Linear Algebra: New Applications
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en-us2016 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemFri, 29 Jul 2016 19:53:51 GMTFri, 29 Jul 2016 19:53:51 GMTNew applications in the Linear Algebra categoryhttp://www.mapleprimes.com/images/mapleapps.gifLinear Algebra: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=144
Kinematics of Our Earth-Moon System
http://www.maplesoft.com/applications/view.aspx?SID=153554&ref=Feed
<p>The purpose of this worksheet is to show the power of Maple in illustrating natural global events, and to show how mathematics can be a fun part of life.</p><img src="/view.aspx?si=153554/a05277b27fbcd36e1df1952c6d5969be.gif" alt="Kinematics of Our Earth-Moon System" align="left"/><p>The purpose of this worksheet is to show the power of Maple in illustrating natural global events, and to show how mathematics can be a fun part of life.</p>153554Wed, 23 Apr 2014 04:00:00 ZAli Abu OamAli Abu OamInternet Page Ranking Algorithms
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In this guest article in the Tips and Techniques series, Dr. Michael Monagan explains how internet pages are ranked.<img src="/view.aspx?si=153532/thumb.jpg" alt="Internet Page Ranking Algorithms" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan explains how internet pages are ranked.153532Thu, 20 Mar 2014 04:00:00 ZProf. Michael MonaganProf. Michael MonaganCollision detection between toolholder and workpiece on ball nut grinding
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<p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p><img src="/view.aspx?si=153477/Collision_Detection_image1.jpg" alt="Collision detection between toolholder and workpiece on ball nut grinding" align="left"/><p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p>153477Mon, 23 Dec 2013 05:00:00 ZGyörgy HegedûsGyörgy HegedûsClassroom 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 LopezApplication of the Modified Gram-Schmidt Algorithm
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<p>Maple's QRDecomposition command basically utilizes one of two routines for generating the Q and R matrices. If the matrix contains only integers and/or symbolic expressions, then Maple performs a QR decomposition using the Classical Gram-Schmidt algorithm. If however, the matrix contains a mixture of integers and floating point decimals or only floating point decimals, then Maple carries out the QR decomposition of the matrix using Householder transformations. My approach below uses a third alternative, the Modified Gram-Schmidt algorithm, which I read about in Chapter 8 of the textbook, NUMERICAL LINEAR ALGEBRA, by Lloyd N. Trefethen and David Bau III.</p><img src="/view.aspx?si=152382/05160ad08a75a6b7948e889b5999f0ea.gif" alt="Application of the Modified Gram-Schmidt Algorithm" align="left"/><p>Maple's QRDecomposition command basically utilizes one of two routines for generating the Q and R matrices. If the matrix contains only integers and/or symbolic expressions, then Maple performs a QR decomposition using the Classical Gram-Schmidt algorithm. If however, the matrix contains a mixture of integers and floating point decimals or only floating point decimals, then Maple carries out the QR decomposition of the matrix using Householder transformations. My approach below uses a third alternative, the Modified Gram-Schmidt algorithm, which I read about in Chapter 8 of the textbook, NUMERICAL LINEAR ALGEBRA, by Lloyd N. Trefethen and David Bau III.</p>152382Tue, 01 Oct 2013 04:00:00 ZDouglas LewitDouglas LewitSymmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE
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<p>Hybrid structures were made by assembling monolayers (MLs) of closely packed colloidal microspheres on a metal-coated glass substrate . In fact, this architecture is one of several realizations of hybrid plasmonic-photonic crystals (PHs), which differ in photonic crystals dimensionality and metal ﬁlm corrugation [1,2].</p>
<p>The main challenge to us were exploring of those properties of structures which are caused by their space symmetry. In particular, it was necessary to establish the so-called "rules of selection", i.e. the list of the allowed transitions between electronic states of different symmetry and energy that can be induced by light of varying polarization. Additional interest for us was to demonstrate the possibilities of MAPLE within this specific field.</p><img src="/view.aspx?si=151383/440fb9a2994e797b26c18564d860131b.gif" alt="Symmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE" align="left"/><p>Hybrid structures were made by assembling monolayers (MLs) of closely packed colloidal microspheres on a metal-coated glass substrate . In fact, this architecture is one of several realizations of hybrid plasmonic-photonic crystals (PHs), which differ in photonic crystals dimensionality and metal ﬁlm corrugation [1,2].</p>
<p>The main challenge to us were exploring of those properties of structures which are caused by their space symmetry. In particular, it was necessary to establish the so-called "rules of selection", i.e. the list of the allowed transitions between electronic states of different symmetry and energy that can be induced by light of varying polarization. Additional interest for us was to demonstrate the possibilities of MAPLE within this specific field.</p>151383Thu, 05 Sep 2013 04:00:00 ZOlga V. DvornikOlga V. DvornikClassroom Tips and Techniques: Least-Squares Fits
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<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 LopezLeast Squares and QP Optimization
http://www.maplesoft.com/applications/view.aspx?SID=129826&ref=Feed
