Hey, I hate M$ as much as the next guy (see, I even used the $ instead of the S, to show what a great anti-M$ guy I am), but they have done something the Linux folks haven't: Released a product that my grandmother can sit down in front of, click a few things, and go. If I hand my mother an XP disk and a new computer and say, 'Install this,' she will first tell me to wash my hands, dinner is almost ready; then she will be able to sit down and do it.
Bullshit. Unless your Grandma is a computer
programmer or just happens to get really lucky
while installing a copy provided by the OEM. The only painless install of Windows
I've had was for Windows 3.1. My site licensed copy of Windows XP didn't
have the correct video driver. I never managed to install Windows ME so it would boot reliably; about 25%
of the time it would boot straight into a crash. Windows 98 crashed during its own install for a record 8 times before I finally got it to install.
Maybe I've had bad luck, but I doubt it. Installing ANY OS on PC hardware is likely to involve a few problems that require more knowledge than the typical grandma.
Since Trent Lott just lost
his job for expressing Dixiecrat sympathy, this
has become current events rather than history.
The Dixiecrats was another
name for the States Rights party, under the banner
of which Strom Thurmond, of whom you seem not to have heard, ran for president in 1948.
The party splintered from the Democrats in opposition to civil rights measures added to the Democratic party platform. Strom Thurmond switched to the Republican party in 1964 and went on to become really old while serving in the senate. At Thurmond's 100th birthday celebration Trent Lott said that if the Dixiecrats had won in 1948, the country would be better off. He then lost his
job. Most other conservative southern Democrats eventually followed Thurmond's 1964 example and became Republicans. Thus, while I
don't want to argue that Republicans are generally racist, mention of the Dixiecrats tends to support the claim that at least some Republicans might not be sympathetic to equal rights for minorities.
"The cookie analogy." Let me get this straight, all those people who love this show think it is one of the best written on TV. And yet, during the final episode, while (as I've been told), Buffy and her friends are in an epic battle to save the whole fucking world, she finds 10 minutes to talk about how her love live is like fucking cookies? Oh yeah, Hamlet's got nothing on this show.
Have you forgotten Polonius's tendency to crap on with all sorts of cliches and dubious analogies? "There---my blessing with thee, and these few precepts in thy memory...". And then it goes on and on, precept after precept. Polonius was supposed to be funny. Someone else has pointed out that some of Hamlet's madness is intended to be funny. Obviously
Shakespeare didn't know if Hamlet was to be a comedy or drama.
I won't claim Buffy is as well written as Hamlet. But I am of the camp that thinks it is better written than most stuff on TV. The reasons:
It had ongoing plots and the characters
developed over an extended period.
It integrated comedy and drama effectively.
This appears to be what you don't like, but for me both worked and both benefited from the mix. It certainly can't be any worse than humorless drama that makes you want to kill the pathetic lifeless characters or constant one-liners delivered from cardboard cutouts and ethnic stereotypes made incarnate.
It poked fun at cliches and managed to do things that were unexpected. The
surprise plot twists actually surprised me. This is rare.
I will say that there were some holes in the plots and some plot lines that weren't followed up on. But this is likely to happen in anything complicated carried out over an extended period---especially if the whole thing isn't planned in advance. Anyway, I can forgive that sort of plotting flaw if there are other features of interest. After all does anyone really think the ending of the Tempest makes sense: Prospero is all set to exact his revenge and the whole thing ends in about a page with a wedding?
A natural logical plot with all loose ends tied up isn't everything.
And if your only tool is a hammer then everything looks like a nail. But that doesn't mean that a hammer is the right tool for every job. If all you need to do is drive in a bolt, why pound it in with a hammer thereby stripping threads and banging your thumb?
If you are writing applications that don't require a language that works at such a low level,
why not use something safer and easier?
True for sufficiently broad definitions of "linear algebra". People who favor a narrower definition might reasonably say that what you are talking about is operator theory or functional analysis. For a lot of people "linear algebra" implies finite dimensional.
