I've had mixed experiences with fixing my laptop. Recently, the hard drive went dead; however, since 2 1/2 inch drives are commodity parts, it was trivial to order a new one, open up the laptop and bung it in.
However, my laptop is also suffering from a broken wireless unit. It uses a GemTek WL-388F mini-USB 802.11b wireless module, which sits in a small compartment accessible via a panel on the underside. I fried the module while flashing its firmware, and I've had no success in finding a replacement anywhere; I guess this is because the module is much less of a commodity part than the hard drive was. If anyone could tell me where to find a replacement module, I would be very grateful!
Of course, the irony is that if someone says to me "pulsar discovery" (and, for the record, I'm a stellar astrophysicist), I immediately think "Jocelyn Bell". I haven't got the first idea who here supervisor was. She may not have won the prize; but she got the credit, well-deservedly.
Wow, you can get an interpolated result for sin(x) in three arithmetic operations? Best chicken--scratch algorithm I can work in pencil and paper in this state of mind is 3 multiplies, 3 add/subtracts and one floor
Oops, I completely mis-wrote. I meant one addition, two subtractions and three multiplies. My standard code is this:
(please allow for my Fortran syntax!). Since for uniform spacing idx is calculated from a floor(), the operation count is identical to yours, and your algorithm is optimal too.
Good point about the x shifting. Another aspect of the algorithm is this: if the interpolation is done as a sequence of monotonic x values, then the w_a calculation need only be performed if idx has changed. This gives an extra speedup (assuming that conditionals do not shaft the pipelineing).
Since division is roughly 10 times the cost of multiply on a PPro/P2/P3 (39 vs 4 cycles) you might be able to come up with an intelligent algorithm for sin/cos using taylor series that is faster and more accurate than interpolation for all but an impractically large LUT
Ah, but interpolations in a uniformly-spaced LUT has no need of division; just 1 subtraction, 1 addition and 1 multiplication. And in a 1024-term table, the error caused by neglecting the x^3 and higher terms of a sin(x) expansion is of the order of 1e-8; i.e., down there with machine precision.
However, I wonder whether something like a rational-function approximation could give a good approximation to sin(x)? Or a Pade approximation? Any thoughts? To be honest, a LUT is aesthetically unappealing to me; I'd love to have a really-fast algorithm in its stead.
You make some good points; I'd overlooked the fact that the factorials can be precalculated. And yes, even with a LUT you are going to have to translate back into an interval of width 2*pi; either [-pi, pi], or [0,2*pi], or whatever. However:
You shouldn't use just 5 terms anyway, writing out the equation for more terms seemed unnecessary.
For sure; however, the further cost of any extra terms, above and beyond the 5 you've already given, makes a single Taylor's series expansion a poor approach compared to a LUT. One way of looking at a N-entry LUT is as N separate first-order Taylor's series expansions, each centered at a different point; I guess this is how LUTs win out, since with a large-enough N the accuracy of series, even at first order, wins out for intervals of width 2*pi/N. And hell, memory is so cheap nowadays that a N=1024-entry LUT is not a problem.
Thanks for your thoughtful reply to my postl; you've just earned yourself a friend.
In honour of the nationality of the scientist involved in the study, and to counter any anti-French sentiment which may have arisen due to unfortunate recent events on the geopolitical scene, maybe we need a new unit for angular measurements: 20 degrees = 1 degree of freedom.
...but the best I ever got was on lake oneida up in NY right before it froze over
Surely, once the lake had frozen the number of skips would have been large? Aren't ice/stone impacts pretty elastic (assuming no chips or instantaneous melting)?
...holds the current Guinness Book of World Records title for a 1992 toss that yielded an impressive 38 bounces across the Blanco River in central Texas.
Wow, I never realized Barnes-Wallace invented bouncing jizz as well. Cool!
Sirius A, for example, is certainly large enough to supernova, and it's 8.6 light-years away.
Not according to any current theory of stellar evolution. Sirius A is spectral type A, which means a mass less than 3 solar masses. Anything under 8 solar masses won't go supernova, so Sirius A is destined for the boring end: expansion into a red giant, loss of envelope as a planetary nebula, and senescence as a white dwarf.
