br />
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On observe ensuite qu'en choisissant une <strong>subdivision de Chebychev</strong> le phénomène de divergence au voisinage des bornes disparait.<br />
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Cette activité a été réalisé dans le cadre de la préparation à l'agrégation interne de mathématiques de Rennes le 10 Mars 2010.<br />
Les nouveaux programmes du concours incitent à proposer des exercices utilisant les TICE. Il semble difficile de proposer une preuve convaincante du phénomène de Runge pour une épreuve orale. Ceci justifie de ne s'en tenir qu'à la seule observation.</p><img src="https://www.maplesoft.com/view.aspx?si=35301/thumb.jpg" alt="Phénomène de Runge - subdivision de Chebychev" style="max-width: 25%;" align="left"/><p>On observe d'abord la divergence du polynôme de Lagrange interpolant la fonction densité de probabilité de la loi de Cauchy lorsque la <strong>subdivision est équirépartie</strong> sur [-1;1]. C'est le <u>phénomène de Runge</u>.<br />
<br />
On observe ensuite qu'en choisissant une <strong>subdivision de Chebychev</strong> le phénomène de divergence au voisinage des bornes disparait.<br />
<br />
Cette activité a été réalisé dans le cadre de la préparation à l'agrégation interne de mathématiques de Rennes le 10 Mars 2010.<br />
Les nouveaux programmes du concours incitent à proposer des exercices utilisant les TICE. Il semble difficile de proposer une preuve convaincante du phénomène de Runge pour une épreuve orale. Ceci justifie de ne s'en tenir qu'à la seule observation.</p>https://www.maplesoft.com/applications/view.aspx?SID=35301&ref=FeedFri, 26 Mar 2010 04:00:00 ZKERNIVINEN SebastienKERNIVINEN SebastienComparison of Multivariate Optimization Methods
https://www.maplesoft.com/applications/view.aspx?SID=1718&ref=Feed
<p>The worksheet demonstrates the use of Maple to compare methods of unconstrained nonlinear minimization of multivariable function. Seven methods of nonlinear minimization of the n-variables objective function f(x1,x2,.,xn) are analyzed:</p>
<p>1) minimum search by coordinate and conjugate directions descent; 2) Powell's method; 3) the modified Hooke-Jeeves method; 4) simplex Nelder-Meed method; 5) quasi-gradient method; 6) random directions search; 7) simulated annealing. All methods are direct searching methods, i.e. they do not require the objective function f(x1,x2,.,xn) to be differentiable and continuous. Maple's Optimization package efficiency is compared with these programs. Optimization methods have been compared on the set of 21 test functions.</p><img src="https://www.maplesoft.com/view.aspx?si=1718/SearchPaths2.PNG" alt="Comparison of Multivariate Optimization Methods" style="max-width: 25%;" align="left"/><p>The worksheet demonstrates the use of Maple to compare methods of unconstrained nonlinear minimization of multivariable function. Seven methods of nonlinear minimization of the n-variables objective function f(x1,x2,.,xn) are analyzed:</p>
<p>1) minimum search by coordinate and conjugate directions descent; 2) Powell's method; 3) the modified Hooke-Jeeves method; 4) simplex Nelder-Meed method; 5) quasi-gradient method; 6) random directions search; 7) simulated annealing. All methods are direct searching methods, i.e. they do not require the objective function f(x1,x2,.,xn) to be differentiable and continuous. Maple's Optimization package efficiency is compared with these programs. Optimization methods have been compared on the set of 21 test functions.</p>https://www.maplesoft.com/applications/view.aspx?SID=1718&ref=FeedTue, 15 Sep 2009 04:00:00 ZOrthogonal Functions, Orthogonal Polynomials, and Orthogonal Wavelets series expansions of function
https://www.maplesoft.com/applications/view.aspx?SID=7256&ref=Feed
