| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1808.1 | | AUSSIE::GARSON | Hotel Garson: No Vacancies | Thu Oct 14 1993 23:58 | 3 |
| re .0
Well it converges to sqrt(3) but that doesn't answer the question.
|
| 1808.2 | Newton-Raphson | HERON::BLOMBERG | Trapped inside the universe | Fri Oct 15 1993 06:02 | 9 |
|
Newton-Raphson for calculating sqrt(3):
x[n+1] = 0.5*(x[n] + 3/x[n])
Do two iterations at a time and you get .0
But is still doesn't answer the question.
/�ke
|
| 1808.3 | is in Hackmem... | GAUSS::ROTH | Geometry is the real life! | Sat Oct 16 1993 09:50 | 12 |
| One of the Hackmem items gives an analytic flow for Newton's method,
which would produce a solution to this problem. My copy isn't
handy or I'd go look it up, I don't remember how the flow was
derived exactly.
An analytic flow is a continuous function of the "iteration number"
for an iterative process - simple examples would be the exponential
expression for the Fibonacci numbers, the Gamma function for the
factorials, etc. but flows have been derived for many other
processes as well, A related subject is Schroeder functions.
- Jim
|
| 1808.4 | | RUSURE::EDP | Always mount a scratch monkey. | Tue Oct 25 1994 14:04 | 31 |
| Solution by Stephen C. Lock, Florida Atlantic University, Boca Raton,
Florida.
We will show that x[n] = p[n]/q[n], where p[n] and q[n] are integers
such that p[n]+q[n]sqrt(3) = (2+sqrt(3)^4^n.
Let x[n] = p[n]/q[n], for integers p[n] and q[n]. Then
p[n+1] p[n]^2+18*p[n]^2q[n]^2+9q[n]^4
------ = ------------------------------.
q[n+1] 4p[n]^3*q[n]+12p[n]q[n]^3
It is natural to set p[n+1]=p[n]^4+18p[n]^2*q[n]^2+9q[n]^4, and
q[n+1]=4p[n]^3*q[n]+12p[n]q[n]^3.
Then
p[n+1]+/-q[n+1]sqrt(3) =
p[n]^4+6p[n]^2*(q[n]sqrt(3))^2+(q[n]sqrt(3)^4 +/-
4p[n]^3*(q[n]sqrt(3) +/- 4 p[n](q[n]sqrt(3)^3 =
(p[n]+/-q[n]sqrt(3)^4.
Since we may choose p[0]=2 and q[0]=1, this establishes
p[n]+q[n]sqrt(3)=(2+sqrt(3))^4^n, p[n]-q[n]sqrt(3)=(2-sqrt(3))^4^n.
Solving for p[n] and q[n] yields
p[n] (2+sqrt(3))^4^n + (2-sqrt(3))^4^n
x[n] = ---- = sqrt(3) ---------------------------------.
q[n] (2+sqrt(3))^4^n - (2-sqrt(3))^4^n
|
| 1808.5 | | RTL::GILBERT | | Sat Nov 12 1994 15:22 | 5 |
| There's a minor typo in .-1, indicated below below the v.
v
p[n+1] p[n]^4+18*p[n]^2q[n]^2+9q[n]^4
------ = ------------------------------.
q[n+1] 4p[n]^3*q[n]+12p[n]q[n]^3
|