mm-79
===
Third, sqrt((double)(n*n)) is exact>>because sqrt is
accurate to 1 ulp on your machine.> This is really the
core of my original question.> How do they do that?>
Someone else has already suggested that> there's something
better than Newton's algorithm.> Perhaps it's related 
to
the old long-division algorithm> that I vaguely remember
from elementary school.> Do you know the details?> I used
to.> I used Google> http://www.google.com/> to search
for> +?g-point +square root +SRT algorithm> I
found lots of stuff> including lots of stuff written by my
mentors --> Milos Ercegovac and Thomas Lang.See: Arbitrary
Precision Square Root Algorithm; Was:Unreasonableaccuracy of
sqrt()TomCee
===
 I am currently studying numerical analysis
and my professor was talkingabout something called minimax.
However, I can not ?d info on it in thenotes that he posts.
I was wondering if somebody has any information on it?I do
know that Mn, the nth minimax polynomial is the best
approximation of apolynomial, but thats about it. Any help
would be appreciated.
===
This is likely a very simple problem
for anyone posting to this group,but I was wondering if
someone might help me out. I should say helpme to help my
son. Here it is:A vacant lot is 100 feet long. The area of
the lot in square feet istwice the perimeter in feet. What is
the width of the lot?Writing it out as shown below with x
being the width of the lot, theonly number I can come up with
that is close is 4.16, but that doesn'tquite work.Can anyone
help? How would you work this to solve for the width?(100x /
2) = (2x+200)dtechtive@verizon.net
===
A simple set-up for
this would begin like:eqn1: A = 100 * X = 2 * P; ==> P =
50Xeqn2: P = 2 * (100 + X) ; ==> X = 25/6 ~ 4.166....So I'd
conclude that your answer is ?almost' correct. Correctly
roundingyou would have reported 4.17 not 4.16, either way
though when you roundyour numbers will not come out quite
right.> This is likely a very simple problem for anyone
posting to this group,> but I was wondering if someone might
help me out. I should say help> me to help my son. Here it
is:>> A vacant lot is 100 feet long. The area of the lot in
square feet is> twice the perimeter in feet. What is the
width of the lot?>> Writing it out as shown below with x
being the width of the lot, the> only number I can come up
with that is close is 4.16, but that doesn't> quite work.>>
Can anyone help? How would you work this to solve for the
width?>> (100x / 2) = (2x+200)>
dtechtive@verizon.net
===
Please clarify doesn't quite work;
your equation appears to match your problem statement.Jerry>
This is likely a very simple problem for anyone posting to
this group,> but I was wondering if someone might help me
out. I should say help> me to help my son. Here it is:> A
vacant lot is 100 feet long. The area of the lot in square
feet is> twice the perimeter in feet. What is the width of
the lot?> Writing it out as shown below with x being the
width of the lot, the> only number I can come up with that is
close is 4.16, but that doesn't> quite work.> Can anyone
help? How would you work this to solve for the width?>
(100x / 2) = (2x+200)> dtechtive@verizon.net
===
>
This is likely a very simple problem for anyone posting to
this group,> but I was wondering if someone might help me
out. I should say help> me to help my son. Here it is:>
> A vacant lot is 100 feet long. The area of the lot in square
feet is> twice the perimeter in feet. What is the width of
the lot?> Writing it out as shown below with x being
the width of the lot, the> only number I can come up with
that is close is 4.16, but that doesn't> quite work.>
Can anyone help? How would you work this to solve for the
width?> (100x / 2) = (2x+200)>
dtechtive@verizon.net> Please clarify doesn't quite work;
your equation appears to> match your problem statement.>
JerryNot quite a match. The equation should be 100x = 2(2x +
100).The answer is 100/99. Very skinny lot!
===
>> This
is likely a very simple problem for anyone posting to this>
> group, but I was wondering if someone might help me out. I
should> say help me to help my son. Here it is:>>
A vacant lot is 100 feet long. The area of the lot in square
feet is> twice the perimeter in feet. What is the width
of the lot?>> Writing it out as shown below with x
being the width of the lot, the> only number I can come
up with that is close is 4.16, but that> doesn't quite
work.>> Can anyone help? How would you work this to
solve for the width?>> (100x / 2) = (2x+200)>>
>> dtechtive@verizon.net>> Please clarify doesn't
quite work; your equation appears to> match your problem
statement.>> Jerry>> Not quite a match. The equation
should be 100x = 2(2x + 100).> The answer is 100/99. Very
skinny lot!I suggest that you attempt to _check_ your answer.
(Your equation isincorrect. Dan's equation was
correct.)David
===
> Please clarify doesn't quite work;I
suppose Dan means that 4.16 feet isn't the _precise_ answer.