The worksheet includes all the best known continuous orthogonal series expansions in the closed form. It demonstrates the use of Maple to evaluate expansion of a function by Fourier, Hartley, Fourier-Bessel, Orthogonal Rational Tangent, Rectangular, Haar Wavelet, Walsh, Slant, Piece-Linear-Quadratic, Associated Legandre, Orthogonal Rational, Generalized sinc, Sinc, Sinc Wavelet, Jacobi, Chebyshev first kind, Chebyshev second kind, Gegenbauer, Generalized Laguerre, Laguerre, Hermite, and classical polynomials orthogonal series. Also the worksheet demonstrates how to create new orthonormal basis of functions by using the Gram-Schmidt orthogonalization process by the example of Slant, and Piece-Linear-Quadratic orthonormal functions creating.<img src="https://www.maplesoft.com/view.aspx?si=7256/thumb.gif" alt="Orthogonal Functions, Orthogonal Polynomials, and Orthogonal Wavelets series expansions of function" style="max-width: 25%;" align="left"/>The worksheet includes all the best known continuous orthogonal series expansions in the closed form. It demonstrates the use of Maple to evaluate expansion of a function by Fourier, Hartley, Fourier-Bessel, Orthogonal Rational Tangent, Rectangular, Haar Wavelet, Walsh, Slant, Piece-Linear-Quadratic, Associated Legandre, Orthogonal Rational, Generalized sinc, Sinc, Sinc Wavelet, Jacobi, Chebyshev first kind, Chebyshev second kind, Gegenbauer, Generalized Laguerre, Laguerre, Hermite, and classical polynomials orthogonal series. Also the worksheet demonstrates how to create new orthonormal basis of functions by using the Gram-Schmidt orthogonalization process by the example of Slant, and Piece-Linear-Quadratic orthonormal functions creating.https://www.maplesoft.com/applications/view.aspx?SID=7256&ref=FeedWed, 18 Feb 2009 00:00:00 ZDr. Sergey MoiseevDr. Sergey MoiseevSimpson and Trapezoidal Methods for Numerical Integeration
https://www.maplesoft.com/applications/view.aspx?SID=6937&ref=Feed
Using this maplet, one can solve a integrate a function numerically using the Simpson and Trapezoidal Methods.<img src="https://www.maplesoft.com/view.aspx?si=6937/Untitled-1.jpg" alt="Simpson and Trapezoidal Methods for Numerical Integeration" style="max-width: 25%;" align="left"/>Using this maplet, one can solve a integrate a function numerically using the Simpson and Trapezoidal Methods.https://www.maplesoft.com/applications/view.aspx?SID=6937&ref=FeedFri, 28 Nov 2008 00:00:00 ZShahzad BhattiShahzad BhattiComparison Between Newton, Householder and Halley Methods
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In this maplet, You can compare your results obtained by the Newton, Householder and Halley methods. You can choose accuracy that you need and can also check the plot of the function etc.<img src="https://www.maplesoft.com/view.aspx?si=6938/Untitled-1.jpg" alt="Comparison Between Newton, Householder and Halley Methods" style="max-width: 25%;" align="left"/>In this maplet, You can compare your results obtained by the Newton, Householder and Halley methods. You can choose accuracy that you need and can also check the plot of the function etc.https://www.maplesoft.com/applications/view.aspx?SID=6938&ref=FeedFri, 28 Nov 2008 00:00:00 ZShahzad BhattiShahzad BhattiTraveling Salesman Problem
https://www.maplesoft.com/applications/view.aspx?SID=6873&ref=Feed
The Traveling Salesman Problem (TSP) is a fascinating optimization problem in which a salesman wishes to visit each of N cities exactly once and return to the city of departure, attempting to minimize the overall distance traveled. For the symmetric problem where distance (cost) from city A to city B is the same as from B to A, the number of possible paths to consider is given by (N-1)!/2. The exhaustive search for the shortest tour becomes very quickly impossible to conduct. Why? Because, assuming that your computer can evaluate the length of a billion tours per second, calculations would last 40 years in the case of twenty cities and would jump to 800 years if you added one city to the tour [1]. These numbers give meaning to the expression "combinatorial explosion". Consequently, we must settle for an approximate solutions, provided we can compute them efficiently. In this worksheet, we will compare two approximation algorithms, a simple-minded one (nearest neighbor) and one of the best (Lin-Kernighan 2-opt).