...not to mention the blast of particle radiation, moving at less than C, that would follow some time later.
Let's assume that Sirius A has a mass of 2 solar masses. Distributing this mass uniformkly over a sphere 8.6 light years in radius results in a mean density of
1.5*10^-18 g cm-3, which is peanuts in astrophysical terms. So, if Sirius A did go supernova (which it won't), the Gamma-ray flux might be nasty, but the particle flux would be irrelevant.
Will give you the first 5 terms of a taylor series for sin(x) at a cost of 9 multiplies, 4 add/subtracts and 1 divide. Additional terms cost 2 multiplies and 1 add/subtract each.
I actually count 19 multiplies; it appears you missed the multiplies which are implicit in raising x to a power.
You can do a taylor expansion with one divide and reasonably low cost for the accuracy.
Except that the expression you gave isn't accurate at all; you have confused high order with high accuracy. Your expression gives sin(2*pi) = 11.9, which is way off the true value of sin(2*pi)=0. At x=pi, your expression is rather more accurate (0.00693, as opposed to the true value of 0), but still not good enough for any numerical calculations.
Guilty as charged, m'lud! I am indeed a pedant. That's why I take pathetic glee in pointing out that, on your resume, you claim to be an expert at HTML; yet the very same resume fails to pass the W3C HTML validator without errors. You're not exactly trying hard to sell your talent, are you?
From Womens Studies class back in college, I'm pretty sure that Oriental is not a desired term by Chinese Americans. We read several stories by Asian American women, who all objected to use of the word. I however am not one such person, so I can't state if it truley has negative conotations. I assume it would be like calling an African American a "colored person"
In the UK, "Asian" refers to ethnic groups from India, Pakistan, and the surrounding countries. People whose ethnic ancestry is from the Pacific rim countries, in particular those whose eyelids are characterized by an epicanthic fold, are termed "Oriential". The English-Chinese population doesn't appear to take offence at this terminology.
Likewise, "mixed race" (argh, hate that term) people in South Africa prefer to refer to themselves as "coloured" rather than "black". So let's remember that much of the terminology used to describe ethnicity or indeed skin pigmentation is very dependent on its cultural context.
It's hard to tell if s/he being purposefully unclear, made a mistake and is now trying to justify themselves, or is just incapable of clearly expressing an idea.
None of the above; in fact, the problem was that many people who responded to my posts were incapable of distinguishing between the absolute spread (varies as SQRT(N)/2, grows with N) and the relative spread (varies as 1/SQRT(N), decreases with N). Nothing to see, move along now...
The physics of the two situations doesn't match up at all though. No two particles can occupy the same space at the same time, but every coin toss is independent of the previous one.
The Pauli exclusion principle has nothing to do with random walks. In an idealized random walk, the diffusing particles do not interact with one another. All that happens is that, every time a particle travels a mean free path, there is a 50% probability (exactly) that its direction of motion will be reversed. In the case of Brownian motion (e.g., diffusion of smoke particles through air), what causes this motion reversal is collision with an air molecule. But the particles themselves behave independently, both of one another and of their previous history.
There's quite the argument going on over this, and I think people might be arguing different things and not realizing it (myself included). Could you clarify how this relates to global warming or is it just a disagreement with the original coin toss post?
I'm afraid it's nothing so controversial as global warming, the argument does indeed relate to the comment I made about coin tossing. Which I think I'm still winning, since the naysayers appear to be going quiet one by one. Although maybe they just don't give a toss....
Looking for an integer value in a real number continuum is an irritating thing to do.
It may be an irritating thing to do, but it is important to to emphasize when demonstrating that the "Law of Averages" (if I toss 20 heads in a row, there is more than a 50% chance that the next toss will be a tail, in order to force convergence toward the expected mean) is utter bunk.
It also helps one understand situations where the expected mean is zero, and all one is left with is the divergent spreading of the probability density function. I am, of couse, describing the diffusion of gas particles away from a point of high concentration, via Brownian random walks. In the latter case, the mean square displacement of an ensemble of particles is given by <r^2> = A t, where A is a constant depending on the temperature and other variables, and t the time (see here for the exact expression, it's right down at the bottom of the page). The standard deviation of the particle distribution obviously varies as sigma = SQRT(<r^2>) = SQRT(A t), and interpreting t as equivalent to the number of coin tosses N then demonstrates the equivalence between the random walk and my arguments above, where sigma varies as SQRT(N).
