Well Brother Jim gave another lecture in August 2008 in Hawaii. I found it by googling around. Anyone who wants to comment on it should contact Jim.
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Beyond Newton’s Method: Generalized Higher-Order Approximation Methods
James D. Turner
Finding the root of an equation f = 0 is a common problem in computational science and
engineering. Many line search and derivative-based methods are available. Newton s
derivative-based Method, discovered in 1669, takes a starting guess, uses first-order
derivative information, and iteratively refines a solution until a specified accuracy level is
achieved. Under well-know conditions Newton’s Method converges quadratically. Higher
order versions of Newton’s Method have been known since Halley’s discovery of a cubically
convergent version of the algorithm in 1694. Chebyshev extended Halley’s cubic algorithm
to vector-systems of equations. The classical higher-order algorithm derivation strategy
develops scalar algorithms and then proposes vector extensions. This paper presents a
three-part strategy for systematically developing and generalizing higher-order versions of
Newton’s method, without introducing ad hoc assumptions or approximations.
Computational differentiation tools are introduced for generating high-order partial
derivative models. New homotopy-based algorithms simultaneously generate differential
correction terms, as well as solution gradients. The solution gradients are exploited in an
analytic continuation version of Davidenko’s method, to provide high-order extrapolations
for the correction update term. Several numerical examples are presented to demonstrate
the effectiveness of these new methods. The resulting math models are compared with
Halley’s and Chebyshev’s algorithms. The proposed high-order successive approximation
strategies are expected to be broadly useful for knowledge discovery applications in science
and engineering where the underlying problems are fundamentally nonlinear.
The PDF form of the Abstract is -> Here
Enjoy
According to Paul. SuperBowl weekend is the official time for Winterfest at the Cabin. The plan is to spend Saturday evening there. We will stoke up the stove in the Cabin and stick Kids in there. The adults will find flat places outside to snore. It will be fun. Espessially if all this snow that is coming tomorrow makes it into a Winter Wonderland there. So be there. Rah.
Well we tried again to make a fast car. We failed. We got 6th place out of 12. But Eddie’s new car design still won 2nd place for show. If the 1st place winner does not want to go to the Reginals then we get to go. We will find out later.
I got to run the software for the pinewood derby race for all 124 scouts in our pack and each scout ran 6 races 3 at a time for for a total of 248 races. The software would time how long each car took to run each race. It was a cool mix of Hardware and Software with a lazer being triggered at the starting gate and another lazer at the finish which would be used to identify the 1st, 2nd and 3rd place winners of each race. Everything went smoothly except for the couple of times that it failed to work. The fun part about that is you get 30-40 parents who are giving you the evil eye while they are wondering if the whole event has just been ruined and they are going to have to go home with their disappointed crying child It made things more interesting. I was able to clear up each of the problems and nobody succeeded in killing me.
Mom has a new kitten which jumped onto my lap and fell asleep. I had to go look at something, so I picked it up and placed its limp sleeping body in the chair. When I was done I came back and sat down. After about 10 seconds I felt something move under me. I looked down and saw a little black paw sticking out from between my legs. It wasn’t making any noise or struggling in any way. I got up and the cat was just laying there on its back with its legs all kinda splayed out. When I picked up the kitten, it was all limp in my hands. I couldn’t tell if the cat was just still asleep or that I had killed it. I shook it some to see if I could just wake it up, but I was worried that if something in the cat was broke, that I would make the cat’s problem worse if I shook it too hard. So after some gentle wiggling of the cat, it seemed to move its legs a little. Eventually after much poking, I woke the cat up and it walked slowly away from me. Now I don’t know if I knocked the cat out or just had a hard time waking it up, or both. But the good thing is that I didn’t kill mom’s cat.