Linear Algebra: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=144
en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 27 Feb 2020 21:30:08 GMTThu, 27 Feb 2020 21:30:08 GMTNew applications in the Linear Algebra categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgLinear Algebra: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=144
Linear Codes and Syndrome Decoding
https://www.maplesoft.com/applications/view.aspx?SID=154536&ref=Feed
Implementation of the encoding and decoding algorithms associated to an error-correcting linear code. Such a code can be characterized by a generator matrix or by a parity-check matrix and we introduce, as examples, the [7, 4, 2] binary Hamming code, the [24, 12, 8] and [23, 12, 7] binary Golay codes and the [12, 6, 6] and [11, 6, 5] ternary Golay codes. We give procedures to compute the minimum distance of a linear code and we use them with the Hamming and Golay codes. We show how to build the standard array and the syndrome array of a linear code and we give an implementation of syndrome decoding. Finally, we simulate a noisy channel and use the Hamming and Golay codes to show how syndrome decoding allows error correction on text messages.<img src="https://www.maplesoft.com/view.aspx?si=154536/Golay3.jpg" alt="Linear Codes and Syndrome Decoding" style="max-width: 25%;" align="left"/>Implementation of the encoding and decoding algorithms associated to an error-correcting linear code. Such a code can be characterized by a generator matrix or by a parity-check matrix and we introduce, as examples, the [7, 4, 2] binary Hamming code, the [24, 12, 8] and [23, 12, 7] binary Golay codes and the [12, 6, 6] and [11, 6, 5] ternary Golay codes. We give procedures to compute the minimum distance of a linear code and we use them with the Hamming and Golay codes. We show how to build the standard array and the syndrome array of a linear code and we give an implementation of syndrome decoding. Finally, we simulate a noisy channel and use the Hamming and Golay codes to show how syndrome decoding allows error correction on text messages.https://www.maplesoft.com/applications/view.aspx?SID=154536&ref=FeedThu, 06 Jun 2019 04:00:00 ZJosé Luis Gómez PardoJosé Luis Gómez PardoSystem of Equations 2x2 and 3x3
https://www.maplesoft.com/applications/view.aspx?SID=154520&ref=Feed
This application solves a set of compatible equations of two or three variables. For two variables, it also graphs the intersection point of the variable "x" and "y". If we want to observe the intersection point closer we will use the zoom button that is activated when manipulating the graph. If we want to change the variable ("x" and "y") we enter the code of the button that solves and graphs. For three variables, the intersecting planes are shown.
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In Spanish.<img src="https://www.maplesoft.com/view.aspx?si=154520/sis_eq_dpd.png" alt="System of Equations 2x2 and 3x3" style="max-width: 25%;" align="left"/>This application solves a set of compatible equations of two or three variables. For two variables, it also graphs the intersection point of the variable "x" and "y". If we want to observe the intersection point closer we will use the zoom button that is activated when manipulating the graph. If we want to change the variable ("x" and "y") we enter the code of the button that solves and graphs. For three variables, the intersecting planes are shown.
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In Spanish.https://www.maplesoft.com/applications/view.aspx?SID=154520&ref=FeedTue, 19 Mar 2019 04:00:00 ZLenin Araujo CastilloLenin Araujo CastilloEigenpairs: What are they and how they are found
https://www.maplesoft.com/applications/view.aspx?SID=154291&ref=Feed
Clearly, Maple can compute eigenpairs (eigenvalues and eigenvectors) for a matrix, but of what help is Maple in getting across the concept of an eigenpair, and relating that insight to the standard algorithms students are expected to use to find them? This application is the companion Maple document to the webinar “Eigenpairs in Maple”, presented by Dr. Robert Lopez. In both the webinar and this application, he demonstrates how Maple can enhance the task of teaching the eigenpair concept, and shows how Maple bridges the gap between the concept and the algorithms.
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<B>View the Recorded Webinar:</B><BR>
<A HREF="/webinars/recorded/featured.aspx?id=1181">Eigenpairs in Maple</A><img src="https://www.maplesoft.com/view.aspx?si=154291/eigenpair.jpg" alt="Eigenpairs: What are they and how they are found" style="max-width: 25%;" align="left"/>Clearly, Maple can compute eigenpairs (eigenvalues and eigenvectors) for a matrix, but of what help is Maple in getting across the concept of an eigenpair, and relating that insight to the standard algorithms students are expected to use to find them? This application is the companion Maple document to the webinar “Eigenpairs in Maple”, presented by Dr. Robert Lopez. In both the webinar and this application, he demonstrates how Maple can enhance the task of teaching the eigenpair concept, and shows how Maple bridges the gap between the concept and the algorithms.
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<B>View the Recorded Webinar:</B><BR>
<A HREF="/webinars/recorded/featured.aspx?id=1181">Eigenpairs in Maple</A>https://www.maplesoft.com/applications/view.aspx?SID=154291&ref=FeedFri, 25 Aug 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezMathematics for Chemistry
https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=Feed
This interactive electronic textbook in the form of Maple worksheets comprises two parts.
