Thread: newton Raphson

1. 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??

2. 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.}

3. Originally Posted by helmszee
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?

$\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

so

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

then what??
For your problem:

$\displaystyle f(x)=1.1 x^{1.4}-\cos(x)$

So the N-R itteration is:

$\displaystyle x_{n+1}=x_n-\frac{1.1 x_n^{1.4}-\cos(x_n)}{1.54 x_n^{0.4}+\sin(x_n)}$

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

RonL