Re: newtons method, notation

• To: mathgroup at smc.vnet.net
• Subject: [mg21197] Re: [mg21084] newtons method, notation
• From: BobHanlon at aol.com
• Date: Fri, 17 Dec 1999 01:24:32 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Newton's method would be written as

Clear[x];

x[n_] := x[n] = x[n - 1] - f[x[n - 1]]/f'[x[n - 1]];

This just uses the slope of the curve evaluated at x[n-1] to determine where
the straight line intersects the x-axis and uses the intercept as the
value of x[n].

Let,

f[x_] := 7*x^2 + 3*x - 5;

Plot[f[x], {x, -1.5, 1.5}];

x[1] = 2.;

Table[x[n], {n, 1, 6}]

{2., 1.06452, 0.722348, 0.659848, 0.657614, 0.657611}

Clear[x];

x[n_] := x[n] = x[n - 1] - f[x[n - 1]]/f'[x[n - 1]];

x[1] = -2.;

Table[x[n], {n, 1, 6}]

{-2., -1.32, -1.1109, -1.08652, -1.08618, -1.08618}

The Fibonacci sequence is

fibTable1 = Table[{n, Fibonacci[n]}, {n, 0, 10}]

{{0, 0}, {1, 1}, {2, 1}, {3, 2}, {4, 3}, {5, 5}, {6, 8}, {7, 13}, {8, 21}, {9,
34}, {10, 55}}

The recursive definition is

a[0] = 0;
a[1] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 2];

fibTable2 = Table[{n, a[n]}, {n, 0, 10}]

{{0, 0}, {1, 1}, {2, 1}, {3, 2}, {4, 3}, {5, 5}, {6, 8}, {7, 13}, {8, 21}, {9,
34}, {10, 55}}

fibTable1 == fibTable2

True

The closed form of a[n] is

expr1 = Simplify[Fibonacci[n] // FunctionExpand, Element[n, Integers]]

(-(-(2/(1 + Sqrt[5])))^n + (1/2*(1 + Sqrt[5]))^n)/Sqrt[5]

The recursive equation can also be solved by Mathematica

Needs["DiscreteMath`RSolve`"]

Clear[a];

expr2 = (a[n] /.
Flatten[RSolve[{a[n] == a[n - 1] + a[n - 2], a[0] == 0, a[1] == 1},
a[n], n]])

-(((1/2*(1 - Sqrt[5]))^n - (1/2*(1 + Sqrt[5]))^n)/Sqrt[5])

FullSimplify[expr1 == expr2]

True

Bob Hanlon

In a message dated 12/13/1999 1:23:46 AM, aka007 at mail.com writes:

>can someone explain a bit the notation for newtons method?
>can someone show me the fibonacci sequence on mathematica?
>
>f(n_):=x(n_)=f(n)/f'(n)
>
>sorry, i don't exactly recall newtons method, nor the mathematica
>to do it.
>
>and then the second part is creating a table of data, based on inital
>guess. graph would be good, too.
>
>but what in the world is the first part doing???
>

```

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