It's a simple cost/benefit analysis. I can spend a week shoe-horning something into the BLAS interface and possibly create obscure bugs in the process, or I can just write a reasonably efficient high-level loop and live with whatever the compiler gives me.
Except for the fact that a large number of
scientific codes come down to solving linear
systems, eigenvalue problems, discretizing and solving a PDE etc. For these sorts of standard problems, the libraries are easier to use than writing your own. They are also more likely to be robust with respect to round-off error. And the people who write such code quite often use BLAS or optimize their code for a particular architecture.
I do agree that if you have a problem for which there are no higher level libraries then it's probably not worth your time to worry about
heavy optimization or use of relatively low-level stuff like BLAS.
You can definitely do the FFT in integer arithmetic. I'm not sure of the details. Fixed
point arithmetic is something I don't think
about very much. It would be more work than
floating point and it doesn't sound very fun to me. But then I'm very lazy, as my presence goofing off here proves. Still, I wish you good luck. A good integer FFT is a noble goal.
Concerning your attempt to get rid of multiplications, the Winograd FFT reduces
the number of multiplications to O(n) but
the number of additions increases (to O(n^2)
if I remember correctly.) For large n this is obviously not useful unless multiplication is much slower than addition (which is not the case on todays computers.) It is however sometimes used
for short length modules in mixed radix FFT algorithms.
If you are interested in efficient
FFT, a web search turned up the following, which
seems to have a ton of references (including the Winograd paper):
burrus-notes There is also a good general
reference book by Charles Van Loan published by the Society for Industrial and Applied Mathematics:
SIAM
It's amazing how much work
is done in this area. The FFT is very important.
Another example is operator overloading. Perhaps operator overloading is useful to the first person writing the code, as it provides a nice little shortcut where they can do foo + bar as opposed to something like foo.add(bar).
In my opinion the problem with this reasoning is that it ignores the fact that specific communities might have very standard ways for overloading operators that won't hinder readability for anyone who would be likely to be reading the code. For example, I do matrix
computations. If you use the operators without much thought for memory allocation, I suppose there can be some performance pitfalls that you
might otherwise avoid. But as
far as readability goes, it's a clear win: Instead of a messy list of function calls you have a direct translation of whatever equation(s) you might be working with in standard algebraic notation. Anyone for whom this isn't more readable is probably not familiar enough with the math to be looking at matrix codes anyway.
Of course this begs the question of whether
general purpose programming languages should be designed to be flexible enough to meet the needs of such specialized groups when we all know that plenty of other people will make stupid and unreadable use of a dangerous feature. But for me lack of this feature can make a language completely unsuitable (or at least highly inconvenient) for what I need to do. Maybe we
should all be using the language that is tailored to our needs (Fortran 90 and later versions overload + -
automatically to work on arrays, along with providing many other features handy for people
writing numerical code). But use of
many different languages also presents problems for people inheriting old code.
It's especially cute that the department at my college that seems the least inclined to grade and return my papers electronically rather than printing them out is the environmental studies department. The most inclined is the Math dept., where some professors won't even accept hardcopies anymore.
I teach math. It completely amazes me that anyone would accept a math homework in electronic form and then make the corrections electronically. None of the word processors I've seen produce anything close to readable equation output. Equations set with MS Word make my eyes bleed.
Most mathematicians still use LaTeX for anything involving equations (departmental documents which are, unfortunately, most likely standardized on Word are another matter). For electronic grading, I'd
either have to correct a student's LaTeX or (worse) fiddle around with the tedious Word equation editor. I'm sure this would double or triple the time I spend grading.
I sometimes have students wondering if I can
put stuff up on the Web via WebCT. This would be a lot of work. This is something people don't think about, but at least for math all the electronic alternatives are a pain in ass.
3. Find your DMR (Daily Burn Rate). Because I sit all day, my caloric DMR is about 2000. Therefore I eat 1850. if you burn 2000, and only feed yourself 1850, where does the other 150 come from? YOUR FAT. It's a beutiful simple concept.