I think you might want to rethink this. "Science" doesn't deal with "truth". "Truth" is for closed formal systems like math or logic or faith.
I have accurately quoted the first paragraph of your post above.
My assertion above can be either true or false (it happens to be true, thanks to cut-n-paste); but it isn't part of a closed formal system. So it appears that your original statement is false.
"...,converges to zero as the number of trials n goes to infinity" == the "chance of difference" approaches zero with higher values of N, so for larger N it is less likely for the value to be N/2 + e, but to approach N/2. This is opposite to the supposition that "more tosses will get you *further* from the mean"( N/2 ).
You are unable to distinguish between the percentage difference, e, and the absolute difference eN. The former tends toward zero in the limit of large N, in accordance with the Law of Large Numbers, but the latter does not. Your expression above should be N/2 + Ne, not N/2 + e; a dimensional analysis shows the latter expression is wrong, since it is adding "heads" (N/2) to a dimensionless fraction (e).
My points are explained well in this article; if you chose to ignore the arguments put forward in the article, then that's your (possibly pecuniary) loss.
If they are claiming that A and B are equaly valid even though one obviosly models things more acurately than the other, then in that case I whole-heartedly agree that they are stupid-heads
And this indeed is what postmodernists such as Baudrillard claim. Hence my original remarks.
ut the sample size is getting bigger than the standard deviation
Of course it is, and I've never claimed otherwise. I'm just trying to point out that the law of averages -- that if I've just tossed 20 heads in a row, then the next toss has a strong probability of being tails -- is just plain false. Furthermore, that the scatter of individual runs about the expected mean *grows* with the sample size. A very lucid discussion of the connection between these two points can be found here.
Slashdot Mirror
The Office is pretentious and boring. Is one of those things that only Brits get I guess.
Nah, nah, no sense of humour!
And like most pendants, you totally miss the point. You're measuring the wrong.
Grammar having deserted you, I declare this dialogue closed.
I've had mixed experiences with fixing my laptop. Recently, the hard drive went dead; however, since 2 1/2 inch drives are commodity parts, it was trivial to order a new one, open up the laptop and bung it in.
However, my laptop is also suffering from a broken wireless unit. It uses a GemTek WL-388F mini-USB 802.11b wireless module, which sits in a small compartment accessible via a panel on the underside. I fried the module while flashing its firmware, and I've had no success in finding a replacement anywhere; I guess this is because the module is much less of a commodity part than the hard drive was. If anyone could tell me where to find a replacement module, I would be very grateful!
The hose was used to equalize pressure and eliminate fog between two panes of a window.
That's a pretty inhumane way to treat ladies of the oldest profession!
Of course, the irony is that if someone says to me "pulsar discovery" (and, for the record, I'm a stellar astrophysicist), I immediately think "Jocelyn Bell". I haven't got the first idea who here supervisor was. She may not have won the prize; but she got the credit, well-deservedly.
Wow, you can get an interpolated result for sin(x) in three arithmetic operations? Best chicken--scratch algorithm I can work in pencil and paper in this state of mind is 3 multiplies, 3 add/subtracts and one floor
Oops, I completely mis-wrote. I meant one addition, two subtractions and three multiplies. My standard code is this:
(please allow for my Fortran syntax!). Since for uniform spacing idx is calculated from a floor(), the operation count is identical to yours, and your algorithm is optimal too.
Good point about the x shifting. Another aspect of the algorithm is this: if the interpolation is done as a sequence of monotonic x values, then the w_a calculation need only be performed if idx has changed. This gives an extra speedup (assuming that conditionals do not shaft the pipelineing).
Since division is roughly 10 times the cost of multiply on a PPro/P2/P3 (39 vs 4 cycles) you might be able to come up with an intelligent algorithm for sin/cos using taylor series that is faster and more accurate than interpolation for all but an impractically large LUT
Ah, but interpolations in a uniformly-spaced LUT has no need of division; just 1 subtraction, 1 addition and 1 multiplication. And in a 1024-term table, the error caused by neglecting the x^3 and higher terms of a sin(x) expansion is of the order of 1e-8; i.e., down there with machine precision.
