bimbo36 Posted November 16, 2015 Author Share Posted November 16, 2015 i am happy that i could organize a part of this vast subject atleast like this ... if those matrix equations and polynomials are all part of the linear system ... i wonder what non linear systems are ... i donno if i should continue in this thread , or start a seperate thread for non linear systems ... because the coversation was a little bit of fun and was very helpfull ... i already read some books , pdfs and got some definitions from it ... let me quote it here ... Nonlinear Systems Nonlinearity is ubiquitous in physical phenomena. Fluid an d plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inherently nonlinear equations. (The one notable exception is quantum mechanics, which is a fundamentally linear theory, although recent attempts at grand unification of all fundamental physical theories, such as string theory and conformal field theory, [ 8 ], are nonlinear.) For this reason, an ever increasing proportion of modern mathematical research is devoted to the analysis of nonlinear systems and nonlinear phenomena Why, then, does one devote so much time studying linear mathematics? The facile answer is that nonlinear systems are vastly more difficult to analyze. In the nonlinear regime, many of the most basic questions remain unanswered: existence and uniqueness of solutions are not guaranteed; explicit formulae are difficult to come by; linear superposition is no longer available; numerical approximations are not always sufficiently accurate; etc., etc. A more intelligent answer is that a thorough understanding of linear phenomena and linear mathematics is an essential prerequisite for progress in the nonlinear arena. Therefore, one must first develop the proper linear foundations in sufficient depth before we can realistically confront the untamed nonlinear wilderness. Moreover, many important physical systems are “weakly nonlinear”, in the sense that, while nonlinear effects do play an essential role, the linear terms tend to dominate the physics, and so, to a first approximation, the system is essentially linear. As a result, such nonlinear phenomena are best understood as some form of perturbation of their linear approximations. The truly nonlinear regime is, even today, only sporadically modeled and even less well understood Nonlinear Equations Definition A value for parameter x that satisfies the equation f(x) = 0 is called a root or a (“zero”) of f(x) . Exact Solutions For some functions, we can calculate roots exactly; e.g., Polynomials up to degree 4 Simple transcendental functions, such as sin x = 0 which has an infinite number of roots x = k π ( k = 0, ± 1, ± 2, . . . ) Numerical Methods Used to estimate roots for nonlinear functions f(x). Bisection Method False Position Method Newton’s Method Secant Method my point is to organize these things in a better way , so that i can have a better understanding of these things .. i am not sure where these are all going right now .... ??? Link to comment Share on other sites More sharing options...
bimbo36 Posted November 19, 2015 Author Share Posted November 19, 2015 this is how my syllabus looked like ... i am trying to organize these in a proper way .. based of my understanding so far , and with a little bit of help from a graph ... i am not sure if am organizing this right ... i thought why not give it a try ... Numerical Methods and errorsInterpolationNumerical Solution of a system of Linear Equations Gauss-Jordan Elimination Gaussian Elimination with Backsubstitution LU Decomposition and Its Applications Tridiagonal and Band Diagonal Systems of Equations Iterative Improvement of a Solution to Linear Equations Singular Value Decomposition Sparse Linear Systems Vandermonde Matrices and Toeplitz Matrices Cholesky Decomposition QR Decomposition Is Matrix Inversion an N 3 Process ? Root Finding and Nonlinear Sets of Equations Bracketing and Bisection Secant Method, False Position Method, and Ridders’ Method Van Wijngaarden–Dekker–Brent Method Newton-Raphson Method Using Derivative Roots of Polynomials Newton-Raphson Method for Nonlinear Systems of Equations Globally Convergent Methods for Nonlinear Systems of Equations Solution of Algebraic and Transcendental EquationsNumerical DifferentiationNumerical IntegrationNumerical Solution of Ordinary differential equationsCurve fittingNumerical Solution of problems associated with Partial Differential Equations Nonlinear Systems Nonlinearity is ubiquitous in physical phenomena. Fluid an d plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inherently nonlinear equations. (The one notable exception is quantum mechanics, which is a fundamentally linear theory, although recent attempts at grand unification of all fundamental physical theories, such as string theory and conformal field theory, [ 8 ], are nonlinear.) For this reason, an ever increasing proportion of modern mathematical