Numerical Analysis: New Applications
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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 19 Apr 2015 12:53:40 GMTSun, 19 Apr 2015 12:53:40 GMTNew applications in the Numerical Analysis categoryhttp://www.mapleprimes.com/images/mapleapps.gifNumerical Analysis: New Applications
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Comparison between LAGRANGE and Spline Interpolation
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<p>In this worksheet a comparison between the <em>LAGRANGE </em>and the <em>Spline Interpolation </em>has been discused based upon eleven interpolating points. In order to arrive at a <em><strong>smooth </strong></em>interpolation a <strong>cubic spline </strong>has been utilized. High-degree splines are similar to LAGRANGE polynomials. A spline of degree = infinity is identical to the LAGRANGE approximation.</p><img src="/view.aspx?si=153574/4eafecf1baae01f83c8a823ac067d4fc.gif" alt="Comparison between LAGRANGE and Spline Interpolation" align="left"/><p>In this worksheet a comparison between the <em>LAGRANGE </em>and the <em>Spline Interpolation </em>has been discused based upon eleven interpolating points. In order to arrive at a <em><strong>smooth </strong></em>interpolation a <strong>cubic spline </strong>has been utilized. High-degree splines are similar to LAGRANGE polynomials. A spline of degree = infinity is identical to the LAGRANGE approximation.</p>153574Mon, 12 May 2014 04:00:00 ZProf. Josef BettenProf. Josef BettenMeasuring Water Flow of Rivers
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In this guest article in the Tips & Techniques series, Dr. Michael Monagan discusses the art and science of measuring the amount of water flowing in a river, and relates his personal experiences with this task to its morph into a project for his calculus classes.<img src="/view.aspx?si=153480/thumb.jpg" alt="Measuring Water Flow of Rivers" align="left"/>In this guest article in the Tips & Techniques series, Dr. Michael Monagan discusses the art and science of measuring the amount of water flowing in a river, and relates his personal experiences with this task to its morph into a project for his calculus classes.153480Fri, 13 Dec 2013 05:00:00 ZProf. Michael MonaganProf. Michael MonaganClassroom Tips and Techniques: Locus of Eigenvalues
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If P(s) is a parameter-dependent square matrix, what is the locus of its eigenvalues as s varies from, say, 0 to 1? For a non-square P, the eigenvalues can become complex, so the loci could exist as curves in the real or complex planes. To avoid these difficulties, consider only real symmetric matrices for which the loci of eigenvalues are real curves, but curves that could intersect. What does it mean to trace an individual eigenvalue of P(0) to P(1) if the eigenvalue has algebraic multiplicity more than 1?<img src="/view.aspx?si=153463/thumb.jpg" alt="Classroom Tips and Techniques: Locus of Eigenvalues" align="left"/>If P(s) is a parameter-dependent square matrix, what is the locus of its eigenvalues as s varies from, say, 0 to 1? For a non-square P, the eigenvalues can become complex, so the loci could exist as curves in the real or complex planes. To avoid these difficulties, consider only real symmetric matrices for which the loci of eigenvalues are real curves, but curves that could intersect. What does it mean to trace an individual eigenvalue of P(0) to P(1) if the eigenvalue has algebraic multiplicity more than 1?153463Fri, 15 Nov 2013 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method
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The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.<img src="/view.aspx?si=149514/thumb.jpg" alt="Classroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method" align="left"/>The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.149514Tue, 16 Jul 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezLAGRANGE Interpolation
http://www.maplesoft.com/applications/view.aspx?SID=139076&ref=Feed
<p>Zur Aufstellung von Interpolationspolynomen n - ten Grades bei n + 1 gegebenen Messpunkten wird die LAGRANGEsche Interpolationsmethode ausführlich diskutiert und mit anderen Verfahren verglichen. Als numerisches Beispiel wird ein Polynom fünften Grades mut sechs Stützpunkten konstruiert.</p>