<p>We will in this worksheet discuss Least Squares (LS) <br /> and its relationship to Quadratic Programming (QP) <br /> when we have a column-dominated matrix. We will <br /> also discuss the normal equation and the problem <br /> with using such equations for a non-square matrix.</p><img src="/view.aspx?si=129826/maple-gf.jpg" alt="Least Squares and QP Optimization" align="left"/><p>We will in this worksheet discuss Least Squares (LS) <br /> and its relationship to Quadratic Programming (QP) <br /> when we have a column-dominated matrix. We will <br /> also discuss the normal equation and the problem <br /> with using such equations for a non-square matrix.</p>129826Thu, 19 Jan 2012 05:00:00 ZMarcus DavidssonMarcus DavidssonzoMbi
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<p>Higher Mathematics for external students of biological faculty.<br />Solver-practicum.<br />1st semester.<br />300 problems (15 labs in 20 variants).<br />mw.zip</p>
<p>Before use - Shake! <br />(Click on the button and activate the program and Maplet).<br />Full version in html: <a href="http://webmath.exponenta.ru/zom/index.html">http://webmath.exponenta.ru/zom/index.html</a></p><img src="/view.aspx?si=129642/zombie_3.jpg" alt="zoMbi" align="left"/><p>Higher Mathematics for external students of biological faculty.<br />Solver-practicum.<br />1st semester.<br />300 problems (15 labs in 20 variants).<br />mw.zip</p>
<p>Before use - Shake! <br />(Click on the button and activate the program and Maplet).<br />Full version in html: <a href="http://webmath.exponenta.ru/zom/index.html">http://webmath.exponenta.ru/zom/index.html</a></p>129642Sun, 15 Jan 2012 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoClassroom Tips and Techniques: An Undamped Coupled Oscillator
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<p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p><img src="/view.aspx?si=129521/thumb.jpg" alt="Classroom Tips and Techniques: An Undamped Coupled Oscillator" align="left"/><p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p>129521Tue, 10 Jan 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezLinear Algebra Example Generator
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<p>One of the challenges in Linear Algebra is in developing problems, projects, and exercises that are both larger dimensional and student-accessible. Indeed, round-off error, computational complexity, difficulty factoring characteristic polynomials of degree 3 or higher, and similar aspects often mean that any problems or applications of rank 3 or higher are approached solely via technology. </p>
<p>
However, that same technology can be used to create student-accessible problems and applications of ranks 4 or 5 or even higher, even allowing the creation -- if desired -- of a technology-free course featuring only hand-calculable problems. In this presentation, we present a freely downloadable Maple worksheet that produces these types of problems. Moreover, it can be used to create hand-calculable applications of arbitrarily large rank involving stochastic matrices, eigenvalues and eigenvectors, Leslie matrix models, the simplex method, and several others.</p>
<p>Two files are included. The first is LinearAlgebraExamples and includes all the example generators. The second is LinearAlgebraExamplesApp and is stripped to only the main component interface of the first. </p><img src="/view.aspx?si=129347/linearnew_sm.jpg" alt="Linear Algebra Example Generator" align="left"/><p>One of the challenges in Linear Algebra is in developing problems, projects, and exercises that are both larger dimensional and student-accessible. Indeed, round-off error, computational complexity, difficulty factoring characteristic polynomials of degree 3 or higher, and similar aspects often mean that any problems or applications of rank 3 or higher are approached solely via technology. </p>
<p>
However, that same technology can be used to create student-accessible problems and applications of ranks 4 or 5 or even higher, even allowing the creation -- if desired -- of a technology-free course featuring only hand-calculable problems. In this presentation, we present a freely downloadable Maple worksheet that produces these types of problems. Moreover, it can be used to create hand-calculable applications of arbitrarily large rank involving stochastic matrices, eigenvalues and eigenvectors, Leslie matrix models, the simplex method, and several others.</p>
<p>Two files are included. The first is LinearAlgebraExamples and includes all the example generators. The second is LinearAlgebraExamplesApp and is stripped to only the main component interface of the first. </p>129347Thu, 05 Jan 2012 05:00:00 ZJeff KnisleyJeff KnisleyClassroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem
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<p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A - lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>-1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p><img src="/view.aspx?si=128444/thumb.jpg" alt="Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem" align="left"/><p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A - lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>-1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p>128444Tue, 06 Dec 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezAlgebraic Riccati Equations in Control Theory
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Algebraic Riccati equations appear in many linear optimal and robust control methods such as in LQR, LQG, Kalman filter, H2 and Hinfinity techniques. Solving these equations is a vital step in designing such controllers and state estimators. "
