bimbo36

well i think i sort of understand what simultaneous equations are studiot , and i think we can use the numerical method gaussian elimination to solve such equations ??

am i getting it wrong somewhere ... ??

ok .. i sort of understand simultaneous equations ... and you use numerical methods such as gaussian elimination to solve it ...

the types of equations were confusing me ...

i thought a bunch of usual polynomial linear equations ...such as these

 

Linear 4x+3

were the simultaneous equations ....

let me arrange a few things for my understandings sake ...

first let me start with .. the usual equations ...

 

When we know the degree we can also give the polynomial a name:
Degree Name Example
0 Constant 7
1 Linear 4x+3
2 Quadratic x2−3x+2
3 Cubic 2x3−5x2
4 Quartic x4+3x−2

then we have simulaneous equations ...which looks like these ...

 

x+2y-3z=10
2x-3y-4z=1
y-3x+z=-8

these two numerical methods can be applied to it if you have to deal with equations like these ...

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

now could somebody please tell me to what sort of equations do i apply the rest of the three methods mentioned below ???

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

are those methods for ??

transcendental functions ??
differentiation ???
intergration ??
differential equation ???

thanks ...

i just got it confused with

 

When we know the degree we can also give the polynomial a name:
Degree Name Example
0 Constant 7
1 Linear 4x+3
2 Quadratic x2−3x+2
3 Cubic 2x3−5x2
4 Quartic x4+3x−2

a linear equation .. ?

and systems of linear equations ??

because i thought the gaussian elimination methods were for ...equations involving many linear equations ...

 

this is a simple linear equation ... Linear 4x+3 ... because the degree of the polynomial is 1 ...
then we have many linear equations ... where the degree of the polynomials is still 1 ...

when we have many linear equations where the degree of the polynomial is 1 .. we have the following methods ...

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

ok ..thanks for the reply .. i have many more basic questions and doubts regarding this numerical methods ...

first of all ,,.

what is the difference between linear equation .. ?

and systems of linear equations ??

 

When we know the degree we can also give the polynomial a name:
0 Constant 7
1 Linear 4x+3
2 Quadratic x^2−3x+2
3 Cubic 2x^3−5x^2
4 Quartic x^4+3x−2

this is a simple linear equation ... Linear 4x+3 ... because the degree of the polynomial is 1 ...
then we have many linear equations ... where the degree of the polynomials is still 1 ...

when we have many linear equations where the degree of the polynomial is 1 .. we have the following methods ...

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

now as for non linear equations ...

 

When we know the degree we can also give the polynomial a name:
Degree Name Example
0 Constant 7
1 Linear 4x+3
2 Quadratic x2−3x+2
3 Cubic 2x3−5x2
4 Quartic x4+3x−2

any polynomial equation of degree greater than one can be called non linear equations ... right ?

and its solving methods starts from the quadratic equation themselves i guess ...

to methods like the mentioned below ...

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

but ...

transcendental functions
differentiation
intergration
differential equation

all are non linear equations ... right ???


do i use these methods to solve all of them ??

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf
http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf
http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

this is a noob question to most of you .. but i would like to have some clarity ...

since i am still following computer oriented numerical methods ....

last time , i opened up a thread about polynomials and numerical methods ...

some of the things got clearer ... and some of the things got lost in the confusion of discussions ...

the last thread was mostly a lot of basic questions about linear and non linear polynomial equations ...

some of the things got clearer ... some not that clear ....

anyway back to the question ...

what are simultaneous equation ?

what are systems of linear equations ?

is it simply a bunch of polynomial equations ??

hello studiot , thanks for all the help , support and questions...

why i dont work with problems right now is because , i was a bit weak in algebra in general ...

i was also looking for small problems to begin working with examples ...

thank god , i found this excellent beginner book on algebra ....

 

Peter Selby, Steve Slavin-Practical Algebra_ A Self-Teaching Guide-John Wiley & Sons (1991)

i am also ... going to buy two more books from an online store ...

 

How to Ace Calculus: The Streetwise Guide

with the help of those two books .. i am going to work on examples of algebra and calculus ....

later , after i get those books... i will start working on differential equations too ... right now i am not touching that subject ....

i am glad i got a better overall picture of this subject ...

now its time to practise from basics.... with the help of those books ....

In y = f(x) do you understand what is meant by the independent variable and the dependent variable; please state which is which?

For the function is y = 5 does my definition make the function linear or nonlinear?