research is devoted to the analysis of nonlinear systems and nonlinear phenomena Why, then, does one devote so much time studying linear mathematics? The facile answer is that nonlinear systems are vastly more difficult to analyze. In the nonlinear regime, many of the most basic questions remain unanswered: existence and uniqueness of solutions are not guaranteed; explicit formulae are difficult to come by; linear superposition is no longer available; numerical approximations are not always sufficiently accurate; etc., etc. A more intelligent answer is that a thorough understanding of linear phenomena and linear mathematics is an essential prerequisite for progress in the nonlinear arena. Therefore, one must first develop the proper linear foundations in sufficient depth before we can realistically confront the untamed nonlinear wilderness. Moreover, many important physical systems are “weakly nonlinear”, in the sense that, while nonlinear effects do play an essential role, the linear terms tend to dominate the physics, and so, to a first approximation, the system is essentially linear. As a result, such nonlinear phenomena are best understood as some form of perturbation of their linear approximations. The truly nonlinear regime is, even today, only sporadically modeled and even less well understood Root Finding and Nonlinear Sets of Equations does this non linear sets of equations simply means , equations involving differentiation and integration ??? Link to comment Share on other sites More sharing options...
studiot Posted November 19, 2015 Share Posted November 19, 2015 It might do. If you have more unknowns than equations, differentiation/integration is one way to get more equations. This occurs in structural engineering where you know the slope is zero so you can differentiate and set to zero. But you should not say nonlinear sets of equations. It is not the set which is linear or nonlinear. That is meaningless. It is the equations in the set or at least one of them which is/are non linear. Link to comment Share on other sites More sharing options...
bimbo36 Posted November 20, 2015 Author Share Posted November 20, 2015 (edited) thanks ... i just keep finding a lot of definitions from a lot of sites ... A linear equation is one related to a straight line, for example f ( x )= mx + c describes a straight line with slope m and the linear equation f ( x )= 0, involving such an f , is easily solved to give x = − c/m (as long as m = 0). If a function f is not related to a straight line in this way we say it is nonlinear The nonlinear equation f ( x )= 0 may have just one solution, like in the linear case, or it may have no solutions at all, or it may have many solutions. For example if f ( x )= x 2 − 9 then it is easy to see that there are two solutions x = − 3 and x =3 . The nonlinear equation f ( x )= x 2 +1 has no solutions at all (unless the application under consideration makes it appropriate to consider complex numbers). Our aim in this Section is to approximate (real-valued) solutions of nonlinear equations of the form f ( x )=0. The following definitions have been gathered together in a key point If the value x is such that f ( x )= 0 we say that x is a root of the equation f ( x )= 0 or that x is a zero of the function f . Find any (real valued) zeros of the following functions. (Give 3 decimal places if you are unable to give an exact numerical value.) (a) f ( x )= x 2 + x − 20 (b) f ( x )= x 2 − 7 x +5 ( c ) f ( x )= 2 x − 3 (d) f ( x )= e x +1 (e) f ( x )= sin( x ) Solution (a) This quadratic factorises easily into f ( x )=( x − 4)( x +5 ) and so the two zeros of this f are x =4, x = − 5. (b) The nonlinear equation x 2 − 7 x + 5=0 requires the quadratic formula and we find that the two zeros of this f are x = 7 ± √ 7 2 − 4 × 1 × 5 2 = 7 ± √ 29 2 which are equal to x =0 . 807 and x =6 . 193, to 3 decimal places. (c ) Using the natural logarithm function we see that x ln(2) = ln(3) from which it follows that x = ln(3) / ln(2) = 1 . 585, to 3 decimal places. (d) This f has no zeros because e x +1 is always positive. (e) sin( x ) has an infinite number of zeros at x =0 , ± π, ± 2 π, ± 3 π ,... .T o3 decimal places this is x =0 . 000 , ± 3 . 142 , ± 6 . 283 , ± 9 . 425 ,... . now if only i could find some examples like these for Numerical differentiation and Numerical integration ...that would make my life a bit more easier ... Edited November 20, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
studiot Posted November 20, 2015 Share Posted November 20, 2015 A linear equation is one related to a straight line, for example f ( x )= mx + c describes a straight line with slope m and the linear equation f ( x )= 0, involving such an f , is easily solved to give x = − c/m (as long as m = 0). If that is what they actually said, I would run a long way from that site and then keep running. Link to comment Share on other sites More sharing options...