<p><em>Keywords: </em>LAGRANGEsche Grundfunktionen, Determinantenmethode, lineare, kubische und Splinefunktionen fünften und höheren Grades, Glättung von Interpolationspolynomen hohen Grades, Formfunktionen vom LAGRANGEschen Typ für finite Elemente</p><img src="/view.aspx?si=139076/b14d4036a98db9e3dc88c1915625b142.gif" alt="LAGRANGE Interpolation" align="left"/><p>Zur Aufstellung von Interpolationspolynomen n - ten Grades bei n + 1 gegebenen Messpunkten wird die LAGRANGEsche Interpolationsmethode ausführlich diskutiert und mit anderen Verfahren verglichen. Als numerisches Beispiel wird ein Polynom fünften Grades mut sechs Stützpunkten konstruiert.</p>
<p><em>Keywords: </em>LAGRANGEsche Grundfunktionen, Determinantenmethode, lineare, kubische und Splinefunktionen fünften und höheren Grades, Glättung von Interpolationspolynomen hohen Grades, Formfunktionen vom LAGRANGEschen Typ für finite Elemente</p>139076Thu, 01 Nov 2012 04:00:00 ZJosef BettenJosef BettenSolving Equations with Maple
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<p>This worksheet is concerned with methods implemented in Maple to solve some equations of several types.</p>
<p> For instance, the procedures <strong>fsolve </strong>and <strong>RootOf </strong>are very effective and should be used in the following examples.</p>
<p> </p>
<p><em>Keywords: </em>fsolve, RootOf, polynomials of degree n > 3, orthopoly, <em>HERMITE, LEGENDRE, </em></p>
<p><em> LAGUERRE, CHEBYSHEV, </em>transcendental equations</p>
<p> </p>
<p><em><br /></em></p><img src="/applications/images/app_image_blank_lg.jpg" alt="Solving Equations with Maple" align="left"/><p>This worksheet is concerned with methods implemented in Maple to solve some equations of several types.</p>
<p> For instance, the procedures <strong>fsolve </strong>and <strong>RootOf </strong>are very effective and should be used in the following examples.</p>
<p> </p>
<p><em>Keywords: </em>fsolve, RootOf, polynomials of degree n > 3, orthopoly, <em>HERMITE, LEGENDRE, </em></p>
<p><em> LAGUERRE, CHEBYSHEV, </em>transcendental equations</p>
<p> </p>
<p><em><br /></em></p>130644Mon, 13 Feb 2012 05:00:00 ZProf. Josef BettenProf. Josef BettenVarInt
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<p><strong>VarInt</strong> - <em>Computer Algebra Aided Design of Variational Integrators</em>.</p>
<p>Create, design and analyse (new) variational integrators for autonomous dynamical systems (with non-conservative forces) up to arbitary order with VarInt, a package for Maple.</p><img src="/view.aspx?si=88830/VarIntSmall.jpg" alt="VarInt" align="left"/><p><strong>VarInt</strong> - <em>Computer Algebra Aided Design of Variational Integrators</em>.</p>
<p>Create, design and analyse (new) variational integrators for autonomous dynamical systems (with non-conservative forces) up to arbitary order with VarInt, a package for Maple.</p>88830Tue, 25 Jan 2011 05:00:00 ZChristian HellströmChristian HellströmClassroom Tips and Techniques: Partial Derivatives by Subscripting
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As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.<img src="/view.aspx?si=100266/thumb.jpg" alt="Classroom Tips and Techniques: Partial Derivatives by Subscripting" align="left"/>As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.100266Wed, 15 Dec 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Diffusion with a Generalized Robin Condition
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<p>The one-dimensonal heat equation with a generalized Robin condition is solved on [0, 1] by a finite-difference scheme and by the Laplace transform, with the inversion implemented numerically. The left end is insulated and the initial temperature is zero. The Robin condition at the right end is driven by a function governed by an ODE, that is in turn, driven by the endpoint temperature.</p><img src="/view.aspx?si=96958/thumb.jpg" alt="Classroom Tips and Techniques: Diffusion with a Generalized Robin Condition" align="left"/><p>The one-dimensonal heat equation with a generalized Robin condition is solved on [0, 1] by a finite-difference scheme and by the Laplace transform, with the inversion implemented numerically. The left end is insulated and the initial temperature is zero. The Robin condition at the right end is driven by a function governed by an ODE, that is in turn, driven by the endpoint temperature.</p>96958Fri, 17 Sep 2010 04:00:00 ZRobert LopezRobert LopezDerivation of Adam's Method Through Lagrange Interpolation,Using Maple
