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Thread: what are all the algorithms involved in finding a solution with numerical methods ?

  1. #1
    Junior Member brandnew's Avatar
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    what are all the algorithms involved in finding a solution with numerical methods ?

    what are all the algorithms involved in finding a solution with numerical methods ?


    an example

    say we were to solve the initial value problem:

    Y′ = 2x

    y(0) = 0

    it's so simple, you could find a formulaic solution in your head, namely y = x2. On the other hand, say we were to use a numerical technique. (yes, i know we don't know how to do this yet, but go with me on this for a second!) the resulting numerical solution would simply be a table of values. To get a better feel for the nature of these two types of solution, let's compare them side by side, along with the graphs we would get based on what we know about each one:

    Notice that the graph derived from the formulaic solution is smoothly continuous, consisting of an infinite number of points on the interval shown. On the other hand, the graph based on the numerical solution consists of just a bare eight points, since the numerical method used apparently only found the value of the solution for x-increments of size 0.2.


    Using numerical solutions

    so what good is the numerical solution if it leaves out so much of the real answer? Well, we can respond to that question in several ways:

    The numerical solution still looks like it is capturing the general trend of the "real" solution, as we can see when we look at the side-by-side graphs. This means that if we are seeking a qualitative view of the solution, we can still get it from the numerical solution, to some extent.

    The numerical solution could even be "improved" by playing "join-the-dots" with the set of points it produces. In fact this is exactly what some solver packages, such as mathematica, do do with these solutions. (mathematica produces a join-the-dots function that it calls interpolatingfunction.)

    when actually using the solutions to differential equations, we often aren't so much concerned about the nature of the solution at all possible points. Think about it! Even when we are able to get formulaic solutions, a typical use we make of the formula is to substitute values of the independent variable into the formula in order to find the values of the solution at specific points. Did you hear that? Let me say it again: To find the values of the solution at specific points. This is exactly what we can still do with a numerical solution

    numerical methods and errors
    interpolation
    numerical differentiation
    numerical integration
    solution of algebraic and transcendental equations
    numerical solution of a system of linear equations
    numerical solution of ordinary differential equations
    curve fitting
    numerical solution of problems associated with partial differential equations
    fixed point iteration method
    bisection and regula false methods
    newton raphson method etc.
    Finite differences operators
    numerical interpolation
    newton’s and lagrangian formulae
    part i
    newton’s and lagrangian formulae
    part ii
    interpolation by iteration
    numerical differentiaton
    numerical integration
    solution of system of linear
    equations
    solution by iterations
    eigen values
    taylor series method
    picard’s iteration method
    euler methods
    runge – kutta methods
    predictor and corrector methods

    my questions sort of looks a bit like this ...







    ??

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  2. #2
    Junior Member brandnew's Avatar
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    Re: what are all the algorithms involved in finding a solution with numerical methods

    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 ...
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