If so,he's right.> your equation appears to match your>
problem statement.>> Jerry> This is likely a very simple
problem for anyone posting to this group,> but I was
wondering if someone might help me out. I should say help>
me to help my son. Here it is:>> A vacant lot is 100 feet
long. The area of the lot in square feet is> twice the
perimeter in feet. What is the width of the lot?>>
Writing it out as shown below with x being the width of the
lot, the> only number I can come up with that is close is
4.16, but that doesn't> quite work.>> Can anyone help?
How would you work this to solve for the width?>> (100x /
2) = (2x+200)50x = 2x + 20048x = 200x = 200/48 = 25/6so the
width is precisely 25/6 feet, which could also be written
as(4 + 1/6) feet or 4' 2 etc.For future reference, an
appropriate newsgroup for such a question isalt.algebra.help
.David
===
>|> This question seems to resurface every few
years; >|>|> Here is that ?elementary school' algorithm
which many of us were>|> taught.>|>|> It is of arbitrary
precision; i.e. limited only by your patience>|> and/or your
machine.>|>|> This method was formerly taught in primary
school, but the algorithm>|> seems to be ?getting lost'
today!!!>Yes, but the real question is why it ever WAS
taught!>It always was a damn-fool method for hand
calculation, though of>some theoretical interest. There are
at least 3 better methods>suitable for hand use or mental
arithmetic. I have no idea why>it was preferred, except that
it is ?elementary' in the sense that>it doesn't use
calculus.>But, considering that both binary chop and linear
interpolation>are ancient, and suitable for using in your
head, that is a feeble>excuse. The method you describe should
be taught in undergraduate>courses and not before!Hmm. Perhaps
I can supply some ancient perspective. I wastaught the hand
method in, IIRC, the 8th or 9th grade. It could be done
reasonably rapidly and I don't recall ever havingany trouble
with it.Why were we not taught any other methods? Because
back thenin the dim recesses of time there were no other
number systems.No binary, no mixed radix, nothing but the
occasional Romannumeral. Nobody knew Newton's method and
there was no way it could havebeen taught. Of course there
were no computers, electroniccalculators, and the mechanical
ones capable of taking square roots were very very new and
very very expensive.I did not see one of them until I was a
graduate student.Slide rules were good only to three digits
at best and thecommon log tables only good to four at
best.The fun thing about the method was demonstrating
algebraicallyhow it worked. If we could do that, we were
considered topstudents.Why did the method persist? Teachers
are very very conservativeas a group. Things are taught
because they were always taught.It takes a while before
things change. For instance, why iscalculus taught before
discrete math? One could argue thatgenerating functions are
far easier to understand than derivatives,yet one is commonly
taught in high school and the other isalmost unknown outside
of the math community.And oh yes, why do we still use logs to
the base 10 and whydon't we teach logs to the base 2 along
with natural logs?Same reason. ----- Paul J. Gans
===
|> |>
Hmm. Perhaps I can supply some ancient perspective. I was|>
taught the hand method in, IIRC, the 8th or 9th grade. It
|> could be done reasonably rapidly and I don't recall ever
having|> any trouble with it.I was taught it a bit earlier,
and had trouble, probably because Icouldn't work out why it
worked, and am bad at learning by rote.So I invented binary
chop. A common reaction.|> Why were we not taught any other
methods? Because back then|> in the dim recesses of time
there were no other number systems.|> No binary, no mixed
radix, nothing but the occasional Roman|> numeral. Eh? None
of those are relevant. Binary chop has nothing to dowith
binary notation, and neither interpolation nor
Newton-Raphsonare discrete methods in that sense.In any case,
with UK-based teaching, we had been using mixed basearithmetic
for years. What is the cost of 3 tons, 7 hundredweight,25
pounds, 6 ounces of something at 2 pounds, 11 shillings and3
pence 3 farthings a stone. All right, it was usually a
BITsimpler :-)|> Nobody knew Newton's method and there was no
way it could have|> been taught. Of course there were no
computers, electronic|> calculators, and the mechanical ones
capable of taking |> square roots were very very new and very
very expensive.|> I did not see one of them until I was a
graduate student.|> Slide rules were good only to three
digits at best and the|> common log tables only good to four
at best.Yer whaa? I thought that you said an ANCIENT
perspective! Whaton earth do you need any of those for? I was
using Newton'smethod for ages before they were allowed in
mathematics teachingor examinations.|> The fun thing about
the method was demonstrating algebraically|> how it worked.
If we could do that, we were considered top|> students.For
that, you need algebra. When people are talking about
itsteaching in elementary school, most of us are referring to
LONGbefore we were taught algebra. If I recall, I was taught
itbefore I was taught about even the use of variables, though
afterI had been started on non-trivial geometry.Nick
Maclaren.