bimbo36 Posted November 20, 2015 Author Share Posted November 20, 2015 (edited) is that definition wrong ... ?? i myself have no idea about the depth of these definitions ... but i am getting a rough overall picture of all these things ... which for me at this stage is very helpful ... but i cannot find much definition or online notes for , numerical differentiation or numerical integration ... not sure where to find a simple description of these ... Edited November 20, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
studiot Posted November 20, 2015 Share Posted November 20, 2015 Not only was the website wrong but the second part of the statement was (I hope) miscopied and even worse x = − c/m (as long as m = 0). I have already said in this thread that a straight line, not through the origin, is not linear, because it does not satistfy both the linearity conditions. Once again it is called affine. Link to comment Share on other sites More sharing options...
bimbo36 Posted November 20, 2015 Author Share Posted November 20, 2015 ok thanks , i have never come across the term affine before ... which is why the confusions ?? where do i get some good online material for numerical differentiation or numerical integration ?? Link to comment Share on other sites More sharing options...
studiot Posted November 20, 2015 Share Posted November 20, 2015 (edited) where do i get some good online material for numerical differentiation or numerical integration ?? Do you understand the difference between differentiation and integration? There is a huge and fundamental difference in the nature of the integral and derivative. Unless you are talking about symbolic manipulations available in certain mathematical programs such as MathCad and Mathematica; The integral of interest in numerical analysis is the definite integral, which is a pure number. The derivative is, as always, a function (hence its full title the derived function). The purpose of numerical differentiation is to solve differential equations by finding the value of the derivatives at specific points. The purpose of numerical integration is to output a result such as the area or work done as a single number. So in asking this question you need to be quite clear what you are seeking. Personally I don't do online courses, but I can recommend books. Edited November 20, 2015 by studiot Link to comment Share on other sites More sharing options...
studiot Posted November 21, 2015 Share Posted November 21, 2015 Perhaps an online start with these people. http://www.esm.psu.edu/courses/emch12/IntDyn/course-docs/Euler-tutorial/ Link to comment Share on other sites More sharing options...
bimbo36 Posted November 21, 2015 Author Share Posted November 21, 2015 (edited) thanks for the reply and the link ... at this moment , even i dont know what i am looking for .. ??? it feels like i am refreshing ...differentiation , integration and differential equations ... because i am not at all familiar with the numerical methods associated with differentiation , integration or differential equations ... but i found something nice online .... Calculus http://www.mathsisfun.com/calculus/index.html Calculus in Context http://www.math.smith.edu/Local/cicintro/ http://www.math.smith.edu/Local/cicintro/book.pdf thought i would share it here ... and also this book ... Tom M. Apostol CALCULUS VOLUME 1 One-Variable Calculus, with an Introduction to Linear Algebra http://www.mif.vu.lt/~stepanauskas/AM1/Tom%20Apostol%20-%20Calculus%20vol.1%20-%20One-variable%20Calculus,%20with%20an%20Introduction%20to%20Linear%20Algebra%20%281975%29.pdf i am not sure which path is the next , because i dont know what the names of the methods are called ? or how many type of methods there are for numerical methods associated with differentiation , integration and differential equation .... Edited November 21, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
mathsme Posted November 23, 2015 Share Posted November 23, 2015 Hi, You want the value of x? First your question is not seems simple, and i have seen many answers that are suggesting matrix use as well. thanks Link to comment Share on other sites More sharing options...