<img src="https://www.maplesoft.com/view.aspx?si=6873/TSgif.gif" alt="Traveling Salesman Problem" style="max-width: 25%;" align="left"/>The Traveling Salesman Problem (TSP) is a fascinating optimization problem in which a salesman wishes to visit each of N cities exactly once and return to the city of departure, attempting to minimize the overall distance traveled. For the symmetric problem where distance (cost) from city A to city B is the same as from B to A, the number of possible paths to consider is given by (N-1)!/2. The exhaustive search for the shortest tour becomes very quickly impossible to conduct. Why? Because, assuming that your computer can evaluate the length of a billion tours per second, calculations would last 40 years in the case of twenty cities and would jump to 800 years if you added one city to the tour [1]. These numbers give meaning to the expression "combinatorial explosion". Consequently, we must settle for an approximate solutions, provided we can compute them efficiently. In this worksheet, we will compare two approximation algorithms, a simple-minded one (nearest neighbor) and one of the best (Lin-Kernighan 2-opt).https://www.maplesoft.com/applications/view.aspx?SID=6873&ref=FeedMon, 10 Nov 2008 00:00:00 ZBruno GuerrieriBruno Guerrieri3D spline interpolation
https://www.maplesoft.com/applications/view.aspx?SID=5644&ref=Feed
This worksheet gives an example of how to use the Maple spline function to create a 3 dimensional spline surface and a function R^2->R, based on discrete values given with respect to their axes in a matrix with the first row containing the x-values and the first column containing the y-values.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="3D spline interpolation" style="max-width: 25%;" align="left"/>This worksheet gives an example of how to use the Maple spline function to create a 3 dimensional spline surface and a function R^2->R, based on discrete values given with respect to their axes in a matrix with the first row containing the x-values and the first column containing the y-values.https://www.maplesoft.com/applications/view.aspx?SID=5644&ref=FeedTue, 05 Feb 2008 05:00:00 ZAndreas SchrammAndreas SchrammClassroom Tips and Techniques: Numeric Solution of a Two-Point BVP
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Maple's dsolve command will solve nonlinear two-point boundary value problems numerically. We investigate one such problem that has multiple solutions, and show how a shooting method can be used to reproduce the solutions.<img src="https://www.maplesoft.com/view.aspx?si=5634/R-30NumericSolutionofaTwo-PointBVP_34.jpg" alt="Classroom Tips and Techniques: Numeric Solution of a Two-Point BVP" style="max-width: 25%;" align="left"/>Maple's dsolve command will solve nonlinear two-point boundary value problems numerically. We investigate one such problem that has multiple solutions, and show how a shooting method can be used to reproduce the solutions.https://www.maplesoft.com/applications/view.aspx?SID=5634&ref=FeedThu, 31 Jan 2008 00:00:00 ZDr. Robert LopezDr. Robert LopezApplications of Numerical Continuation, ODE-IVP Approach
https://www.maplesoft.com/applications/view.aspx?SID=5631&ref=Feed
Numerical continuation technique is applied to determine intersection of two surfaces and to perform kinematic analysis of fourbar.mechanism.<img src="https://www.maplesoft.com/view.aspx?si=5631/numericalcontinuation.jpg" alt="Applications of Numerical Continuation, ODE-IVP Approach" style="max-width: 25%;" align="left"/>Numerical continuation technique is applied to determine intersection of two surfaces and to perform kinematic analysis of fourbar.mechanism.https://www.maplesoft.com/applications/view.aspx?SID=5631&ref=FeedMon, 28 Jan 2008 00:00:00 ZHakan TiftikciHakan TiftikciScalar Volterra Integro-Differential Equations
https://www.maplesoft.com/applications/view.aspx?SID=5148&ref=Feed
This worksheet can be used as an experimental tool to investigate the general behavior of solutions of scalar Volterra integro-differential equations of the form
z'(t) = a(t) z(t) + int(b(t, s, z(s)), s = 0 .. t) + g(t)