In Maple 15, the CARE and DARE solvers for continuous and discrete algebraic Riccati equations are enhanced with high-precision solvers that allow you to get solutions beyond IEEE double precision.<img src="/view.aspx?si=103818/thumb.jpg" alt="Algebraic Riccati Equations in Control Theory" align="left"/>Algebraic Riccati equations appear in many linear optimal and robust control methods such as in LQR, LQG, Kalman filter, H2 and Hinfinity techniques. Solving these equations is a vital step in designing such controllers and state estimators. "
In Maple 15, the CARE and DARE solvers for continuous and discrete algebraic Riccati equations are enhanced with high-precision solvers that allow you to get solutions beyond IEEE double precision.103818Wed, 06 Apr 2011 04:00:00 ZMaplesoftMaplesoftTerminator circle with animation
http://www.maplesoft.com/applications/view.aspx?SID=100509&ref=Feed
<p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p><img src="/view.aspx?si=100509/thumb.jpg" alt="Terminator circle with animation" align="left"/><p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p>100509Tue, 28 Dec 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyGeneration of correlated random numbers
http://www.maplesoft.com/applications/view.aspx?SID=99806&ref=Feed
<p>This application is an extension of an earlier document on multivariate distributions and demonstrates how Maple can be used to generate random samples from such distribution. In a narrow sense, it presents the tool for generation of correlated samples. The sampling need for multi-factor random variables (RV) with a given correlation structure arises in many applications in economics, finance, but also in natural sciences such as genetics, physics etc. and here we show that such task can be accomplished with ease using Maple’s <em>Statistic</em>s and <em>Linear Algebra</em> packages.</p><img src="/view.aspx?si=99806/maple_icon.jpg" alt="Generation of correlated random numbers" align="left"/><p>This application is an extension of an earlier document on multivariate distributions and demonstrates how Maple can be used to generate random samples from such distribution. In a narrow sense, it presents the tool for generation of correlated samples. The sampling need for multi-factor random variables (RV) with a given correlation structure arises in many applications in economics, finance, but also in natural sciences such as genetics, physics etc. and here we show that such task can be accomplished with ease using Maple’s <em>Statistic</em>s and <em>Linear Algebra</em> packages.</p>99806Fri, 03 Dec 2010 05:00:00 ZI. HlivkaI. HlivkaVectors and Matrices
http://www.maplesoft.com/applications/view.aspx?SID=99816&ref=Feed
<p>This worksheet shows how Maple performs vector and matrices computations</p><img src="/view.aspx?si=99816/maple_icon.jpg" alt="Vectors and Matrices" align="left"/><p>This worksheet shows how Maple performs vector and matrices computations</p>99816Fri, 03 Dec 2010 05:00:00 ZAli Abu OamAli Abu OamClassroom Tips and Techniques: Fitting Circles in Space to 3-D Data
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<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 LopezEigenvalues and Eigenvectors
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<p><span>The purpose of this worksheet is to introduce the concepts of eigenvalues and eigenvectors from both algebraic and geometric point of view </span></p><img src="/applications/images/app_image_blank_lg.jpg" alt="Eigenvalues and Eigenvectors" align="left"/><p><span>The purpose of this worksheet is to introduce the concepts of eigenvalues and eigenvectors from both algebraic and geometric point of view </span></p>87625Mon, 10 May 2010 04:00:00 ZAli Abu OamAli Abu OamMatrix Algebra
http://www.maplesoft.com/applications/view.aspx?SID=87614&ref=Feed
<p> </p>
<p align="left">T he purpose of this worksheet is to illustrate some of interesting properties of matrix algebra, and to show that how maple can give an interest fun and make mathematics so lovely field as will be shown among this worksheet.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Matrix Algebra" align="left"/><p> </p>
<p align="left">T he purpose of this worksheet is to illustrate some of interesting properties of matrix algebra, and to show that how maple can give an interest fun and make mathematics so lovely field as will be shown among this worksheet.</p>87614Sat, 08 May 2010 04:00:00 ZAli Abu OamAli Abu OamClassroom Tips and Techniques: A Note on Parametric Plotting
http://www.maplesoft.com/applications/view.aspx?SID=87605&ref=Feed
<p>Under suitable conditions, the equations f(x, y, t) = g(x, y, t) and g(x, y, t) = 0 define the curve x = x(t), y = y(t) parametrically. If the algebra permits an explicit solution, the resulting curve can be drawn with Maple's basic parametric plotting functionality. However, when the algebra is recalcitrant, the equations must be solved numerically for a sequence of points, or converted to differential equations that are solved numerically. All three of these alternatives are explored in this month's article.</p><img src="/view.aspx?si=87605/thumb.jpg" alt="Classroom Tips and Techniques: A Note on Parametric Plotting" align="left"/><p>Under suitable conditions, the equations f(x, y, t) = g(x, y, t) and g(x, y, t) = 0 define the curve x = x(t), y = y(t) parametrically. If the algebra permits an explicit solution, the resulting curve can be drawn with Maple's basic parametric plotting functionality. However, when the algebra is recalcitrant, the equations must be solved numerically for a sequence of points, or converted to differential equations that are solved numerically. All three of these alternatives are explored in this month's article.</p>87605Wed, 05 May 2010 04:00:00 ZDr. Robert LopezDr. Robert Lopez