bimbo36 Posted November 23, 2015 Author Share Posted November 23, 2015 (edited) differentiation or integration , which comes first ? many books starts with integration , then moves on to explain differentiation ... i was reading this book roughly ... A Graduate Introduction to Numerical Methods_ From the Viewpoint of Backward Error Analysis when i tried to copy paste a few definitions some of the things are not showing up properly , due to this Latex issues ... even the book starts with integration , then moves to differentiation ... i thought differentiation was the easy part and integration the hard part ... anyway .. let me try to copy paste a few things from the book .. starting with differentiation itself ... i like these definitons at one place ... atleast its helping me ... let me see if this thread can get all the definitions i want atleast roughly .... http://www.personal.soton.ac.uk/jav/soton/HELM/helm_workbooks.html http://www.nucleartheory.net/Essential_Maths/helm_workbooks_jan2008.pdf Introducing Differentiation http://www.personal.soton.ac.uk/jav/soton/HELM/workbooks/workbook_11/11_1_intro_diffn.pdf Using a table of derivatives http://www.personal.soton.ac.uk/jav/soton/HELM/workbooks/workbook_11/11_2_use_tbl_derivs.pdf Basic Concepts of Integration http://www.personal.soton.ac.uk/jav/soton/HELM/workbooks/workbook_13/13_1_basic_cncpts_intgrn.pdf Numerical Differentiation and Finite Differences Numerical differentiation can be described in nearly the same terms as we described quadrature, simply by replacing three words: The basic idea of numerical differentiation is to replace f ( x ) with a slightly different function, call it f ( x )+ Δ f ( x ) or ( f + Δ f )( x ) , and differentiate the second function instead. This is our engineered problem (see Chap. 1 ). We will choose Δ f so that it’s not too large, and so that f + Δ f is simple to differentiate exactly Numerical Integration Numerical integration (also known as quadrature ) consists in using numerical methods to approximate the value of a definite integral The basic idea of numerical quadrature is to replace f ( x ) with a slightly different function, call it f ( x )+ Δ f ( x ) or ( f + Δ f )( x ) , and integrate the second function instead. This is our engineered problem (see Chap. 1 ). We will choose Δ f so that it’s not too large, and so that f + Δ f is simple to integrate exactly. That this idea handles all cases above will be seen in the examples that follow. Edited November 23, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
studiot Posted November 23, 2015 Share Posted November 23, 2015 Fancy books and fancy formulae? Why do you think folks study numerical techniques? What are the uses of integration and differentiation? Here is an everyday simple example. The construction of a concrete ground slab for a new building involves digging out ground for the slab to a depth of 0.5, along with a ring beam under the entire perimeter of the 30m x 30m slab. The ringbeam is an extra 1.0m deep below the underside of the slab, 1.2m wide at the top and 0.5 m wide at the bottom in trapezoidal section. How much earth has to be removed and replaced with concrete. This is an exercise in numerical integration that involves two of the three principle uses of integration. Link to comment Share on other sites More sharing options...
bimbo36 Posted November 25, 2015 Author Share Posted November 25, 2015 (edited) i was studying this because my syllabus had it ... we had computer oriented numerical methods in c ...i am trying to making it a bit simpler with the help of this thread , and the conversations we are having is very helpful ... i havent solved much integration or differentiation problems before ... because i was busy trying to understand the code and the languages .. back in the days , i had no idea what i was trying to solve ... there was not much depth for this subject ... it was a bit uninteresting too ... which is why i am trying to learn this from scratch .... you replies are helping ... because i can reply back with something even though its drifiting away from the subject sometimes .... i wish i could start from the basics and go to the methods of differentiation and integration ... differentiation rules integration rules the only thing i can think of is that , i get a bunch of polynomials again after differentiation or integration ???? where to from it ??? i know i am going away and away from the original topic ... that is because i have not found any example problems with explanations to work with ... or even look at ... i tried reading a lot of books ... but none of it had proper examples .... Edited November 25, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
studiot Posted November 25, 2015 Share Posted November 25, 2015 i was studying this because my syllabus had it ... we had computer oriented numerical methods in c ...i am trying to making it a bit simpler with the help of this thread , and the conversations we are having is very helpful So what is the title of your course? Link to comment Share on other sites More sharing options...
