Optimization: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=1600
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 30 Jul 2014 03:05:23 GMTWed, 30 Jul 2014 03:05:23 GMTNew applications in the Optimization categoryhttp://www.mapleprimes.com/images/mapleapps.gifOptimization: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=1600
Optimizing the Design of a Coil Spring
http://www.maplesoft.com/applications/view.aspx?SID=153608&ref=Feed
<p>The design optimization of helical springs is of considerable engineering interest, and demands strong solvers. While the number of constraints is small, the coil and wire diameters are raised to higher powers; this makes the optimization difficult for gradient-based solvers working in standard floating-point precision; a larger number of working digits is needed.</p>
<p>Maple lets you increase the number of digits used in calculations; hence numerically difficult problems, like this, can be solved.</p>
<p>This application minimizes the mass of a helical spring. The constraints include the minimum deflection, the minimum surge wave frequency and the maximum stress, and a loading condition.</p>
<ul>
<li>the minimum deflection, </li>
<li>the minimum surge wave frequency, </li>
<li>the maximum stress, </li>
<li>and a loading condition.</li>
</ul>
<p>The design variables are the</p>
<ul>
<li>diameter of the wire, </li>
<li>the outside diameter of the spring,</li>
<li>and the number of coils</li>
</ul>
<p> Reference: "Introduction to Optimum Design", Jasbir S. Arora, 3<sup>rd</sup> Edition 2012.</p><img src="/view.aspx?si=153608/695d991fff8fb4975d1e1dcd90bb771d.gif" alt="Optimizing the Design of a Coil Spring" align="left"/><p>The design optimization of helical springs is of considerable engineering interest, and demands strong solvers. While the number of constraints is small, the coil and wire diameters are raised to higher powers; this makes the optimization difficult for gradient-based solvers working in standard floating-point precision; a larger number of working digits is needed.</p>
<p>Maple lets you increase the number of digits used in calculations; hence numerically difficult problems, like this, can be solved.</p>
<p>This application minimizes the mass of a helical spring. The constraints include the minimum deflection, the minimum surge wave frequency and the maximum stress, and a loading condition.</p>
<ul>
<li>the minimum deflection, </li>
<li>the minimum surge wave frequency, </li>
<li>the maximum stress, </li>
<li>and a loading condition.</li>
</ul>
<p>The design variables are the</p>
<ul>
<li>diameter of the wire, </li>
<li>the outside diameter of the spring,</li>
<li>and the number of coils</li>
</ul>
<p> Reference: "Introduction to Optimum Design", Jasbir S. Arora, 3<sup>rd</sup> Edition 2012.</p>153608Tue, 17 Jun 2014 04:00:00 ZSamir KhanSamir KhanCircle Packing in an Ellipse
http://www.maplesoft.com/applications/view.aspx?SID=153598&ref=Feed
<p>This application optimizes the packing of circles in an ellipse, such that the area of the ellipse is minimized. A typical solution is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 35 circles generates 666 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p>
<p>The constraints and ellipse parameterization are taken from "Packing circles within ellipses", Birgin et al., International Transactions in Operational Research , Volume 20, Issue 3, pages 365–389, May 2013.</p><img src="/view.aspx?si=153598/5f52383daddaeb53aec548d14ebd6ce0.gif" alt="Circle Packing in an Ellipse" align="left"/><p>This application optimizes the packing of circles in an ellipse, such that the area of the ellipse is minimized. A typical solution is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 35 circles generates 666 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p>
<p>The constraints and ellipse parameterization are taken from "Packing circles within ellipses", Birgin et al., International Transactions in Operational Research , Volume 20, Issue 3, pages 365–389, May 2013.</p>153598Wed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanPacking Circles into a Triangle
http://www.maplesoft.com/applications/view.aspx?SID=153596&ref=Feed