You might want to check your math on your
metabolic rate. A lb. of
fat is about 3000 calories. A 150 calorie daily deficit is 1050 per week or about 1/3 lb. of fat.
You say you are losing 1-2 lbs. per week. Sounds to me like your daily deficit is closer to 500 calories. This is a good amount.
Also, exercise is good. Most people want to lose fat, not muscle. If you don't exercise 30%-50% of your weight loss will be lean body mass, not fat. This will reduce your metabolism, something the body is inclined to do anyway when you are restricting calories. Lifting weights is good to help off-set this. Most people who keep weight off take up some sort of sustainable exercise program.
I must be an expert. I needed to solve a matrix equation involving a 1700x1700 autocorrelation matrix.
Algorithms for such matrices which usually have Toeplitz or Toeplitz-like structure (depending on whether it is an actual autocorrelation matrix or computed from samples) is my field of research. Is this matrix structured?
The fastest algorithms are then O(nlog^2(n)) using O(n) storage. O(n^2) Schur or generalized Schur algorithms are well known and easy to code. Using general system solvers is not the way to go. If you want references, just let me know here. Otherwise do a search on the "generalized Schur algorithm". Even inefficient Matlab implementations of these algorithms should be fine. 1700x1700 is really small if the matrix is structured.
I hacked it up in C, used gsl to solve the matrix, and had results MUCH faster than Matlab can provide them. And, the kicker in this case was that the memory usage of C was trivial compared to that of matlab - gsl used about 25 MBytes, Matlab threw the 256 M RAM machine into swap.
It's easy to make poor use of memory with Matlab. If you don't put some thought into memory management, you can easily write code that allocates lots of unecessary temporary storage. Without seeing your Matlab code, I can't really comment. If you are just using the "\" operator for solution, it should go to LAPACK for
Gaussian elimination. And I stand by my claim that you probably aren't going to do better than that in C. I will concede that Matlab won't do an in-place factorization, but 1700x1700 isn't particularly huge. So one extra 1700x1700 matrix shouldn't be your problem with swapping. In fact, I just used "\" to solve a 1700x1700 autocorrelation system (without exploiting Toeplitz structure) and it took my system less than 45 seconds. There was no swapping and I also have 256M. Whatever is making your code run with problems it isn't solution of a 1700x1700 system.
Matlab is a toy scripting language that is good for small data sets, where memory dwarfs the problem at hand, and when the speed of prototyping the algorithm is the limiting factor (which is really really often).
Matlab is a scripting language for putting together quite good matrix solvers. If you use it for anything else, I agree it will not be efficient. But the mistake made is usually using it incorrectly or for the wrong problem. Other than the fact that you might be not exploiting matrix structure, it doesn't sound like your problem is something Matlab couldn't handle.
But perhaps there is more to your problem than the autocorrelation matrix.
But still, get down to serious crunching, and people are rewriting algorithms in C or ForTran to save time.
Sure, especially if they have an even somewhat novel algorithm that isn't based on basic matrix operations. But not if they value their time and their code is just glueing together some system solvers or FFTs or the like.
Matlab has evolved considerably from a Matrix Laboratory to a general all purpose numerical programming and plotting scripting language. Kinda like a blunt swiss army knife.
People do use it like that, perhaps when they shouldn't. But strictly speaking if you use it like a scripting language it is fine (i.e. if you use it for glueing together LAPACK solvers, or
other optimized internal Matlab functions that do most of the work). What it sucks at is being a general purpose programming language; that I agree with.
As the poster said, this is based on the assumption that matrix operations are used. If you make heavy use of matrix operations on large matrices via the builtin functions, which is what Matlab is intended for, then the speed will be largely determined by the code that Matlab uses. (LAPACK I believe for matrix operations.) Unless you are an expert at this sort of thing and have a lot of free time you aren't going to write C code to beat LAPACK.
On the other hand, if you write a lot of
your own loops, work with lots of small problems or force Matlab to do lots of stupid allocations for matrices to contain temporary results, then it will be very slow. However this shouldn't be a problem if your problem is at all suited to MATLAB; if your code doesn't require pretty heavy use of matrices, a MATrix LABoratory might not be the right tool for the job.