Is this true for any y = a constant?

 

x is independent ,y is dependent

y=5 is linear and for any value of a yes it is linear

for any value of 'a', yes it is linear

let me rearrange few things one more time ....

f(x) = polynomial ??

f(x) = transcendental ??

f(x) = polynomial of degree one ?? therefore linear ? Solution of Linear Equations? Direct Methods ? http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

f(x) = polynomial of degree one ?? therefore linear ? Solution of Linear Equations? Indirect Methods ? http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

f(x) = polynomial greater than degree one ?? therefore non linear ?? solving nonlinear equations ? Fixed Point ?http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

f(x) = polynomial greater than degree one ?? therefore non linear ?? solving nonlinear equations ? Bisection ? http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

f(x) = polynomial greater than degree one ?? therefore non linear ?? solving nonlinear equations ? Newton ? http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

is this right so far ??

its better than getting stuck .... atleast things are going to get better in 2016 ....

 

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 ???

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.
polynomial

When we know the degree we can also give the polynomial a name:
Degree Name Example
0 Constant 7
1 Linear 4x+3
2 Quadratic x2−3x+2
3 Cubic 2x3−5x2
4 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 ???

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 .... ???

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 Method
Method of false position
Iteration method
Newton-Raphson Method
Ramanujan's method
The 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 series
Picard's method of successive approximations
Euler's method
Modified Euler's Method
Runge-Kutta method
Predictor-Corrector Methods
Adams-Moulton Method
Milne's method

i guess this is really it ... ????

no more numerical methods left to learn .... ???

Computer Oriented Numerical Methods

https://books.google.com.sa/books/about/Computer_Oriented_Numerical_Methods_1E.html?id=J1J9W7EDdmcC&redir_esc=y

this one ...

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 ....

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.

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 ....

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 ??

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 ...

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 ...

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 errors

Interpolation

Numerical 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 Equations

Numerical Differentiation
Numerical Integration


Numerical Solution of Ordinary differential equations
Curve fitting
Numerical 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 ???

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 .... ???

the problem was , back in the days ... programming was a bit hard ... especially trying to code something for numerical methods in c ...

i had this book ...

Numerical Recipes

Flannery, Press, Teukolsky and Vetterling

but i lost it , somehow ... i think i lost interest too ... since both the programming and the maths involved in it was hard ...

noawadays , programming doesnt look that hard ... but the maths is ...

this one for example ...

this one with a little bit of effort and time ... can be turned into a program in C ...

for example a matrix operation with c program ...

i need to figure out how to turn a polynomial into a good matrix equation ... with its coefficients and all ...

this will take a bit of time ....

i think the coefficients becomes the elements of the matrix , in this case ...

 

how to write c program to solve a system of linear equations

http://www.slideshare.net/SendashPangambam/solvng-linear-equations-37190053

 

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

 

#include <stdio.h>
int main(){
float a[2][2], b[2][2], c[2][2];
int i,j;
printf("Enter the elements of 1st matrix\n");
/* Reading two dimensional Array with the help of two for loop. If there was an array of 'n' dimension, 'n' numbers of loops are needed for inserting data to array.*/
for(i=0;i<2;++i)
for(j=0;j<2;++j){
printf("Enter a%d%d: ",i+1,j+1);
scanf("%f",&a[j]);
}
printf("Enter the elements of 2nd matrix\n");
for(i=0;i<2;++i)
for(j=0;j<2;++j){
printf("Enter b%d%d: ",i+1,j+1);
scanf("%f",&b[j]);
}
for(i=0;i<2;++i)
for(j=0;j<2;++j){
/* Writing the elements of multidimensional array using loop. */
c[j]=a[j]+b[j]; /* Sum of corresponding elements of two arrays. */
}
printf("\nSum Of Matrix:");
for(i=0;i<2;++i)
for(j=0;j<2;++j){
printf("%.1f\t",c[j]);
if(j==1) /* To display matrix sum in order. */
printf("\n");
}
return 0;
}

 

$ gcc inverse_matrix.c -o inverse_matrix
$ ./inverse_matrix

Enter the order of the Square Matrix : 3

Enter the elements of 3X3 Matrix : 3 5 2 1 5 8 3 9 2

The inverse of matrix is :
0.704545 -0.090909 -0.340909
-0.250000 -0.000000 0.250000
0.068180 0.136364 -0.113636

let me see if i can focus on the ...

