A few more questions about fixed point iteration ....?

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Discussion Overview

The discussion revolves around the fixed point iteration method, particularly its application to solving transcendental functions. Participants express their challenges with numerical methods and seek resources to improve their understanding of both trigonometry and fixed point iteration. The scope includes theoretical concepts, practical applications, and personal learning paths.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • Some participants express a desire to learn numerical methods, specifically fixed point iteration, and seek guidance on foundational topics like trigonometry.
  • There is a discussion on the nature of transcendental functions, with some noting that they include not only trigonometric functions but also exponential and logarithmic functions.
  • One participant emphasizes the difficulty of learning numerical methods without a background in calculus, while others suggest it may be possible to learn certain aspects without it.
  • Participants share resources, including links to books and PDFs that cover numerical methods and fixed point iteration.
  • There is a question about the relationship between linear equations, systems of linear equations, and simultaneous equations, indicating some confusion about terminology.
  • Concerns are raised about the process of choosing initial values for iterations, with suggestions that starting values can be based on educated guesses or rough estimates.
  • One participant expresses a need for clarification on the terms used in fixed point iteration, such as \(x_n\) and the significance of rearranging equations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of calculus for learning numerical methods. There are differing opinions on the feasibility of mastering fixed point iteration without a calculus background, and some participants express confusion over terminology related to equations.

Contextual Notes

Participants mention various resources and links, but there is an acknowledgment of the prerequisites needed to fully grasp numerical methods. Some participants express feelings of being overwhelmed by the complexity of the subject.

Who May Find This Useful

This discussion may be useful for individuals seeking to learn about numerical methods, particularly fixed point iteration, and those needing foundational support in trigonometry and calculus.

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first of all i simply don't want to give up learning numerical methods ...

i am trying to follow fixed point iteration method from this link ...

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf fixed point iteration can be used to solve transcendental functions ...

if i am not wrong ,transcendental functions are functions containing trigonometric equations ...

i need a bit of help with trigonometry too .. as i am a bit weak in trigonometry ...

this is a great book to begin maths .. if you are out of touch like myself... i am going to follow this book too ...

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

after that i have plans to move to learn some trigonometry ...

my understanding of trigonometry looks a bit like this only ... unfortunately ...

syw1so.jpg


2pydxkw.jpg
solving transcendental functions with the help of numerical methods such as fixed point iteration isn't exactly an easy task for me ...

so i would like to start somewhere ...

please help ...

can somebody suggest me a good book to learn trigonometry ...?
 
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rosekidcute said:
first of all i simply don't want to give up learning numerical methods ...

i am trying to follow fixed point iteration method from this link ...

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdffixed point iteration can be used to solve transcendental functions ...

if i am not wrong ,transcendental functions are functions containing trigonometric equations ...
trig functions are transcendental functions but so are exponential functions, logarithms, Bessel functions, etc.

i need a bit of help with trigonometry too .. as i am a bit weak in trigonometry ...

this is a great book to begin maths .. if you are out of touch like myself... i am going to follow this book too ...
What book?

after that i have plans to move to learn some trigonometry ...

my understanding of trigonometry looks a bit like this only ... unfortunately ...

syw1so.jpg


2pydxkw.jpg
I have no idea what those pictures have to do with trigonometry!

solving transcendental functions with the help of numerical methods such as fixed point iteration isn't exactly an easy task for me ...

so i would like to start somewhere ...

please help ...

can somebody suggest me a good book to learn trigonometry ...?
Here are some you can download for free:
http://www.freebookcentre.net/Mathematics/Trigonometry-Books-Download.html
 
ok thanks ...

I somehow have to learn this fixed point iteration method properly ...

Geometry and Trigonometry for Calculus: A Self-Teaching Guide
Peter H. Selby

learning it has become the aim of my life ...

I am not going to touch calculus for a very long time too ... I made a path for myself to follow ...
which would be to follow these links ...