bimbo36 Posted November 25, 2015 Author Share Posted November 25, 2015 (edited) Computer Oriented Numerical Methods https://books.google.com.sa/books/about/Computer_Oriented_Numerical_Methods_1E.html?id=J1J9W7EDdmcC&redir_esc=y this one ... Edited November 25, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
studiot Posted November 25, 2015 Share Posted November 25, 2015 Seems to contain all the standard stuff. But before you start to calculate some values in mathematics you need to know some maths to want to have the results. This is where you seem to need strengthening. Link to comment Share on other sites More sharing options...
studiot Posted November 26, 2015 Share Posted November 26, 2015 What I don't understand is why you don't respond directly to material I post, specifically designed to help you. For instance you responded to my post #39 by copying tables of standard integrals and derivatives. I have no idea what that achieved. I repeat my conviction that you need to connect mathematics to its applications to progress, particularly in numical methods for they are all about applications. So why don't you try the question in my post#39? A very good series of books at your level for your purposes is the Edward Arnold Modular Mathematics series. In particular the book by Berry and Houston " Mathematical Modelling" Is all about connecting mathematics to its applications. Link to comment Share on other sites More sharing options...
bimbo36 Posted December 6, 2015 Author Share Posted December 6, 2015 (edited) i am sorry ... i am currently working as an accountant somewhere ... and i dont get so much free time to work on example problems ... i am only trying to get an overalll picture of my syllabus .... i might work on examples after i get an overall grip on this whole subject .... A very good series of books at your level for your purposes is the Edward Arnold Modular Mathematics series. In particular the book by Berry and Houston " Mathematical Modelling" Is all about connecting mathematics to its applications sorry i could not find both the books ... its unavailable at local online stores ... in the meanwhile ... i managed to put together a book with the help of nitro pro 10 ... i joined all the pdf files from HELM , helping engineers learn mathematics website ... https://learn-pilot.lboro.ac.uk/archive/olmp/olmp_resources/pages/wbooks_fulllist.html http://www21.zippyshare.com/v/eMWFsoho/file.html here is the link to the free content .... currently i am refering to that book ... because the book has nice simple explanations .... one thing i am looking at right now , is numerical differentiation https://learn-pilot.lboro.ac.uk/archive/olmp/olmp_resources/pages/workbooks_1_50_jan2008/Workbook31/31_3_num_diff.pdf the one-sided (forward) difference the one-sided (backward) difference the central difference and numerical integration .... https://learn-pilot.lboro.ac.uk/archive/olmp/olmp_resources/pages/workbooks_1_50_jan2008/Workbook31/31_2_num_int.pdf trapezium rule Simpson’s rule Gaussian quadrature i think i have only two more sections to looks into ... after this ... that is differential equations and "algebraic and transcendental equations" .... i dont know what both of it is .... right now ??? http://www.universityofcalicut.info/SDE/BSc_maths_numerical_methods.pdf Algebraic and Transcendental Equations Bisection MethodMethod of false positionIteration methodNewton-Raphson MethodRamanujan's methodThe Secant Method f(x) = 0 is called an algebraic equation if the corresponding f (x) is a polynomial An example is 7x2 + x - 8 = 0 f (x) 0 is called transcendental equation if the f (x) contains trigonometric, or exponential or logarithmic functions . Examples of transcendental equations are sin x – x = 0, tan x - x = 0 and 7x3 + log(3x - 6) + 3ex cos x+ tan x = 0. Numerical Solutions of Ordinary Differential Equations Solution by Taylor's seriesPicard's method of successive approximationsEuler's methodModified Euler's MethodRunge-Kutta methodPredictor-Corrector MethodsAdams-Moulton MethodMilne's method i guess this is really it ... ???? no more numerical methods left to learn .... ??? Edited December 6, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
studiot Posted December 6, 2015 Share Posted December 6, 2015 (edited) More than 100 views over the 1000 on this thread, yet no one else seems to have anything to say? I am sorry you could not locate the books locally, part of the reason I mentioned them is that they readily are availble secondhand at reasonable prices from international online supplies such as abe. i am sorry ... i am currently working as an accountant somewhere ... and i dont get so much free time to work on example problems ... i am only trying to get an overalll picture of my syllabus .... i might work on examples after i get an overall grip on this whole subject .... I would council against avoiding examples until the end. Numerical mathematics is an enormous subject and trying to learn