NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Review of the Longhand Haversine Sight Reductionmethod
From: Hanno Ix
Date: 2015 Jun 28, 19:44 -0700
Truncating an operation +,-,*,/ will by necessity cause errors. "Error" doesn't mean mistake but a small deviation from the accurate result that we are willing to accept for the sake of efficiency and speed. You do that virtually all the time in numerical math e.g. when planning a space mission or an u-processor! Otherwise you'd never get to a practical result.
From: Hanno Ix
Date: 2015 Jun 28, 19:44 -0700
Dave,
I wonder if that Vedic multiplication diagram I showed is understandable as far as the process goes. If not I am glad to elaborate whatever detail causing a difficulty.
The whole scheme is predicated by the observation that 4-digit calculations are sufficient for the kind of CelNav we do. Now, please look again at the Vedic multiplication diagram: It requires the addition of 5 columns ( not 3! ). The accurate multiplication would require 7 or 8 columns but we stop here since we can show the the difference to the exact value is small enough. That's called truncating.
When accepting an error range you have to find out what that range is. I did that for Vedic with a Monte - Carlo simulation of 1 million random 4-digit integers. I found that the truncated product is, in average, too small by 6 units in the 5th digit counted from the left. This error varies from case to case but in 90% of the cases is between 4.5 units and 7.5 units of that 5th digit.
Therefore, we compensate, partially, by assuming a carry of 6 into the 5th column. We know that 6 is in average the carry which an accurate multiplication would produce. We might well have over-compensated or under-compensated by +-1.5 units of in the 5th column in a particular case but may safely do so because that carry of 6 will create an sufficient accurate sum of column 5. After all, the purpose of the 5th column is just having a good reason for rounding up or rounding down the leading 4 digits that we need. Beyond that we don't need it.
This is in virtually all detail what is going on.
H
On Sat, Jun 27, 2015 at 4:06 PM, Dave Walden <NoReply_Walden@fer3.com> wrote:
The usefullness of the vedic method is being able stop calculating when the have the precison you need. It of course gives the exact answer when carried to completion.
I admit the remembering and adding of three two digit numbers generates errors for me.