<p>This application finds the best packing and largest radius of equal-sized circles, such that they fit in a pre-defined triangle. One solution, as visualized by this application, is given below.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="http://www.maplesoft.com/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 310 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations. The vertices of the triangle can also be modified</p>
<p>Applications like this are used to stress-test global optimizers.</p><img src="/view.aspx?si=153596/2ac6ca1378717b3d939f3d8107616b35.gif" alt="Packing Circles into a Triangle" align="left"/><p>This application finds the best packing and largest radius of equal-sized circles, such that they fit in a pre-defined triangle. One solution, as visualized by this application, is given below.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="http://www.maplesoft.com/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 310 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations. The vertices of the triangle can also be modified</p>
<p>Applications like this are used to stress-test global optimizers.</p>153596Wed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanWelded Beam Design Optimization
http://www.maplesoft.com/applications/view.aspx?SID=153592&ref=Feed
<p>A rigid member is welded onto a beam, with a load applied to the end of the member. The total cost of production is equal to the labor costs (a function of the weld dimensions) plus the cost of the weld and beam material.</p>
<p>The design of the beam is optimized to minimize the production costs by varying the weld and member dimensions.</p>
<p>The constraints include limits on the shear stress, bending stress, buckling load and end deflection, and several size constraints.</p>
<p>The application uses Maple’s non-linear optimizers</p><img src="/view.aspx?si=153592/0621a9aba622112f66506495e21f68d9.gif" alt="Welded Beam Design Optimization" align="left"/><p>A rigid member is welded onto a beam, with a load applied to the end of the member. The total cost of production is equal to the labor costs (a function of the weld dimensions) plus the cost of the weld and beam material.</p>
<p>The design of the beam is optimized to minimize the production costs by varying the weld and member dimensions.</p>
<p>The constraints include limits on the shear stress, bending stress, buckling load and end deflection, and several size constraints.</p>
<p>The application uses Maple’s non-linear optimizers</p>153592Fri, 30 May 2014 04:00:00 ZSamir KhanSamir KhanOptimizing the Design of a Fuel Pod with NX and Maple
http://www.maplesoft.com/applications/view.aspx?SID=153573&ref=Feed
<p>A manufacturer has designed a fuel pod in NX. The fuel pod has a hemispherical and conical end, and a cylindrical mid-section. To minimize material costs, the manufacturer wants to minimize the surface area of the fuel pod while maintaining the existing volume.</p>
<p>This application:</p>
<ul>
<li>pulls the current dimensions of the fuel pod (radius of the hemispherical end, length of the cylindrical midsection, and height of the conical end) from the NX CAD model, </li>
<li>calculates the current volume of the fuel pod,</li>
<li>optimizes the dimensions to minimize the surface area while maintaining the existing volume,</li>
<li>and pushes the optimized dimensions back into the NX CAD model.</li>
</ul>
<p>NOTE: To use this application, you must</p>
<ul>
<li>have a supported version of NX installed, </li>
<li>load canisterOptimization.prt in NX (this is the CAD model of the fuel pod),</li>
<li>ensure the NX-Maple link works correctly.</li>
</ul><img src="/view.aspx?si=153573/fuelpod.jpg" alt="Optimizing the Design of a Fuel Pod with NX and Maple" align="left"/><p>A manufacturer has designed a fuel pod in NX. The fuel pod has a hemispherical and conical end, and a cylindrical mid-section. To minimize material costs, the manufacturer wants to minimize the surface area of the fuel pod while maintaining the existing volume.</p>
<p>This application:</p>
<ul>
<li>pulls the current dimensions of the fuel pod (radius of the hemispherical end, length of the cylindrical midsection, and height of the conical end) from the NX CAD model, </li>
<li>calculates the current volume of the fuel pod,</li>
<li>optimizes the dimensions to minimize the surface area while maintaining the existing volume,</li>
<li>and pushes the optimized dimensions back into the NX CAD model.</li>
</ul>
<p>NOTE: To use this application, you must</p>
<ul>
<li>have a supported version of NX installed, </li>