by numerically solving them with the implicit trapezoidal rule and Newton's method for nonlinear systems---and then graphing their solutions.<img src="https://www.maplesoft.com/view.aspx?si=5148/IntegroDE_209.jpg" alt="Scalar Volterra Integro-Differential Equations" style="max-width: 25%;" align="left"/>This worksheet can be used as an experimental tool to investigate the general behavior of solutions of scalar Volterra integro-differential equations of the form
z'(t) = a(t) z(t) + int(b(t, s, z(s)), s = 0 .. t) + g(t)
by numerically solving them with the implicit trapezoidal rule and Newton's method for nonlinear systems---and then graphing their solutions.https://www.maplesoft.com/applications/view.aspx?SID=5148&ref=FeedWed, 22 Aug 2007 00:00:00 ZDr. Leigh BeckerDr. Leigh BeckerBack-Solving Assistant
https://www.maplesoft.com/applications/view.aspx?SID=5066&ref=Feed
In Maple 11, one of the new additions to the Assistants list is the Back-Solving Assistant. This Tips and Techniques will show you how to get the most out of this new tool.
The Back-Solving Assistant automatically generates a back-solver for your equation, allowing you to instantly solve for any variable in your formula, given the values of the other parameters. You can also plot the behavior of your formula as parameters vary.<img src="https://www.maplesoft.com/view.aspx?si=5066/BackSolver3.jpg" alt="Back-Solving Assistant" style="max-width: 25%;" align="left"/>In Maple 11, one of the new additions to the Assistants list is the Back-Solving Assistant. This Tips and Techniques will show you how to get the most out of this new tool.
The Back-Solving Assistant automatically generates a back-solver for your equation, allowing you to instantly solve for any variable in your formula, given the values of the other parameters. You can also plot the behavior of your formula as parameters vary.https://www.maplesoft.com/applications/view.aspx?SID=5066&ref=FeedTue, 03 Jul 2007 04:00:00 ZMaplesoftMaplesoftRomberg Algorithm for Integration
https://www.maplesoft.com/applications/view.aspx?SID=4895&ref=Feed
Implementation of Romberg Algorithm for estimating the integral of a Riemann-integrable function over an interval.<img src="https://www.maplesoft.com/view.aspx?si=4895//applications/images/app_image_blank_lg.jpg" alt="Romberg Algorithm for Integration" style="max-width: 25%;" align="left"/>Implementation of Romberg Algorithm for estimating the integral of a Riemann-integrable function over an interval.https://www.maplesoft.com/applications/view.aspx?SID=4895&ref=FeedTue, 10 Apr 2007 00:00:00 ZJay PedersenJay PedersenNaive Gauss Elimination Method
https://www.maplesoft.com/applications/view.aspx?SID=4863&ref=Feed
The worksheet shows the step by step procedure for solving set of simultaneous linear equations using Naive Gauss Elimination Method.<img src="https://www.maplesoft.com/view.aspx?si=4863//applications/images/app_image_blank_lg.jpg" alt="Naive Gauss Elimination Method" style="max-width: 25%;" align="left"/>The worksheet shows the step by step procedure for solving set of simultaneous linear equations using Naive Gauss Elimination Method.https://www.maplesoft.com/applications/view.aspx?SID=4863&ref=FeedFri, 26 Jan 2007 00:00:00 ZDr. Autar KawDr. Autar KawLU Decomposition
https://www.maplesoft.com/applications/view.aspx?SID=4862&ref=Feed
This worksheet shows the algorithm for the LU Decomposition of solving Simultaneous Linear Equations.<img src="https://www.maplesoft.com/view.aspx?si=4862//applications/images/app_image_blank_lg.jpg" alt="LU Decomposition" style="max-width: 25%;" align="left"/>This worksheet shows the algorithm for the LU Decomposition of solving Simultaneous Linear Equations.https://www.maplesoft.com/applications/view.aspx?SID=4862&ref=FeedFri, 26 Jan 2007 00:00:00 ZDr. Autar KawDr. Autar KawLibLip - multivariate scattered data interpolation and smoothing
https://www.maplesoft.com/applications/view.aspx?SID=4854&ref=Feed
LibLip is a Maple toolbox, which provides many methods to interpolate scattered data (with or without preprocessing) by using only the data itself and one additional parameter - the Lipschitz constant (which is basically the upper bound on the slope of the function). The Lipschitz constant can be automatically estimated from the data.