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Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019<img src="https://www.maplesoft.com/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" style="max-width: 25%;" align="left"/>This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=FeedTue, 30 May 2017 04:00:00 ZProf. John OgilvieProf. John OgilvieKinematics of Our Earth-Moon System
https://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="https://www.maplesoft.com/view.aspx?si=153554/a05277b27fbcd36e1df1952c6d5969be.gif" alt="Kinematics of Our Earth-Moon System" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=153554&ref=FeedWed, 23 Apr 2014 04:00:00 ZAli Abu OamAli Abu OamInternet Page Ranking Algorithms
https://www.maplesoft.com/applications/view.aspx?SID=153532&ref=Feed
In this guest article in the Tips and Techniques series, Dr. Michael Monagan explains how internet pages are ranked.<img src="https://www.maplesoft.com/view.aspx?si=153532/thumb.jpg" alt="Internet Page Ranking Algorithms" style="max-width: 25%;" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan explains how internet pages are ranked.https://www.maplesoft.com/applications/view.aspx?SID=153532&ref=FeedThu, 20 Mar 2014 04:00:00 ZProf. Michael MonaganProf. Michael MonaganCollision detection between toolholder and workpiece on ball nut grinding
https://www.maplesoft.com/applications/view.aspx?SID=153477&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=153477/0320a66eb812382755a045a5251b1390.gif" alt="Collision detection between toolholder and workpiece on ball nut grinding" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=153477&ref=FeedMon, 23 Dec 2013 05:00:00 ZGyörgy HegedûsGyörgy HegedûsClassroom Tips and Techniques: Locus of Eigenvalues
https://www.maplesoft.com/applications/view.aspx?SID=153463&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=153463/thumb.jpg" alt="Classroom Tips and Techniques: Locus of Eigenvalues" style="max-width: 25%;" 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?https://www.maplesoft.com/applications/view.aspx?SID=153463&ref=FeedFri, 15 Nov 2013 05:00:00 ZDr. Robert LopezDr. Robert LopezApplication of the Modified Gram-Schmidt Algorithm
https://www.maplesoft.com/applications/view.aspx?SID=152382&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=152382/05160ad08a75a6b7948e889b5999f0ea.gif" alt="Application of the Modified Gram-Schmidt Algorithm" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=152382&ref=FeedTue, 01 Oct 2013 04:00:00 ZDouglas LewitDouglas LewitSymmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE
https://www.maplesoft.com/applications/view.aspx?SID=151383&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=151383/440fb9a2994e797b26c18564d860131b.gif" alt="Symmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=151383&ref=FeedThu, 05 Sep 2013 04:00:00 ZOlga V. DvornikOlga V. DvornikClassroom Tips and Techniques: Least-Squares Fits
https://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="https://www.maplesoft.com/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=140942&ref=FeedWed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezLeast Squares and QP Optimization
https://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="https://www.maplesoft.com/view.aspx?si=129826/maple-gf.jpg" alt="Least Squares and QP Optimization" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=129826&ref=FeedThu, 19 Jan 2012 05:00:00 ZMarcus DavidssonMarcus DavidssonzoMbi
https://www.maplesoft.com/applications/view.aspx?SID=129642&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=129642/zombie_3.jpg" alt="zoMbi" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=129642&ref=FeedSun, 15 Jan 2012 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoClassroom Tips and Techniques: An Undamped Coupled Oscillator
https://www.maplesoft.com/applications/view.aspx?SID=129521&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=129521/thumb.jpg" alt="Classroom Tips and Techniques: An Undamped Coupled Oscillator" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=129521&ref=FeedTue, 10 Jan 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezLinear Algebra Example Generator
https://www.maplesoft.com/applications/view.aspx?SID=129347&ref=Feed
<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>
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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="https://www.maplesoft.com/view.aspx?si=129347/linearnew_sm.jpg" alt="Linear Algebra Example Generator" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=129347&ref=FeedThu, 05 Jan 2012 05:00:00 ZJeff KnisleyJeff KnisleyClassroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem
https://www.maplesoft.com/applications/view.aspx?SID=128444&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=128444/thumb.jpg" alt="Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=128444&ref=FeedTue, 06 Dec 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezAlgebraic Riccati Equations in Control Theory
https://www.maplesoft.com/applications/view.aspx?SID=103818&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=103818/thumb.jpg" alt="Algebraic Riccati Equations in Control Theory" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=103818&ref=FeedWed, 06 Apr 2011 04:00:00 ZMaplesoftMaplesoftTerminator circle with animation
https://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="https://www.maplesoft.com/view.aspx?si=100509/thumb.jpg" alt="Terminator circle with animation" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=100509&ref=FeedTue, 28 Dec 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyGeneration of correlated random numbers
https://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="https://www.maplesoft.com/view.aspx?si=99806/maple_icon.jpg" alt="Generation of correlated random numbers" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=99806&ref=FeedFri, 03 Dec 2010 05:00:00 ZI. HlivkaI. HlivkaVectors and Matrices
https://www.maplesoft.com/applications/view.aspx?SID=99816&ref=Feed
<p>This worksheet shows how Maple performs vector and matrices computations</p><img src="https://www.maplesoft.com/view.aspx?si=99816/maple_icon.jpg" alt="Vectors and Matrices" style="max-width: 25%;" align="left"/><p>This worksheet shows how Maple performs vector and matrices computations</p>https://www.maplesoft.com/applications/view.aspx?SID=99816&ref=FeedFri, 03 Dec 2010 05:00:00 ZAli Abu OamAli Abu Oam