<li>load canisterOptimization.prt in NX (this is the CAD model of the fuel pod),</li>
<li>ensure the NX-Maple link works correctly.</li>
</ul>153573Wed, 07 May 2014 04:00:00 ZSamir KhanSamir KhanClassroom Tips and Techniques: Bivariate Limits - Then and Now
http://www.maplesoft.com/applications/view.aspx?SID=145979&ref=Feed
An introductory overview of the functionalities in Maple's GraphTheory package.<img src="/view.aspx?si=145979/thumb.jpg" alt="Classroom Tips and Techniques: Bivariate Limits - Then and Now" align="left"/>An introductory overview of the functionalities in Maple's GraphTheory package.145979Wed, 17 Apr 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Introduction to Maple's GraphTheory Package
http://www.maplesoft.com/applications/view.aspx?SID=142357&ref=Feed
An introductory overview of the functionality in Maple's GraphTheory package.<img src="/view.aspx?si=142357/thumb.jpg" alt="Classroom Tips and Techniques: Introduction to Maple's GraphTheory Package" align="left"/>An introductory overview of the functionality in Maple's GraphTheory package.142357Thu, 17 Jan 2013 05:00:00 ZProf. Michael MonaganProf. Michael MonaganClassroom Tips and Techniques: Least-Squares Fits
http://www.maplesoft.com/applications/view.aspx?SID=140942&ref=Feed
<p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p><img src="/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" align="left"/><p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p>140942Wed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Best Taylor-Polynomial Approximations
http://www.maplesoft.com/applications/view.aspx?SID=136471&ref=Feed
In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.<img src="/view.aspx?si=136471/image.jpg" alt="Classroom Tips and Techniques: Best Taylor-Polynomial Approximations" align="left"/>In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.136471Tue, 14 Aug 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: An Inequality-Constrained Optimization Problem
http://www.maplesoft.com/applications/view.aspx?SID=135904&ref=Feed
<p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p><img src="/view.aspx?si=135904/thumb.jpg" alt="Classroom Tips and Techniques: An Inequality-Constrained Optimization Problem" align="left"/><p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p>135904Mon, 16 Jul 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezStreet-fighting Math
http://www.maplesoft.com/applications/view.aspx?SID=129226&ref=Feed
This interactive Maple document contains a simple street-fighting game and performs a mathematical analysis of it, involving probability and game theory. The document is suitable for presentation in an undergraduate course on operations research, probability or linear programming. No knowledge of Maple is required.<img src="/view.aspx?si=129226/fighter_sm.jpg" alt="Street-fighting Math" align="left"/>This interactive Maple document contains a simple street-fighting game and performs a mathematical analysis of it, involving probability and game theory. The document is suitable for presentation in an undergraduate course on operations research, probability or linear programming. No knowledge of Maple is required.129226Thu, 29 Dec 2011 05:00:00 ZDr. Robert IsraelDr. Robert IsraelGreat Expectations
http://www.maplesoft.com/applications/view.aspx?SID=127116&ref=Feed
<p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p><img src="/view.aspx?si=127116/expectation_thum.png" alt="Great Expectations" align="left"/><p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p>127116Thu, 27 Oct 2011 04:00:00 ZClassroom Tips and Techniques: Steepest-Ascent Curves
http://www.maplesoft.com/applications/view.aspx?SID=123985&ref=Feed
Steepest-ascent curves are obtained for surfaces defined analytically and digitally.<img src="/view.aspx?si=123985/thumb.jpg" alt="Classroom Tips and Techniques: Steepest-Ascent Curves" align="left"/>Steepest-ascent curves are obtained for surfaces defined analytically and digitally.123985Tue, 19 Jul 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Nonlinear Fit, Optimization, and the DirectSearch Package
http://www.maplesoft.com/applications/view.aspx?SID=122760&ref=Feed