all the theory before application is about the most difficult way I can imagine. The example I offered take no more than 5 minutes and the back of a small envelope to figure out. But it does provided the opportunity to discuss many of the principles underlying numerical techniques without difficult numbers getting in the way. Incidentally it is always a good idea to try out a technique on a known easy example. The posh word for this is calibration and is one of the subject areas missing from your list. Talking of lists, yours seems largely to be a list of methods or techniques and I have to tell you that it is far from complete. i guess this is really it ... ???? no more numerical methods left to learn .... ??? The old fashioned (I called it standard before) presentation of numerical methods was just this an ever growing list of methods and techniques. An alternative is to divide into subject areas, here is the contents page from the book by Brebbia "Computational Methods of Solution of Engineering Problems". You will note that almost nothing here is on your list. Yet even Brebbia's list is far from complete. Some further subject areas include Optimisation Maximum and Minimum problems Variational methods. Numerical transform methods Numerical Harmonic analysis Potential Theory Exploiting the link between boundary and body elements using numerical versions of Green's and Gauss' theorems, Linear programming Non linear programming Computational Efficiency and Accuracy and Error Theory. Here is another problem for an accountant to ponder. Edited December 6, 2015 by studiot Link to comment Share on other sites More sharing options...
bimbo36 Posted December 8, 2015 Author Share Posted December 8, 2015 (edited) thanks for the suggestions ... sorry i could not find those books anywhere ... currently i dont have a CC ... and i dont think i can make use of cash on delivery even if i ordered that book , since its a bit far away .... by this time i managed to read a lot of pdf and ebooks ... most of it lacked worked out examples ... but this thread has turned out to be something very usefull for me , even though it looks a bit messy ... that is because in my humble opinion , this is a big subject and organizing these subjects properly in my head was a bit of a hard task ... let me revise these things i have managed to learn and organize into this one post ... first of all , there is this HELM workbooks ...i find it very usefull for a person like me , who is not a maths expert ... but who is willing to spent time learning ... https://learn-pilot.lboro.ac.uk/archive/olmp/olmp_resources/pages/wbooks_fulllist.html http://www21.zippyshare.com/v/eMWFsoho/file.html and my confusions about linear , non linear maths is a bit less now ... https://books.google.com.sa/books/about/Computer_Oriented_Numerical_Methods_1E.html?id=J1J9W7EDdmcC BSc maths numerical methods http://www.universityofcalicut.info/SDE/BSc_maths_numerical_methods.pdf i also found these two links , which i find very related to my syllabus ... it is a bit more organized than many of the books ... http://www.caee.utexas.edu/prof/mckinney/ce311k/assign.html http://www.ce.utexas.edu/prof/mckinney/ce311k/assign_ce311k.html ... this is how much i have managed to organize so far ... now to work on some example problems... let me see if i can find few examples for all these methods .... ??? Edited December 8, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
bimbo36 Posted December 9, 2015 Author Share Posted December 9, 2015 (edited) this once again has messed up my head succesfully ... and i am confused again .. about the linear and non linear ... Polynomials are just about the simplest mathematical functions that exist, requiring only multiplications and additions for their evaluation. Yet they also have the flexibility to represent very general nonlinear relationships. Approximation of more complicated functions by poly- nomials is a basic building block for a great many numerical techniques A polynomial is a function that can be written in the form p(x) = c 0 + c 1 x +···+ c n x n , for some coefficients c 0 ,...,c n .If c n = 0, then the polynomial is said to be of order n . A first-order (linear) polynomial is just the equation of a straight line, while a second-order (quadratic) polynomial describes a parabola "Solving" means finding the "roots" ...... a "root" (or "zero") is where the function is equal to zero:How do we solve polynomials? That depends on the Degree!The first step in solving a polynomial is to find its degree.The Degree of a Polynomial with one variable is ...... the largest exponent of that variable.polynomialWhen we know the degree we can also give the polynomial a name:Degree Name Example0 Constant 71 Linear 4x+32 Quadratic x2−3x+23 Cubic 2x3−5x24 Quartic x4+3x−2 does this mean that only a polynomial of degree 1 is linear in nature and a quadratic equation of degree 2 , is non linear in nature ???? the numerical methods for linear equations and