Re:What is an example that can't run in parallel?
on
Forget Moore's Law?
·
· Score: 2, Informative
Inversion is always a bad idea due to the effect of round-off error. What you really want to do is to solve the linear system Ax=b using a stable method without actually computing the inverse of A. For solving Ax=b it is clearer to speak of parallelizable algorithms instead whether solving a general system is parallelizable; there are too many different types of matrices A.
Some algorithms work well on some A and not on others.
Guassian elimination is the one method that works on any system but it does not exploit any structure in A (presence of zero elements, etc.). Also, assuming that is what you mean by "matrix inversion", you are correct. It is not terribly parallelizable:
the amount of communication required increases pretty quickly as you increase the number of processors until you don't really gain anything at all. And since it doesn't exploit the structure of A it is typically not very fast. Nevertheless, it's the slow steady method that will eventually give you a solution to a problem on which other methods fail.
Iterative methods can easily exploit the structure of matrices A that come from partial differential equations. (This covers fluid flow and many other physics/engineering simulations). In some cases there are methods (e.g. domain decomposition) which can make the methods highly parallel. But whether iterative methods work well in parallel, or even whether they work at all, depends on the problem. These methods can be efficient when they work, but can sometimes require an expert to make them to work at all.
In practice if you pick the right algorithm and have a bit of luck you might get a good speedup solving a linear system on a parallel computer. More often than not, unless you put a lot of work into matching your algorithm to a very specific matrix A, the gains from parallel solution will not be amazingly impressive. Having a faster computer that doesn't depend on parallelism is better. This was true even in the old days when supercomputers had fewer processors; in my experience the gains from vectorization were easier to get and more dramatic than gains from parallelization. Now that we try to divide up the problems on supercomputers with many more processors, the limits imposed by communication are even more significant.
But I have no problem paying for something that works well.
There are plenty of people with specialized needs for whom it doesn't work well. Last time I used the Word equation editor it crashed on me, in addition to being a not-very-convenient interface. Everything I have to do with Word is mercifully short, but I still
hear people complaining about what a mess it is for
large documents. But since it's a closed format
and a de facto standard, we're forced to use it anyway (in my case only occasionally). That sucks.
Body analysis devices all have a certain degree of error. Some methods have so high a degree of error as to be almost worthless (those scales that claim to electrically measure bodyfat for example) Calipers are OK
for body fat, but don't even pretend to
measure water weight. They also have to be used by someone who knows what they are doing. Hydrostatic testing
is the most accurate, but even it relies on
assumptions on average body composition. (Note that the specific
gravity of muscle is 1.08 which means that it
is not easy to tell the difference between
muscle and water weight loss by hydrostatic
testing.)
Weigh that against that numbers: The figure of 3000 calories/lb.
of fat is correct. If you really lost 20lbs.
of fat in two weeks then you had to be burning
4286 calories per day. If you were running a lot and not eating anything at all and your
metabolism didn't crash, that might
be possible. Otherwise, something doesn't add up.
Then consider that losing water weight upon entering ketosis is normal and think about what this all suggests about the likely
accuracy of your body composition test.
Weight lifting is mostly upper body work, with the exceptions of a few lifts such as squats.
Those "exceptions" are the most important lifts. IMO a proper weight lifting routine should
be mostly lower body and back work.
The real benefits from weight lifting are more
likely to come from squats, deadlifts and cleans.
Just ask any power lifter or olympic lifter. Just because gyms are filled with guys doing 20 sets of bicep curls doesn't mean that is the smart way to lift weights.
I don't recall saying not to read later writers like Riemann, Cauchy, or Weierstrass.
Perhaps not, but you said that Newton and Leibniz were great for "understanding what
calculus really is", but very poor for practical calculus. Considering that from
Newton and Leibniz up to the 19th century
calculus had very little theoretical basis and
was in large part a method of formal manipulation (with many practical applications in physics), it seems to me that you have things backward. You did not say Newton, Leibniz, Cauchy, Riemann and Weirstrass are good for understanding what calculus really is. I would have agreed with that statement.