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<p>this simple program will return the adam's method by suitable changing of the variable s in the program to get the desire numerical method for solving ordinary differential equations numerically.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Derivation of Adam's Method Through Lagrange Interpolation,Using Maple" align="left"/><p>this simple program will return the adam's method by suitable changing of the variable s in the program to get the desire numerical method for solving ordinary differential equations numerically.</p>96648Sat, 04 Sep 2010 04:00:00 ZProf.M.O.IBRAHIMProf.M.O.IBRAHIMPhénomène de Runge - subdivision de Chebychev
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<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><img src="/view.aspx?si=35301/thumb.jpg" alt="Phénomène de Runge - subdivision de Chebychev" 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>35301Fri, 26 Mar 2010 04:00:00 ZKERNIVINEN SebastienKERNIVINEN SebastienComparison of Multivariate Optimization Methods
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<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="/view.aspx?si=1718/SearchPaths2.PNG" alt="Comparison of Multivariate Optimization Methods" 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>1718Tue, 15 Sep 2009 04:00:00 ZFixed Point Iteration
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<p>This worksheet is concerned with finding numerical solutions of non-linear equations in a single unknown. Using MAPLE 12 the <em>fixed-point iteration </em>has been applied to some examples.</p>
<p> </p><img src="/view.aspx?si=33176//applications/images/app_image_blank_lg.jpg" alt="Fixed Point Iteration" align="left"/><p>This worksheet is concerned with finding numerical solutions of non-linear equations in a single unknown. Using MAPLE 12 the <em>fixed-point iteration </em>has been applied to some examples.</p>
<p> </p>33176Tue, 30 Jun 2009 04:00:00 ZProf. Josef BettenProf. Josef BettenFixed Point Iteration
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<p>This worksheet is concerned with finding numerical solutions of non-linear equations in a single unknown. Using MAPLE 12 the <em>fixed-point iteration </em>has been applied to some examples.</p><img src="/view.aspx?si=33175//applications/images/app_image_blank_lg.jpg" alt="Fixed Point Iteration" align="left"/><p>This worksheet is concerned with finding numerical solutions of non-linear equations in a single unknown. Using MAPLE 12 the <em>fixed-point iteration </em>has been applied to some examples.</p>33175Tue, 30 Jun 2009 04:00:00 ZProf. Josef BettenProf. Josef BettenOrthogonal Functions, Orthogonal Polynomials, and Orthogonal Wavelets series expansions of function
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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="/view.aspx?si=7256/thumb.gif" alt="Orthogonal Functions, Orthogonal Polynomials, and Orthogonal Wavelets series expansions of function" 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.7256Wed, 18 Feb 2009 00:00:00 ZDr. Sergey MoiseevDr. Sergey MoiseevComparison 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="/view.aspx?si=6938/Untitled-1.jpg" alt="Comparison Between Newton, Householder and Halley Methods" 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.6938Fri, 28 Nov 2008 00:00:00 ZShahzad BhattiShahzad BhattiSimpson and Trapezoidal Methods for Numerical Integeration
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Using this maplet, one can solve a integrate a function numerically using the Simpson and Trapezoidal Methods.<img src="/view.aspx?si=6937/Untitled-1.jpg" alt="Simpson and Trapezoidal Methods for Numerical Integeration" align="left"/>Using this maplet, one can solve a integrate a function numerically using the Simpson and Trapezoidal Methods.6937Fri, 28 Nov 2008 00:00:00 ZShahzad BhattiShahzad BhattiTraveling Salesman Problem
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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="/view.aspx?si=6873/TSgif.gif" alt="Traveling Salesman Problem" 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).6873Mon, 10 Nov 2008 00:00:00 ZBruno GuerrieriBruno Guerrieri3D spline interpolation
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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="/applications/images/app_image_blank_lg.jpg" alt="3D spline interpolation" 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.5644Tue, 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="/view.aspx?si=5634/R-30NumericSolutionofaTwo-PointBVP_34.jpg" alt="Classroom Tips and Techniques: Numeric Solution of a Two-Point BVP" 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.5634Thu, 31 Jan 2008 00:00:00 ZDr. Robert LopezDr. Robert Lopez