===
In?ite Dimensional Analysis, Quantum
Probability and Related TopicsView table-of-contents and
abstracts athttp://www.worldscinet.com/idaqp.htmlIntegral On
A ManifoldRemi LeandreOptimal Consumption And Portfolio In A
BlackScholes Market Driven ByFractional Brownian
MotionYaozhong Hu, Bernt Oksendal And Agnes SulemCentral
Limit Theorems For Coxeter Systems And Artin Systems Of
ExtraLarge TypeGero FendlerSupport Of A Joint Resolution Of
Identity And The Projection SpectralTheoremArtem D.
PulemyotovMartingale Characterizations Of Increment Processes
In A LocallyCompact GroupHerbert Heyer And Gyula PapShort
NotesA Degenerate Stochastic Partial Differential Equation
For The PurelyAtomic Superprocess With Dependent Spatial
MotionDonald A. Dawson, Zenghu Li and Hao WangEffective
Determination Of Poisson NoiseSi SiThe Combinatorics Of The
R-Free ConvolutionSurveyCentral Gaussian Convolution
Semigroups On Compact Groups: A SurveyA. Bendikov and L.
Saloff-CosteFor more information, go to
http://www.worldscinet.com/idaqp.htmlX-Received: (from
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===
Here is the algorithm you are seeking using
a symbolic packagecalled Macsyma
v.2.2:[xx:taylor(sum(sum(z^b?^sqrt(q))/(z^b?^sqrt(q
))-1),p,2,5),q,2,5),z,inf,5),b?akelist(powers(denom(
part(xx,j)),z),j,1,length(xx)))];Sample of results:[16 +
(1/(z^(29736144583/11157424511)))
+(1/(z^(593022861439/178513961987))) + (1/(z^4)) +
(1/(z^(36140041546/7671231941))) +
(1/(z^(40691659899/8605063048)))+ . . ., [[0.0b0],
[2.6651441426902251887b0], [3.3219970854839128052b0],
[4.0b0], [4.7111131333214371389b0],
[4.7288043878374149479b0]]]Of course I can't compute to
million of ?g point place withthis pacakge but if you
have a supercomputer this is how thisis done. You can get
millions, or trillions of suchtranscendental numbers.Hope
this is what you are looking for. If not pleas referdr.Huen
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===
Dear All,I am
trying to solve a system of nonlinear equations which is
verycomplicated. The resulting function is a stochastic
variable, which Ihave to approximate using the Monte Carlo
methods. Thereforeevaluation of the function is very time
consuming (one evaluation isabout one minute). My problem is
to ?d an appropriate algorithmwhich solves the nonlinear
system of equations without approximationof the
Jacobian/Hessian matrix. Due to the complexity of
thestochastic function it does not make sense to use methods
like Newton,Newton-Raphson, etc. I was thinking about
something like bisections or regula falsi,eventually secant
method. They work ?e in one dimensional space. ButI do not
know about higher dimensions.Pshem
===
> Dear All,>> I am
trying to solve a system of nonlinear equations which is
very> complicated. The resulting function is a stochastic
variable, which I> have to approximate using the Monte Carlo
methods. Therefore> evaluation of the function is very time
consuming (one evaluation is> about one minute). My problem
is to ?d an appropriate algorithm> which solves the
nonlinear system of equations without approximation> of the
Jacobian/Hessian matrix. Due to the complexity of the>
stochastic function it does not make sense to use methods
like Newton,> Newton-Raphson, etc.>> I was thinking about
something like bisections or regula falsi,> eventually secant
method. They work ?e in one dimensional space. But> I do not
know about higher dimensions.> PshemTry doing a Google
search on Broyden's method.--There are two things you must
never attempt to prove: the unprovable --and the
obvious.--Democracy: The triumph of popularity over
principle.--http://www.crbond.com
===
> The resulting
function is a stochastic variable, which I> have to
approximate using the Monte Carlo methods. Therefore>
evaluation of the function is very time consuming (one
evaluation is> about one minute). My problem is to ?d an
appropriate algorithm> which solves the nonlinear system of
equations without approximation> of the Jacobian/Hessian
matrix. Due to the complexity of the> stochastic function
it does not make sense to use methods like Newton,>
Newton-Raphson, etc.> Try doing a Google search on
Broyden's method.This is unlikely to work well for noisy
equations.There is no really cheap way for problems which are
both expensive and noisy.Try the software
fromhttp://www4.ncsu.edu/eos/users/c/ctkelley/www/tim.htmlBut
probably you'll need to write your own routine which ?s
successive linearmodels to the equations and then solves
theresulting linear systems.Arnold Neumaier