LibLip also provides approximation methods using locally Lipschitz functions.
If the data contains noise, it can be smoothened using special techniques which rely on linear programming. Lipschitz constant can
also be estimated from noisy data by using sample splitting and cross-validation.
In addition LibLip also accommodates monotonicity and range constraints. It is useful for approximation of functions that are known to be monotone with respect to all or a subset of variables, as well as monotone only on parts of the domain. Range constraints accommodate non-constant bounds on the values of the data and the interpolant.<img src="https://www.maplesoft.com/view.aspx?si=4854/image.jpg" alt="LibLip - multivariate scattered data interpolation and smoothing" style="max-width: 25%;" align="left"/>LibLip is a Maple toolbox, which provides many methods to interpolate scattered data (with or without preprocessing) by using only the data itself and one additional parameter - the Lipschitz constant (which is basically the upper bound on the slope of the function). The Lipschitz constant can be automatically estimated from the data.
LibLip also provides approximation methods using locally Lipschitz functions.
If the data contains noise, it can be smoothened using special techniques which rely on linear programming. Lipschitz constant can
also be estimated from noisy data by using sample splitting and cross-validation.
In addition LibLip also accommodates monotonicity and range constraints. It is useful for approximation of functions that are known to be monotone with respect to all or a subset of variables, as well as monotone only on parts of the domain. Range constraints accommodate non-constant bounds on the values of the data and the interpolant.https://www.maplesoft.com/applications/view.aspx?SID=4854&ref=FeedFri, 29 Dec 2006 00:00:00 ZDr. Gleb BeliakovDr. Gleb BeliakovRanLip - black-box non-uniform random variate generator
https://www.maplesoft.com/applications/view.aspx?SID=4849&ref=Feed
RanLip is a toolbox for generation of nonuniform random variates from arbitrary Lipschitz-continuous distributions in Maple environment. It uses acceptance/ rejection approach, which is based on approximation of the probability density function from above with a "hat" function. RanLip provides very fast preprocessing and generation times, and yields small rejection constant. It exhibits good performance for up to five variables, and provides the user with a black box nonuniform random variate generator for a large class of distributions, in particular, multimodal distributions.<img src="https://www.maplesoft.com/view.aspx?si=4849/ranlib.jpg" alt="RanLip - black-box non-uniform random variate generator" style="max-width: 25%;" align="left"/>RanLip is a toolbox for generation of nonuniform random variates from arbitrary Lipschitz-continuous distributions in Maple environment. It uses acceptance/ rejection approach, which is based on approximation of the probability density function from above with a "hat" function. RanLip provides very fast preprocessing and generation times, and yields small rejection constant. It exhibits good performance for up to five variables, and provides the user with a black box nonuniform random variate generator for a large class of distributions, in particular, multimodal distributions.https://www.maplesoft.com/applications/view.aspx?SID=4849&ref=FeedWed, 13 Dec 2006 00:00:00 ZDr. Gleb BeliakovDr. Gleb BeliakovPolynomial Regression through Least Square Method
https://www.maplesoft.com/applications/view.aspx?SID=4845&ref=Feed