matrices http://ads.harvard.edu/books/1990fnmd.book/ http://ads.harvard.edu/books/1990fnmd.book/chapt2.pdf We saw in the previous chapter that linear equations play an important role in transformation theory and that these equations could be simply expressed in terms of matrices However, this is only a small segment of the importance of linear equations and matrix theory to the mathematical description of the physical world. Thus we should begin our study of numerical methods with a description of methods for manipulating matrices and solving systems of linear equations. However, before we begin any discussion of numerical methods, we must say something about the accuracy to which those calculations can be made. Nonlinear Systems Nonlinearity is ubiquitous in physical phenomena. Fluid an d plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inherently nonlinear equations. (The one notable exception is quantum mechanics, which is a fundamentally linear theory, although recent attempts at grand unification of all fundamental physical theories, such as string theory and conformal field theory, [ 8 ], are nonlinear.) For this reason, an ever increasing proportion of modern mathematical research is devoted to the analysis of nonlinear systems and nonlinear phenomena Why, then, does one devote so much time studying linear mathematics? The facile answer is that nonlinear systems are vastly more difficult to analyze. In the nonlinear regime, many of the most basic questions remain unanswered: existence and uniqueness of solutions are not guaranteed; explicit formulae are difficult to come by; linear superposition is no longer available; numerical approximations are not always sufficiently accurate; etc., etc. A more intelligent answer is that a thorough understanding of linear phenomena and linear mathematics is an essential prerequisite for progress in the nonlinear arena. Therefore, one must first develop the proper linear foundations in sufficient depth before we can realistically confront the untamed nonlinear wilderness. Moreover, many important physical systems are “weakly nonlinear”, in the sense that, while nonlinear effects do play an essential role, the linear terms tend to dominate the physics, and so, to a first approximation, the system is essentially linear. As a result, such nonlinear phenomena are best understood as some form of perturbation of their linear approximations. The truly nonlinear regime is, even today, only sporadically modeled and even less well understood so everything other than a polynomial of degree one ... is considered non linear in nature ??? then the method to solve these sort of problem ... must be like this ... right ??? Solution of Linear Equations Direct Methods http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf Solution of Linear Equations Indirect Methods http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf ??? then what does it mean to solve non linear equations ?? does that mean any polynomial other than degree 1 is non linear ??? Edited December 9, 2015 by bimbo36 Link to comment Share on other sites More sharing options...
studiot Posted December 9, 2015 Share Posted December 9, 2015 (edited) Sigh. You do not seem to want to work with me but keep posting great tracts and links trawled from the web. I will try one last time to help. Linear mathematics is about the idea that if you have a function y = f(x), if you double x you double y and if you multiply x by 55 then y increases by a factor of 55. If this is true then the funtion is linear. If it is not true then the function is non linear. Now to help both of us please answer the following questions. 1) In y = f(x) do you understand what is meant by the independent variable and the dependent variable; please state which is which? 2) For the function is y = 5 does my definition make the function linear or nonlinear? 3) Is this true for any y = a constant? Edited December 9, 2015 by studiot Link to comment Share on other sites More sharing options...
bimbo36 Posted December 9, 2015 Author Share Posted December 9, 2015 In y = f(x) do you understand what is meant by the independent variable and the dependent variable; please state which is which? x is independent, y dependent having a symbol like y to stand for f(x), is most useful when you look at the graph of f, and think about rises over runs (small changes in x, the corresponding change in y, the ratio of the latter to the former), and dy/dx defined as lim_{Delta x -> 0} (Delta y)/(Delta x), that sort of thing. All of that can be rewritten using f(x) instead of y. It's just convention. ??? f(x) = polynomial ?? f(x) = transcendental ?? f(x) = polynomial of degree one ?? therefore linear ? f(x) = polynomial greater than degree one ?? therefore non linear ?? also all the above mentioned stuff about a linear change against a non linear change ??? Link to comment Share on other sites More sharing options...
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