 

The Numerical Methods for Linear Equations and Matrices

anyway thanks again for the book recomendations...

actually the subject i was following was called computer oriented numerical method ... with some examples of c programming language trying to solve some methods of numerical methods ...

there was lots of different types of numerical methods in it ... like newton raphson method , bisection method ...secant method ...regula falsi method ...mullers method ..horners method ...trapezoidal formula ..trapezoidal rule ... etc etc ...

and in the end i am supposed to write computer programs for it in C language ... which was and still is a difficult task for me ...

but it also had this one ...

 

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.

but it also had these ones,with matrix operations ...

which looked like a bit more easier ones to put it into computer programs in C ...

which is what i was looking for ...

the simpler ones ...

the rest seems like a mess ...

anyway thanks a lot for taking the time to reply ...

its this part from that website where i will be focusing on ...

http://ads.harvard.edu/books/1990fnmd.book/chapt2.pdf

 

The Numerical Methods for Linear Equations and Matrices

 

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.

because i think without much difficulty , this part can be made into a program in C ...

yea , i read that .. still a bit difficult for me to grasp everything written in it ..

the pdf book has some simpler explanations ...

for example ...

 

Suppose that the function to be approximated, f(x) , is observed at a series of values x 1 ,...,x N . In general, we will observe y i = f(x i ) + ε i ,wherethe ε i are unknown errors. The task is to estimate f(x) for new values of x . If the new x is within the range of the observed abscissae, then the problem is interpolation . If it is outside, then the problem is extrapolation

 

. Polynomials are useful for interpolation, but notoriously poor at extrapolation

right now am not sure how many methods there are for polynomial approximations ...

i just googled " methods of polynomial approximation " .. and found this site ...

http://ads.harvard.edu/books/1990fnmd.book/

 

the numerical methods for linear equations and matrices

 

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 begi n our study of numerical methods with a description of methods for manipulating matrices and solving syst ems 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.

isnt this a bit like the noob friendly part of the polynomials and numerical method ???

i am not that familiar with other methods ...

its the amount of terms , that is currently confusing me ... it does not seem to end ...

sorry i feel very new to numerical methods .. even though we had this subject in college , along with computer programs in c ...

let me change a few things , in your reply ... so that it makes a little bit more sense to me ... for my understanding ....

 

Beware of jumping ahead too quickly or you will find yourself increasingly tangled up.

 

I identified that you misunderstand the word linear ( a very common misunderstanding, shared with the author of your last reference).

 

A function, p(x), (including polynomials) is linear if and only if

 

1) For any x₁ and x₂

 

p(x₁ + x₂) = p(x₁) + f(x₂)

 

2) For any x and any coefficient a

 

ap(x) = p(ax)

 

Try this with p(x) = 1+x and see if it works.

 

You will find it does not.

 

p(x) = 1 + x is called affine not linear.

 

If you add the constant to a polynomial it changes things.

 

I did suggest you need to look at linear mathematics and also offered some links about basic terms in numerical mathematics.

 

Did you look at them?

i had to refresh a lot of terms ... to understand this one ...

including terms like ...

monomial

binomial

trinomial

polynomial

degree

coeficients

linear

quadratic

cubic

quartic ...

base , now more stuffs ...

is this learning of terms ever going to end ? sorry i am a noob ..

how does a simple question look like ?

i have seen polynomial coefficients arranged into matrix equations for solving purposes ...

i dont remember what it was called now ...

because i dont have that book with me right now ... i might have to collect it from someone else ....

 

Consider height as a function of age for 318 girls who were seen in a disease study [6] in East Boston in 1980 (Figure 2). Height might be described roughly by a straight line over a short range of ages – say, ages 5 – 10 – but over wider age ranges a more general function is needed. We initially fit a sixth-order polynomial

with the intention of decreasing the order later if a simpler polynomial is found to fit just as well. This leads to a multiple linear regression

problem for the coefficients β 0 ,...,β 6 , in which the design matrix is

 

y i ≈ β 0 + β 1 x i + β 2 x 2 i + β 3 x 3 i + β 4 x 4 i + β 5 x 5 i + β 6 x 6 i + ε i ,

this conversation is helping by the way ... even though i keep getting lost ....

i found a book online ... which explains a lot about what i was looking for ...

http://www.statsci.org/webguide/smyth/pubs/EoB/bap064-.pdf

this is something i found when i searched Polynomial Approximation , in google

 

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

can somebody tell me , where to from here ??