------ Forwarded Message
From: Manfred Von Thun <m.vonthun@...>
Date: Mon, 12 Dec 2005 17:37:15 +1100
To: Manfred Von Thun <m.vonthun@...>
Subject: Re: [stack] More annoying quadratic...

On 12/12/05 5:14 PM, "Manfred Von Thun" <m.vonthun@...> wrote:

>
>
>
> On 7/12/05 2:06 PM, "William Tanksley, Jr" <wtanksleyjr@...> wrote:
>
>> [..]
>>
>> The challenge was to use a single-variable polynomial (18th order) to
>> approximate a function simulating something in the real world. The
>> equation was something like
>>
>> y = 1.5 * (1.2 - 0.03x^2 - 0.02x^4 - 0.0076x^6 ... );
>>
>> [..]

I'm sorry about the extra >> which this eudora abomination put in
specifically to annoy me. The text below is what I'm trying to send now

>> I can¹t access our much loved Alpha under VMS any more, and I have
>> not yet been fully transplanted onto the linux machine where I shall
>> live in future. So I can¹t really test my ideas, but here we go:
>>
>> The idea of producing a list of coefficient/powers list and then
>> taking the sum is obviously correct, at least as a first draft.
>> (An optimisation might improve on this by using an accumulator
>> to avoid creating the list in the first place).
>>
>> So my first idea was to use the same technique as for the quadratic formula:
>> assume that the value of x is second on the stack, and on top is a list of
>> quotations, where each quotation computes a term of the polynomial:
>>
>> x [ ... [coefficient power foo] ...] [i] map sum popd
>>
>> The [i] map should produce the list as for the quadratic, take the sum, and
>> finally pop the 3 which is still sitting there, inspected but unchanged by
>> each call of foo. We still need to define foo, which takes a power of
>> x and multiplies by the coefficient.
>>
>> But then we have to write foo as many times as there are terms in the
>> polynomial. Much nicer if we could define a different foo and write
>> the details as a list of coefficient/power pairs, in the form [c p]
>>
>> DEFINE poly == [foo] map sum popd
>>
>> Then a call will look like this:
>>
>> x [ ... [c p] ... ] poly
>>
>> for example
>>
>> x [[1.1 7] [2.2 18]] poly
>>
>> should compute y = 1.1 * x^7 + 2.2 * x^18
>>
>> We stiil have to define this foo function. Every time it is called by map
>> it will find a pair [c p] and below that the x which it can consume
>> for itself but which will be restored by map for the next pair. So foo must
>> put the two numbers in the pair on top of the stack, like this
>> x c p
>> and that can be done by either uncons uncons pop, or more simply
>> by just the i combinator. Now swap the x and the c by the swapd
>> operator:
>> c x p
>> and take the p-th power of x with the power operator pow. Finally, to
>> multiply with the coefficient, just use the * operator. That makes
>> foo == i swapd pow *
>> However, this is such an idiosyncratic operator that one would not want to
>> define it at all, but just inline it into the definition of poly:
>>
>> DEFINE poly == [i swapd pow *] map sum popd
>>
>> The intermediate list could probably be avoided by using the step
>> combinator instead of the map combinator, and use an accumulator
>> during the stepping. But I suspect the extra dips that would be needed
>> will cost as much as the construction of the list.
>>
>> So, that is my design, untested unfortunately.
>> I quite like the notation of a list of [c p] pairs, which is as close as
>> one can get to the ordinary mathematical notation.
>>
>> - Manfred
>>
>>
>>
>>
>>
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>>
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>>
>>
>>
>>
>

------ End of Forwarded Message

FW: [stack] More annoying quadratic...

Manfred Von Thun — 2005-12-12 06:44:57