However, I wonder whether something like a rational-function approximation could give a good approximation to sin(x)? Or a Pade approximation? Any thoughts? To be honest, a LUT is aesthetically unappealing to me; I'd love to have a really-fast algorithm in its stead.
or snow, or imperfections made by wind on the water, or warping for other reasons.
Yes, yes, yes, but if we just assume a spherical Lake Oneida in free space, then...
You make some good points; I'd overlooked the fact that the factorials can be precalculated. And yes, even with a LUT you are going to have to translate back into an interval of width 2*pi; either [-pi, pi], or [0,2*pi], or whatever. However:
You shouldn't use just 5 terms anyway, writing out the equation for more terms seemed unnecessary.
For sure; however, the further cost of any extra terms, above and beyond the 5 you've already given, makes a single Taylor's series expansion a poor approach compared to a LUT. One way of looking at a N-entry LUT is as N separate first-order Taylor's series expansions, each centered at a different point; I guess this is how LUTs win out, since with a large-enough N the accuracy of series, even at first order, wins out for intervals of width 2*pi/N. And hell, memory is so cheap nowadays that a N=1024-entry LUT is not a problem.
Thanks for your thoughtful reply to my postl; you've just earned yourself a friend.
Rich
In honour of the nationality of the scientist involved in the study, and to counter any anti-French sentiment which may have arisen due to unfortunate recent events on the geopolitical scene, maybe we need a new unit for angular measurements: 20 degrees = 1 degree of freedom.
Surely, once the lake had frozen the number of skips would have been large? Aren't ice/stone impacts pretty elastic (assuming no chips or instantaneous melting)?
Wow, I never realized Barnes-Wallace invented bouncing jizz as well. Cool!
Sirius A, for example, is certainly large enough to supernova, and it's 8.6 light-years away.
Not according to any current theory of stellar evolution. Sirius A is spectral type A, which means a mass less than 3 solar masses. Anything under 8 solar masses won't go supernova, so Sirius A is destined for the boring end: expansion into a red giant, loss of envelope as a planetary nebula, and senescence as a white dwarf.
Let's assume that Sirius A has a mass of 2 solar masses. Distributing this mass uniformkly over a sphere 8.6 light years in radius results in a mean density of 1.5*10^-18 g cm-3, which is peanuts in astrophysical terms. So, if Sirius A did go supernova (which it won't), the Gamma-ray flux might be nasty, but the particle flux would be irrelevant.
Oh, and you also forgot the multiplies needed to calculate the factorials...
Will give you the first 5 terms of a taylor series for sin(x) at a cost of 9 multiplies, 4 add/subtracts and 1 divide. Additional terms cost 2 multiplies and 1 add/subtract each.
I actually count 19 multiplies; it appears you missed the multiplies which are implicit in raising x to a power.
You can do a taylor expansion with one divide and reasonably low cost for the accuracy.
Except that the expression you gave isn't accurate at all; you have confused high order with high accuracy. Your expression gives sin(2*pi) = 11.9, which is way off the true value of sin(2*pi)=0. At x=pi, your expression is rather more accurate (0.00693, as opposed to the true value of 0), but still not good enough for any numerical calculations.
Instead, you were just being pedantic.
Guilty as charged, m'lud! I am indeed a pedant. That's why I take pathetic glee in pointing out that, on your resume, you claim to be an expert at HTML; yet the very same resume fails to pass the W3C HTML validator without errors. You're not exactly trying hard to sell your talent, are you?
From Womens Studies class back in college, I'm pretty sure that Oriental is not a desired term by Chinese Americans. We read several stories by Asian American women, who all objected to use of the word. I however am not one such person, so I can't state if it truley has negative conotations. I assume it would be like calling an African American a "colored person"
In the UK, "Asian" refers to ethnic groups from India, Pakistan, and the surrounding countries. People whose ethnic ancestry is from the Pacific rim countries, in particular those whose eyelids are characterized by an epicanthic fold, are termed "Oriential". The English-Chinese population doesn't appear to take offence at this terminology.
