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I The algorithms involved in finding a solution with numerical methods

  1. Jul 28, 2016 #1
    what are all the algorithms involved in finding a solution with numerical methods ?

     
    Last edited by a moderator: Aug 11, 2016
  2. jcsd
  3. Jul 28, 2016 #2

    BvU

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    Well, you've listed quite a bunch of them already. Do you expect someone in PF to complete your list? Or is it in fact domething else you want to know ?
     
  4. Jul 28, 2016 #3
    BvU ,

    thanks for the reply ...

    i am so confused right now...

    what is this initial guess ??
    the initial guess of what ??
     
  5. Jul 28, 2016 #4

    BvU

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    A numerical integration has to start with some value (the initial value, e.g. y(0) = 0) and can then proceed. Good for initial value problems.

    When you solve a boundary problem (e.g. y(10) = 100 as a given) using numerical integration, you still have to start somewhere, so you 'guess' the initial value and see where you end up with the integration. (Look up 'shooting method')
     
  6. Jul 28, 2016 #5
    these numerical type questions are a bit hard usually ...
    i really thought first we had to try to solve the equation in an ,analytic , symbolic or formulaic way ...before trying the numerical way ...

    so it means, when i encounter a question like this .. i have to take an initial value ...






    then i have to use the algorithm , depending on the type of the question ....





    this is surely going to take some time to get used to ... because i am not that familiar with many of the terms ...
     
    Last edited by a moderator: Jul 29, 2016
  7. Jul 28, 2016 #6

    BvU

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    I see two two-foot long posts and I don't see a difference. Nor a question I can answer ...
     
  8. Jul 28, 2016 #7
    i am sorry for the long posts ... as i learn more , the size of the post seems to be getting smaller ...

    i really need to understand how these works ...



    lagrange's interpolation formula
    newton raphson method
    bisection method
    eulers method
    newtons forward
    runge kutta method
    trapezoidal rule


    and i dont have one text that explains it all ...
     
  9. Jul 28, 2016 #8

    BvU

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    Here ?
    And I see a lot of your goodies here as well ! Should keep you off the streets for a while .... :smile:
     
  10. Jul 28, 2016 #9
    BvU ,

    thanks a lot for that book .. there is a lot to learn from that book ...
    i really need to follow at least this much amount of books to really understand all these ...

    but before i delve a bit deeper into the books alone ...

    i have two more doubts ...

    one is ...

    how to find a proper initial guess ...?

    second is ...

    i understand the steps involved here ...

    ccc_numerical_methods_png_2.png

    is this the same thing we did here ??

    ccc_numerical_methods_png_3.png
    ccc_numerical_methods_png_1.png

    ccc_numerical_methods_1.png
    ccc_numerical_methods_2.png



    first we are given a gradient of the curve or an instantaneous rate of change
    then we are trying to find a function that is giving this gradient of the curve or this instantaneous rate of change

    then we change the dy/dx , the gradient of the curve or an instantaneous rate of change to ... delta y /delta x
    which is an even smaller gradient of the curve or a smaller instantaneous rate of change ...

    we start with x = 0 , we get y =0


    when we increment x by 0.1 , we get the value of y around -0.299

    which is another gradient of the curve for delta y / delta x , the smaller instantaneous rate of change


    why are we adding up these gradients of the curve or the smaller instantaneous rate of change for these small small change in delta y / delta x?

    this means that these smaller gradients of the curve or these smaller instantaneous rate of change has certain property at each value of x ..., the gradient of the curve with respect to x which is indeed "the rate" at which y is changing as x is changing ....

    so the function has the property of this "rate" ???

    usually , when we try to solve ...

    we get a function that has this instantaneous rate of change ...

    but when we try to solve it with numerical methods , we only get a " rate " which is the property of that function ???
     
  11. Jul 29, 2016 #10

    BvU

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    Again, several feet of post.
    Good thing you already prepared an answer. General answer: use all your knowledge, intuition and luck. As if you were a TV detective. Stay critical.

    Numerically, we use the derivative (which changes from point to point) to approximate the function as linear. Which is good up to order (step size)2. After a step, we re-evaluate the derivative. We make errors of a certain order, but still get an estimate of the function values.

    Start reading the book :smile:
     
  12. Jul 29, 2016 #11

    Mark44

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    I deleted the duplicated images
     
  13. Jul 29, 2016 #12

    Mark44

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    This is advice that has been given in previous threads to you (rosiekidcute). Have you obtained a book, yet?
     
    Last edited: Jul 29, 2016
  14. Jul 29, 2016 #13
    BvU ,

    thanks ...

    Mark44 ,

    yes , i have this book ... numerical analysis and i am reading it ...
     
  15. Jul 29, 2016 #14

    Mark44

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    It would make more sense to start with a textbook that deals with ordinary differential equations.

    You have recently started threads with these titles:
    • Few basic questions about differential equations...
    • A few differential equation questions
    • differential equations and numerical methods questions
    • first order differential equations ??
    • few more numerical methods question...?
    Any good differential equations textbook would provide answers for all of the questions you've asked here.
     
    Last edited: Jul 29, 2016
  16. Jul 30, 2016 #15
    Mark44,

    thanks ...

    can you recommend some books on ordinary differential equations ??

    i have few more doubts from the previous posts ...
    i don't know if i am understanding this properly ...
    this thread , post # 9 ...
    picture two ..

    that question is about finding the "function" that has that instantaneous rate of change ... right ??

    we found the function ... that has that rate of change ...

    same thread
    picture one ,

    we are given a function and we are trying to solve it numerically ...

    if you take dy/dx of that instantaneous rate of change
    you get certain properties of that function at certain points or places ...
    then if you increment x ... you get properties of that function again at certain points or places

    if you add all the function values ... you get the overall properties of a function ...
    and that is the function with all those properties that has given that instantaneous rate of change ...


    ??
     
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