You are giving short shrift to Newton and Leibniz. The "incorrect" or "incomplete" ideas of the past are what informed the "correct" and "complete" ideas of the present.
And you are giving short shrift to Bishop Berkeley and his "ghosts of departed quantities".
Newton and Leibniz did not
have a rigorous basis for the calculus. I'm not saying they were anything other than amazingly original geniuses. However we've built on their ideas, extending and strengthening them. Ideas advance and this can't be ignored.
What you and others are reflexively attempting to say to me in reponse is that your method of study of mathematics is the only valid method, and the approach I am recommending is clearly inferior to yours. Given that I am not making an apparently chauvinistic argument about my own preference, and you are, I suspect that the bias lies with you.
I'm not sure how you make the determination of what constitutes a chauvanistic argument. I took your implication that Newton and Leibniz were
sufficient without Cauchy, etc. to be a chauvanistic dismissal of the importance of studying later work. It looks to me like you
have done nothing more than assert that my position is chauvanistic without any basis.
In any event, I am no longer sure what you mean by a "historical" and "general"
approach. I'm certainly not going to argue against the need for comprehension or intuition. However I should point out that conceptual intuition
is not sufficient by itself for doing mathematics. As a researcher I've seen too many conceptual ideas for a proof vanish when confronted with details. If the point of
studying mathematics is not to do mathematics or to use mathematics in practice, then I'm not sure what it is. Since your approach seems to be dismissive of practical applications and hostile to the idea that mathematics should always be taught at a level of detail sufficient for research, perhaps you could expound on the use of learning mathematics your way?
I won't say much about the other stuff conerning "revelatory visions". I
have never supported such a view and I would have thought my call for detailed study would
in fact suggest that I hold the opposite view. IMO understanding is hard won. It also seems to me that the
highly formal approach commonly used to teach math is
contrary to this "revelatory" approach that you claim is in conflict with your "conceptual" but surprisingly not "revelatory" approach.
I have an intensive classic liberal arts education. Calculus directly from Newton and Leibniz, for example. This is great for understanding what the calculus really is, but very poor for doing the kind of calculus that people do as a practical matter.
It's also not a very good basis for
understanding the theory behind calculus. The theoretical background for calculus came much later than Newton or Leibniz: think instead of Cauchy and Riemann. If you've
studied only Newton and Leibniz, you've studied a small part of the history and origin of calculus---not its theory, not its practical use and not even its full history.
The concept of a limit was fuzzy at best and Leibniz worked with infinimetesimals.
They didn't really understand "what the calculs really is." This was something that took a couple of centuries to figure out. The idea that these later technical refinements are not relevant to "a deep comprehension of the subject matter in general" is nonsense IMO.
Which is not to say that the larger point of studying generalities first is bad. But
ultimately math is about details. Dismissing
these details as being irrelevant to a deep understanding is misleading.
The faculty and courses don't remain nearly the same if they expect the upgrade to work credibly. This change usually involves a plan for new courses. Subsequent hiring will also take the change into account. College and University are not arbitrary labels.
I am a math professor at a four year university. I'll make the ad hominem note that a lot of people most
critical of math teaching are flunking math
because: 1. they aren't doing the work 2.
they aren't coming to class 3. they get caught
cheating (the most common way to fail my course last semester). Which is not to say
that teaching isn't bad sometimes; but there
are also a lot of people who just won't do the work and then act angry and blame the teacher when the natural results follow.
Obviously teaching varies from place to place. The comment that at major universities the professors are mostly concerned with grad students and research is
probably true for a significant number of professors---but not all professors for all
courses. This is worse the lower the course:
it's a cold truth, but very few people with a PhD relish teaching high-school level math.
In fact, even at a major university I'd wager very
few people teaching such courses have a PhD.