Likewise, "mixed race" (argh, hate that term) people in South Africa prefer to refer to themselves as "coloured" rather than "black". So let's remember that much of the terminology used to describe ethnicity or indeed skin pigmentation is very dependent on its cultural context.
It's hard to tell if s/he being purposefully unclear, made a mistake and is now trying to justify themselves, or is just incapable of clearly expressing an idea.
None of the above; in fact, the problem was that many people who responded to my posts were incapable of distinguishing between the absolute spread (varies as SQRT(N)/2, grows with N) and the relative spread (varies as 1/SQRT(N), decreases with N). Nothing to see, move along now...
The physics of the two situations doesn't match up at all though. No two particles can occupy the same space at the same time, but every coin toss is independent of the previous one.
The Pauli exclusion principle has nothing to do with random walks. In an idealized random walk, the diffusing particles do not interact with one another. All that happens is that, every time a particle travels a mean free path, there is a 50% probability (exactly) that its direction of motion will be reversed. In the case of Brownian motion (e.g., diffusion of smoke particles through air), what causes this motion reversal is collision with an air molecule. But the particles themselves behave independently, both of one another and of their previous history.
There's quite the argument going on over this, and I think people might be arguing different things and not realizing it (myself included). Could you clarify how this relates to global warming or is it just a disagreement with the original coin toss post?
I'm afraid it's nothing so controversial as global warming, the argument does indeed relate to the comment I made about coin tossing. Which I think I'm still winning, since the naysayers appear to be going quiet one by one. Although maybe they just don't give a toss....
Looking for an integer value in a real number continuum is an irritating thing to do.
It may be an irritating thing to do, but it is important to to emphasize when demonstrating that the "Law of Averages" (if I toss 20 heads in a row, there is more than a 50% chance that the next toss will be a tail, in order to force convergence toward the expected mean) is utter bunk.
It also helps one understand situations where the expected mean is zero, and all one is left with is the divergent spreading of the probability density function. I am, of couse, describing the diffusion of gas particles away from a point of high concentration, via Brownian random walks. In the latter case, the mean square displacement of an ensemble of particles is given by <r^2> = A t, where A is a constant depending on the temperature and other variables, and t the time (see here for the exact expression, it's right down at the bottom of the page). The standard deviation of the particle distribution obviously varies as sigma = SQRT(<r^2>) = SQRT(A t), and interpreting t as equivalent to the number of coin tosses N then demonstrates the equivalence between the random walk and my arguments above, where sigma varies as SQRT(N).
I think you might want to rethink this. "Science" doesn't deal with "truth". "Truth" is for closed formal systems like math or logic or faith.
I have accurately quoted the first paragraph of your post above.
My assertion above can be either true or false (it happens to be true, thanks to cut-n-paste); but it isn't part of a closed formal system. So it appears that your original statement is false.
"...,converges to zero as the number of trials n goes to infinity" == the "chance of difference" approaches zero with higher values of N, so for larger N it is less likely for the value to be N/2 + e, but to approach N/2. This is opposite to the supposition that "more tosses will get you *further* from the mean"( N/2 ).
You are unable to distinguish between the percentage difference, e, and the absolute difference eN. The former tends toward zero in the limit of large N, in accordance with the Law of Large Numbers, but the latter does not. Your expression above should be N/2 + Ne, not N/2 + e; a dimensional analysis shows the latter expression is wrong, since it is adding "heads" (N/2) to a dimensionless fraction (e).
My points are explained well in this article; if you chose to ignore the arguments put forward in the article, then that's your (possibly pecuniary) loss.
If they are claiming that A and B are equaly valid even though one obviosly models things more acurately than the other, then in that case I whole-heartedly agree that they are stupid-heads
And this indeed is what postmodernists such as Baudrillard claim. Hence my original remarks.
ut the sample size is getting bigger than the standard deviation
Of course it is, and I've never claimed otherwise. I'm just trying to point out that the law of averages -- that if I've just tossed 20 heads in a row, then the next toss has a strong probability of being tails -- is just plain false. Furthermore, that the scatter of individual runs about the expected mean *grows* with the sample size. A very lucid discussion of the connection between these two points can be found here.