In this month's article, I revisit a nonlinear curve-fitting problem that appears in my Advanced Engineering Mathematics ebook, examine the role of Maple's Optimization package in that problem, and then explore the DirectSearch package from Dr. Sergey N. Moiseev.<img src="/view.aspx?si=122760/thumb.jpg" alt="Classroom Tips and Techniques: Nonlinear Fit, Optimization, and the DirectSearch Package" align="left"/>In this month's article, I revisit a nonlinear curve-fitting problem that appears in my Advanced Engineering Mathematics ebook, examine the role of Maple's Optimization package in that problem, and then explore the DirectSearch package from Dr. Sergey N. Moiseev.122760Wed, 15 Jun 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezDirectSearch optimization package, version 2
http://www.maplesoft.com/applications/view.aspx?SID=101333&ref=Feed
<p> The DirectSearch package is a collection of commands to numerically compute local and global minimums (maximums) of nonlinear multivariate function with (without) constraints. The package optimization methods are universal derivative-free direct searching methods, i.e. they do not require the objective function and constraints to be differentiable and continuous.<br /> The package optimization methods have quadratic convergence for quadratic functions.<br /><br /> The package also contains commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data.<br /><br />The following is a summary of the major improvements in DirectSearch v.2.<br /><br />-- Three new derivative-free optimization methods are added.<br />-- The new global optimization command GlobalOptima is added.<br />-- The commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data are added.<br />-- Mixed integer-discrete-continuous optimization is now supported.<br />-- You can now specify inequality constraints as any Boolean expressions.<br />-- You can now set bound inequality constraints x>=a, x<=b as: x=a..b.<br />-- Assume and assumption commands are supported for inequality constraints.<br />-- You can now specify problem variables as Vector.<br />-- High dimensional optimization problem are now solved a much faster.<br />-- Search in space curve direction is added to all algorithms.<br />-- Penalty function method is added for optimization with inequality constraints<br />-- Improved optimization algorithm for equality constraints is faster and more reliable.<br />-- The feasible initial point searching is improved.<br />-- Now the package is compatible with Maple 12 and above.<br />-- Detailed description of CDOS method in .pdf format is added.<br />-- Russian version of the package is now available.<br /><br /></p><img src="/view.aspx?si=101333/maple_icon.jpg" alt="DirectSearch optimization package, version 2" align="left"/><p> The DirectSearch package is a collection of commands to numerically compute local and global minimums (maximums) of nonlinear multivariate function with (without) constraints. The package optimization methods are universal derivative-free direct searching methods, i.e. they do not require the objective function and constraints to be differentiable and continuous.<br /> The package optimization methods have quadratic convergence for quadratic functions.<br /><br /> The package also contains commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data.<br /><br />The following is a summary of the major improvements in DirectSearch v.2.<br /><br />-- Three new derivative-free optimization methods are added.<br />-- The new global optimization command GlobalOptima is added.<br />-- The commands for multiobjective optimization, solving system of equations, fitting nonlinear function to data are added.<br />-- Mixed integer-discrete-continuous optimization is now supported.<br />-- You can now specify inequality constraints as any Boolean expressions.<br />-- You can now set bound inequality constraints x>=a, x<=b as: x=a..b.<br />-- Assume and assumption commands are supported for inequality constraints.<br />-- You can now specify problem variables as Vector.<br />-- High dimensional optimization problem are now solved a much faster.<br />-- Search in space curve direction is added to all algorithms.<br />-- Penalty function method is added for optimization with inequality constraints<br />-- Improved optimization algorithm for equality constraints is faster and more reliable.<br />-- The feasible initial point searching is improved.<br />-- Now the package is compatible with Maple 12 and above.<br />-- Detailed description of CDOS method in .pdf format is added.<br />-- Russian version of the package is now available.<br /><br /></p>101333Tue, 01 Feb 2011 05:00:00 ZDr. Sergey MoiseevDr. Sergey MoiseevPortfolio Simulation and Quadratic Programming
