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Math Help - newton Raphson

  1. #1
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    newton Raphson

    cosx = 1.1x^1.4 (x=>0, radians)

    write out the formula for newton raphson metho applied to this equation tabulat successive values of netwon raphson approximation for this equation, with starting solution x = 1 untill u have the root to at least 5 deciamal places..

    ..ahh help!?

    formula

    x - f(x)/f'(x) right?

    so

    1 - 1.1x^1.4 - cosx / 1.54x^.4 + sinx .......... ? right?

    then what??
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  2. #2
    MHF Contributor
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    formula

    x - f(x)/f'(x) right?

    so

    1 - 1.1x^1.4 - cosx / 1.54x^.4 + sinx .......... ? right?

    then what??


    The "formula" is
    x2 = x1 -[f(x1) / f'(x1)]
    x3 = x2 -[f(x2) / f'(x2)]
    and so on....

    where
    x1, x2, x3 are read x sub 1, x sub 2, x sub 3, respectively.

    x2 is nearer to the correct root than x1.
    x3 is nearer to the correct root than x2.
    and so on...

    So your "formula" for your Problem should be
    x2 = x1 -[(1.1(x1)^(1.4) -cos(x1)) / (1.54(x1)^(0.4) +sin(x1))]

    {Too many parentheses and brackets? Yes. You need to practice that. }

    Then just substitute the seed root, or starting solution, x1 = 1 in the formula to get x2.
    x2 = 1 -[(1.1(1)^(1.4) -cos(1)) / (1.54(1)^(0.4) +sin(1))]

    {In the cos(1) and sin(1), the 1 is 1 radian. So use your calculator's angle mode in radians.}
    x2 = 1 -[(1.1(1) -0.540302306) / (1.54(1) +0.841470985)]
    x2 = 1 -0.235021841
    x2 = 0.764978159

    Then, for the second iteration,
    x3 = 0.764978159 -[(1.1(0.764978159)^(1.4) -cos(0.764978159)) / (1.54(0.764978159)^(0.4) +sin(0.764978159))]

    Whoa, if you can navigate that, you are good.

    Go up to x4 or x5 or x6, if you like.
    {Normally, if x1 is close enough to the correct solution, you need up to x4 only.}
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by helmszee View Post
    cosx = 1.1x^1.4 (x=>0, radians)

    write out the formula for newton raphson metho applied to this equation tabulat successive values of netwon raphson approximation for this equation, with starting solution x = 1 untill u have the root to at least 5 deciamal places..

    ..ahh help!?

    formula

    x - f(x)/f'(x) right?

    <br />
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}<br />

    so

    1 - 1.1x^1.4 - cosx / 1.54x^.4 + sinx .......... ? right?

    then what??
    For your problem:

    <br />
f(x)=1.1 x^{1.4}-\cos(x)<br />

    So the N-R itteration is:

    <br />
x_{n+1}=x_n-\frac{1.1 x_n^{1.4}-\cos(x_n)}{1.54 x_n^{0.4}+\sin(x_n)}<br />

    Now you need an initial guess at a root try x_0=0.5 or x_0=0.7, and away you go.

    RonL
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