The goals of this document are to show the approximation of a Point Dispersion through Quadratic Regression Polynomials using the Least Square Method and Maple 10 tools.<img src="https://www.maplesoft.com/view.aspx?si=4845/PolyReg.htm_67.gif" alt="Polynomial Regression through Least Square Method" style="max-width: 25%;" align="left"/>The goals of this document are to show the approximation of a Point Dispersion through Quadratic Regression Polynomials using the Least Square Method and Maple 10 tools.https://www.maplesoft.com/applications/view.aspx?SID=4845&ref=FeedTue, 21 Nov 2006 00:00:00 ZProf. David Macias FerrerProf. David Macias FerrerGlobal and nonsmooth optimization toolbox
https://www.maplesoft.com/applications/view.aspx?SID=4840&ref=Feed
GANSO is a programming library for multivariate global and non-smooth nonlinear optimization. It implements the Extended Cutting Angle method, the Derivative Free Bundle method, Dynamical systems based heuristic and multistart local search. Unlike most nonlinear optimization tools, GANSO algorithms are not trapped in shallow local minima. This Maple toolbox contains the user manual and a sample maple worksheet. The programming library should be downlodaded separately from
http://www.ganso.com.au/libs/gansomapledlls.zip
and its contents extracted into a directory on the path. GANSO homepage is www.ganso.com.au<img src="https://www.maplesoft.com/view.aspx?si=4840/ganso.jpg" alt="Global and nonsmooth optimization toolbox" style="max-width: 25%;" align="left"/>GANSO is a programming library for multivariate global and non-smooth nonlinear optimization. It implements the Extended Cutting Angle method, the Derivative Free Bundle method, Dynamical systems based heuristic and multistart local search. Unlike most nonlinear optimization tools, GANSO algorithms are not trapped in shallow local minima. This Maple toolbox contains the user manual and a sample maple worksheet. The programming library should be downlodaded separately from
http://www.ganso.com.au/libs/gansomapledlls.zip
and its contents extracted into a directory on the path. GANSO homepage is www.ganso.com.auhttps://www.maplesoft.com/applications/view.aspx?SID=4840&ref=FeedFri, 03 Nov 2006 00:00:00 ZCentre for Informatics and Applied Optimization CIAOCentre for Informatics and Applied Optimization CIAOFunctional Approximation through Finite Fourier Series
https://www.maplesoft.com/applications/view.aspx?SID=4841&ref=Feed
The principal goals of this worksheet are To aproximate a Piecewise Continuous Function through Trigonometric Polynomials commonly called Fourier Partial Sums or Finite Fourier Series, to show the convergence of these aproximation via Bessel's Inequality and using Maple spreadsheets and to show Maple’s powerful graphics tools to visualize the application of Weierstrass's Theorem.
The attached .zip file contains both the original Maple 8 .mws file and a Maple 10 .mw version of the worksheet.<img src="https://www.maplesoft.com/view.aspx?si=4841/image.jpg" alt="Functional Approximation through Finite Fourier Series" style="max-width: 25%;" align="left"/>The principal goals of this worksheet are To aproximate a Piecewise Continuous Function through Trigonometric Polynomials commonly called Fourier Partial Sums or Finite Fourier Series, to show the convergence of these aproximation via Bessel's Inequality and using Maple spreadsheets and to show Maple’s powerful graphics tools to visualize the application of Weierstrass's Theorem.
The attached .zip file contains both the original Maple 8 .mws file and a Maple 10 .mw version of the worksheet.https://www.maplesoft.com/applications/view.aspx?SID=4841&ref=FeedFri, 03 Nov 2006 00:00:00 ZProf. David Macias FerrerProf. David Macias Ferrer