On the other hand, although I've known
some pretty dedicated community college teachers, the chances of getting a decent
course in multivariable calculus or differential equations much less abstract algebra or analysis at a community college
is not good. I'd say make up any pre-college level math at a community college
and then switch to a university if you want to follow it through or past calculus.
For someone who is self motivated and just needs a refresher on rusty math skills
in algebra, self-study with any decent text-book should also work. Just make yourself work problems with the discipline you
would if you had to turn them in.
Lot's of people don't want to (or can't) work. In my experience policies that allow students to get help from other students end up being abused. I'm currently teaching a numerical analysis class in which the assignments are half programming and half math. Students think nothing of copying another student's code or downloading some code found via a web search. My stated policy was that they could discuss general
methods of solution but not duplicate another person's code. Unfortunately this leaves a grey area that makes it harder to enforce a consistent policy. ("But Professor, I didn't copy. My
code is similar to Steve's because he explained to me how Romberg integration works.") In the future I will require that they work completely independently.
On the other hand, if the policy at Georgia Tech really does prevent students from referencing anything outside the course materials, that is ridiculous. Students who take the initiative to do a little outside research should be encouraged.
Geez - 4 digits sums ! This is really an undergraduate project that would involve two microphones, a low pass filter, an amplifier, an audio amp, and a speaker.
I don't believe it is quite that simple.
Adaptive noise cancellation uses a reference
microphone (e.g. a microphone on the headband,
or someplace away from the ear), two microphones in the ear and an adaptive
filter. The filter processes the signal
from the reference microphone and adds it to the
signal going out through the speakers. The
filter is tuned (e.g. using the LMS algorithm)
to try to drive the signal at the ear microphones
to zero.
Without the reference signal and the
adaptive filter, this won't work too well.
You need something to cancel the signal with
(the reference signal) and a way to adjust the
cancellation (to take into account changes in
the phase of the signal.)
This might also be a senior level undergrad project if the undergrad in question was really strong on DSP. But it's not just a lowpass filter.
And what about newer FORTRANs, like F90? Well, if you're going to retrain on that rats-nest of a language, you might as well just stick with the industry-standard rats-nest, and use C++. For an F77 programmer, the learning curve will be the same.
Except that the industry standard rats
nest in engineering and science is Fortran.
Compatability with people writing GUI programs
is less important than the fact that F90 compiles
the enormous base of F77 numerical code. Also
F90 has features of special interest to those
writing numerical code. With the features
for handling arrays, F90 matrix code is very easy to write.
Slashdot Mirror
Bullshit. Unless your Grandma is a computer programmer or just happens to get really lucky while installing a copy provided by the OEM. The only painless install of Windows I've had was for Windows 3.1. My site licensed copy of Windows XP didn't have the correct video driver. I never managed to install Windows ME so it would boot reliably; about 25% of the time it would boot straight into a crash. Windows 98 crashed during its own install for a record 8 times before I finally got it to install. Maybe I've had bad luck, but I doubt it. Installing ANY OS on PC hardware is likely to involve a few problems that require more knowledge than the typical grandma.
Since Trent Lott just lost his job for expressing Dixiecrat sympathy, this has become current events rather than history. The Dixiecrats was another name for the States Rights party, under the banner of which Strom Thurmond, of whom you seem not to have heard, ran for president in 1948. The party splintered from the Democrats in opposition to civil rights measures added to the Democratic party platform. Strom Thurmond switched to the Republican party in 1964 and went on to become really old while serving in the senate. At Thurmond's 100th birthday celebration Trent Lott said that if the Dixiecrats had won in 1948, the country would be better off. He then lost his job. Most other conservative southern Democrats eventually followed Thurmond's 1964 example and became Republicans. Thus, while I don't want to argue that Republicans are generally racist, mention of the Dixiecrats tends to support the claim that at least some Republicans might not be sympathetic to equal rights for minorities.
Have you forgotten Polonius's tendency to crap on with all sorts of cliches and dubious analogies? "There---my blessing with thee, and these few precepts in thy memory...". And then it goes on and on, precept after precept. Polonius was supposed to be funny. Someone else has pointed out that some of Hamlet's madness is intended to be funny. Obviously Shakespeare didn't know if Hamlet was to be a comedy or drama.