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

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf
in this way I can learn some important numerical methods without touching calculus ...

its like my new aim in life ...
 
rosekidcute said:
in this way I can learn some important numerical methods without touching calculus ...
In my opinion learning numerical methods without even "touching" calculus is almost impossible, particularly when these methods concern fixed point iteration. Maybe you can learn some numerical linear algebra without calculus, but then I still have my sincere doubts. Actually, my doubts are so serious that I take the liberty to offer you this link: https://www.physicsforums.com/insights/self-study-calculus/

The author of that text offers good advice on learning mathematics, so you might find other texts by him useful, too.
rosekidcute said:
its like my new aim in life ...
I don't know... Learning numerical methods is maybe not the best thing to adopt as your "aim of life". Why don't you aim to be happy and content, alone or together with others, whichever you prefer? Numerical methods will come along the way, if you make a well-planned effort.
 
thanks for the advice ... and the link for calculus ..

actually i was having plans to buy this book too this year for some reference ...

How to Ace Calculus The Streetwise Guide
i found this nice link .. its about numerical methods and it has some examples of fixed point iteration method for different types of equations ...

it was a bit hard to get some sort of flow for this subject .. which is why learning this was a bit depressive ... there are always more pre requisites to fullfill to understand these methods properly ...http://www.universityofcalicut.info/SDE/BSc_maths_numerical_methods.pdf

from page 11 , of that pdf ... they describe the fixed point iteration method ...

Fixed Point Iteration Method

d1e4fb0.png


5cc216e.png




i need to understand all those somehow ... it feels better because it has some simple looking equations where this fixed point iteration method can be applied ... even though i don't completely understand the definition .. or why the phi is used ... or what it actually mean by "take an arbitrary x0...anyone has time for a rough explanation ?
 
i managed to narrow it down to few methods ...

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
simultaneous equations ...

x+2y-3z=10
2x-3y-4z=1
y-3x+z=-8
i have a doubt if i am confusing simultaneous equations with systems of linear equations ??
or all these three things .. the linear equation , the systems of linear equations and simultaneous equations different ?and i think the methods described in these two links below , could be used to solve ... the linear equation , the systems of linear equations and simultaneous equations ...

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.pdfnon 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
fixed point iteration
the bisection method
the Newton raphson method
the secant method
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 re

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

http://www.maths.dit.ie/~dmackey/lectures/Roots.pdf

these fixed point iteration type methods always involve re arranging equations ... to fit certain formulas ... right ??

do we call the first re arranged form of the equation x_n
because i don't understand the terms

x_0
x_1
x_2
x_3
x_{n+1}

how do i choose this starting value x_n
is it a common procedure for most of the iteration type numerical methods ??

fixed point iteration
the bisection method
the Newton raphson method
the secant method
is it something we get when the equations are re arranged to fit certain formulas ?o_O
 
can somebody help me with the graph of this ones answer ... ??


i am not sure how the graph of the answer looks like ...??
 
Last edited by a moderator:
rosekidcute said:
how do i choose this starting value
You can start with a wild guess, or with a more reasoned guess, or with a rough estimate after trying a few values on your calculator or after sketching a graph.

As a general rule, for a given rearrangement the closer your initial value is to the exact solution, the fewer the required number of iterations to achieve the accuracy desired.

I suggest that you set yourself some exercises using equations for which you already know the exact solution, so you can see how iteration converges towards the solution (or not, as the case may be). For example, you can see in advance the solutions to this quadratic: (x-7)(x+3)=0
which can be written x2 - 4x - 21 = 0
 
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the things i were following were these numerical methods ...

fixed point iteration
the bisection method
the Newton raphson method
the secant method

then i found a simple example to work with ...

2ec01a2.png


2ab997c.png


2e49ce0.png


then i was trying to figure out a way to solve it ...

i have a little bit of confusions here ...i think i figured it out that the initial guess ,(the initial root )... the answer of the question ... is like f(1.618) ... which makes the equation close to zero ... ??

this positive root is called a golden ratio ... right ??

is this like an answer to the question already ?

should i be looking for more values which makes the equation closer to zero ?

is that why we use numerical methods such as fixed point iteration ... ??

anyway then you put the values to fit the formulas of iteration methods ...

to get more approximate values which makes the equation closer to zero ?
 
Last edited by a moderator:
  • #10
What are these photos of a cricket match??
 
  • #11
yea..lol ... i was just trying to figure out the steps .. the syllabus wants me to put the numerical methods to a computer program in c ...so i was just confused about the steps involved ...

sorry about putting too much pictures ...
 

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