http://www.maplesoft.com/applications/view.aspx?SID=100604&ref=Feed
<p>We will in this maple worksheet explore portfolio theory and quadratic optimization.<br />We will start by simulating some data for 50 stocks and then optimize the portfolio.<br />We will also use empirical data to backtest our portfolio strategy.</p><img src="/view.aspx?si=100604/maple_icon.jpg" alt="Portfolio Simulation and Quadratic Programming" align="left"/><p>We will in this maple worksheet explore portfolio theory and quadratic optimization.<br />We will start by simulating some data for 50 stocks and then optimize the portfolio.<br />We will also use empirical data to backtest our portfolio strategy.</p>100604Mon, 03 Jan 2011 05:00:00 ZMarcus DavidssonMarcus DavidssonClassroom Tips and Techniques: Partial Derivatives by Subscripting
http://www.maplesoft.com/applications/view.aspx?SID=100266&ref=Feed
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: Fitting Circles in Space to 3-D Data
http://www.maplesoft.com/applications/view.aspx?SID=1644&ref=Feed
<p>In "A Project on Circles in Space," Carl Cowen provided an algebraic solution for the problem of fitting a circle to a set of points in space. His technique used the singular value decomposition from linear algebra, and was recast as a project in the volume ATLAST: Computer Exercises for Linear Algebra. Both versions of the problem used MATLAB® for the calculations. In this worksheet, we implement the algebraic calculations in Maple, then add noise to the data to test the robustness of the algebraic method. Next, we solve the problem with an analytic approach that incorporates least squares, and appears to be more robust in the face of noisy data. Finally, the analytic approach leads to explicit formulas for the fitting circle, so we end with graphs of the data, fitting circle, and plane lying closest to the data in the least-squares sense.</p>
<p><em><sub>Simulink is a registered trademark of The MathWorks, Inc.</sub></em></p><img src="/view.aspx?si=1644/thumb3.jpg" alt="Classroom Tips and Techniques: Fitting Circles in Space to 3-D Data" align="left"/><p>In "A Project on Circles in Space," Carl Cowen provided an algebraic solution for the problem of fitting a circle to a set of points in space. His technique used the singular value decomposition from linear algebra, and was recast as a project in the volume ATLAST: Computer Exercises for Linear Algebra. Both versions of the problem used MATLAB® for the calculations. In this worksheet, we implement the algebraic calculations in Maple, then add noise to the data to test the robustness of the algebraic method. Next, we solve the problem with an analytic approach that incorporates least squares, and appears to be more robust in the face of noisy data. Finally, the analytic approach leads to explicit formulas for the fitting circle, so we end with graphs of the data, fitting circle, and plane lying closest to the data in the least-squares sense.</p>
<p><em><sub>Simulink is a registered trademark of The MathWorks, Inc.</sub></em></p>1644Mon, 17 May 2010 04:00:00 ZDr. Robert LopezDr. Robert LopezDirect search optimization package
http://www.maplesoft.com/applications/view.aspx?SID=87637&ref=Feed
<p>The DirectSearch package is a collection of commands to numerically computes local and global minimums (maximums) of nonlinear multivariate function with (without) constraints. The package optimization methods are direct searching methods, i.e. they do not require the objective function to be differentiable and continuous.</p><img src="/view.aspx?si=87637/Fig2.jpg" alt="Direct search optimization package" align="left"/><p>The DirectSearch package is a collection of commands to numerically computes local and global minimums (maximums) of nonlinear multivariate function with (without) constraints. The package optimization methods are direct searching methods, i.e. they do not require the objective function to be differentiable and continuous.</p>87637Wed, 12 May 2010 04:00:00 ZDr. Sergey MoiseevDr. Sergey MoiseevPhénomène de Runge - subdivision de Chebychev
http://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 />
<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 Sebastien