Few noob questions and numerical methods help....

[awholenumber]

i was trying to work with few problems for my computer oriented numerical method ...which i had in college ...

i am not in college anymore ... but i failed that subject ... i am trying to improve this maths field as much as i possibly can , because its a difficult subject for me ...

let met start with a few things i am familiar 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

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

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

i think 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[/PLAIN]

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[/PLAIN]



are those methods for ??

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

 

[S.G. Janssens]

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.pdfare those methods for ??

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

These methods are for determining solutions of (systems of) nonlinear equations. The equations may involve transcendental functions, but for example also higher order polynomials. (Note that the system of three equations that you wrote down above consists of linear equations only.) Newton's method is a good default choice, but there are caveats. Nonlinear root finders may also occur as part of the numerical solution of other problems, for example involving differential equations.

 

[awholenumber]

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.pdfso these three methods can be used to solve ...polynomials with degree greater than one , which is not linear ...and when the usual factorisation methods won't work ? for my understandings sake ...

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

also to ...

transcendental functions
differentiation
intergration
differential equation

for some reason .. i thought i could narrow it down to few types of equations and its numerical methods ...
but this subject again .. even though its just the three methods ... is still confusing ...

so this

fixed point iteration

will work for ...

non linear higher order polynomials ...??
transcendental functions
differentiation
intergration
differential equation ...

bisection method

works for ...non linear higher order polynomials ...??
transcendental functions
differentiation
intergration
differential equation ...

Newtons method

works for ...

non linear higher order polynomials ...??
transcendental functions
differentiation
intergration
differential equation ...

 

[pbuk]

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

These are all polynomial expressions, for example we write $x^2 - 3x + 2 = 0$ and call it a second degree polynomial equation, or a quadratic equation. First degree polynomial equations are also called linear equations.

then we have simulaneous equations ...which looks like these ...x+2y-3z=10
2x-3y-4z=1
y-3x+z=-8

Note that these are all linear equations, and together we call them a system of linear equations, or a linear system.

i think 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

Yes, as the titles imply these methods are used to find numerical solutions to linear systems.

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

Again as the titles imply these methods are used to find numerical solutions to non-linear systems. Quadratic systems e.g. $x = 3t; y = 10t - \frac {9.8}2 t^2; y = 0$ are non-linear, and so is any system that contains terms other than a variable multiplied by a constant e.g. $x = \cos t; y = \sin t; t = 2$.

are those methods for ??
differentiation ?
intergration ??
differential equation ?

No, they are all for solving systems of equations i.e. working out what values of the variables solve all the constraints. (Numerical) methods for differentiation and integration solve different classes of problems, although if you look at Newton's method in your last link, this can also be used for solving differential equations.

 

[S.G. Janssens]

for some reason .. i thought i could narrow it down to few types of equations and its numerical methods ...but this subject again .. even though its just the three methods ... is still confusing ...

There are many numerical methods for many different problems. Even when we restrict ourselves to systems of linear equations, there are various different methods that have different benefits and drawbacks, depending on the structure of the particular problem at hand.

I think you may benefit from a well-structured introductory book with many examples. Would you like that? I have my own favorites on numerical analysis, but maybe in this case it is better when others make some recommendations first. Or did your course already come with prescribed literature?

 

[awholenumber]

thanks for the detailed explanation MrAnchovyKrylov , this basically means i have to find atleast one proper simple example ... for

non linear higher order polynomials ...?? fixed point , bisection , Newton
transcendental functions ?? fixed point , bisection , Newton
differentiation ? fixed point , bisection ,Newton
intergration ?? fixed point , bisection , Newton
differential equation ... ?? fixed point , bisection , Newton

to apply these methods to ...

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

that makes it atleast 15 problems to look into ...

i myself have gone through many books roughly ... which had algorithms in it in programming languages ...
most of it were a bit hard to follow...

could you suggest me some simple books ??

 

[S.G. Janssens]

could you suggest me some simple books ??

I enjoy https://www.amazon.com/dp/0471624896/?tag=pfamazon01-20 by Atkinson, but I'm not sure whether the level is appropriate for you. It does not give codes, but presents the basic algorithms clearly enough so you can do your own coding in the language of your choice. Atkinson, together with Han, has also written Elementary Numerical Analysis, but I don't know this book.

There is also the book by Cleve Moler, Numerical Computing with MATLAB, which is available for free. It is much more applied than Atkinson's book and is tied to the MATLAB environment, which is a good but non-free software package suitable for the exploration of numerical methods. Octave is a free alternative, but I don't know how well it goes together with Moler's book.

Many other textbooks exist, so I also encourage you to browse an online bookstore and have a look at the reviews. It depends a lot on your personal style.

 

[awholenumber]

thanks for the books suggestion ... i will try to buy the books from some online store ...

let me see if i can apply the

fixed point iteration

tonon linear higher order polynomials
transcendental functions
differentiation
intergration
differential equation ...to begin with ...