I won't claim Buffy is as well written as Hamlet. But I am of the camp that thinks it is better written than most stuff on TV. The reasons:
I will say that there were some holes in the plots and some plot lines that weren't followed up on. But this is likely to happen in anything complicated carried out over an extended period---especially if the whole thing isn't planned in advance. Anyway, I can forgive that sort of plotting flaw if there are other features of interest. After all does anyone really think the ending of the Tempest makes sense: Prospero is all set to exact his revenge and the whole thing ends in about a page with a wedding? A natural logical plot with all loose ends tied up isn't everything.
And if your only tool is a hammer then everything looks like a nail. But that doesn't mean that a hammer is the right tool for every job. If all you need to do is drive in a bolt, why pound it in with a hammer thereby stripping threads and banging your thumb?
If you are writing applications that don't require a language that works at such a low level, why not use something safer and easier?
True for sufficiently broad definitions of "linear algebra". People who favor a narrower definition might reasonably say that what you are talking about is operator theory or functional analysis. For a lot of people "linear algebra" implies finite dimensional.
I'll second this. DVDShrink is an amazingly wonderful program. It's fast, easy and (so far for me) stable.
Except for the fact that a large number of scientific codes come down to solving linear systems, eigenvalue problems, discretizing and solving a PDE etc. For these sorts of standard problems, the libraries are easier to use than writing your own. They are also more likely to be robust with respect to round-off error. And the people who write such code quite often use BLAS or optimize their code for a particular architecture.
I do agree that if you have a problem for which there are no higher level libraries then it's probably not worth your time to worry about heavy optimization or use of relatively low-level stuff like BLAS.
You can definitely do the FFT in integer arithmetic. I'm not sure of the details. Fixed point arithmetic is something I don't think about very much. It would be more work than floating point and it doesn't sound very fun to me. But then I'm very lazy, as my presence goofing off here proves. Still, I wish you good luck. A good integer FFT is a noble goal.
Concerning your attempt to get rid of multiplications, the Winograd FFT reduces the number of multiplications to O(n) but the number of additions increases (to O(n^2) if I remember correctly.) For large n this is obviously not useful unless multiplication is much slower than addition (which is not the case on todays computers.) It is however sometimes used for short length modules in mixed radix FFT algorithms.
If you are interested in efficient FFT, a web search turned up the following, which seems to have a ton of references (including the Winograd paper): burrus-notes There is also a good general reference book by Charles Van Loan published by the Society for Industrial and Applied Mathematics: SIAM It's amazing how much work is done in this area. The FFT is very important.
In my opinion the problem with this reasoning is that it ignores the fact that specific communities might have very standard ways for overloading operators that won't hinder readability for anyone who would be likely to be reading the code. For example, I do matrix computations. If you use the operators without much thought for memory allocation, I suppose there can be some performance pitfalls that you might otherwise avoid. But as far as readability goes, it's a clear win: Instead of a messy list of function calls you have a direct translation of whatever equation(s) you might be working with in standard algebraic notation. Anyone for whom this isn't more readable is probably not familiar enough with the math to be looking at matrix codes anyway.
Of course this begs the question of whether general purpose programming languages should be designed to be flexible enough to meet the needs of such specialized groups when we all know that plenty of other people will make stupid and unreadable use of a dangerous feature. But for me lack of this feature can make a language completely unsuitable (or at least highly inconvenient) for what I need to do. Maybe we should all be using the language that is tailored to our needs (Fortran 90 and later versions overload + - automatically to work on arrays, along with providing many other features handy for people writing numerical code). But use of many different languages also presents problems for people inheriting old code.
I teach math. It completely amazes me that anyone would accept a math homework in electronic form and then make the corrections electronically. None of the word processors I've seen produce anything close to readable equation output. Equations set with MS Word make my eyes bleed.
Most mathematicians still use LaTeX for anything involving equations (departmental documents which are, unfortunately, most likely standardized on Word are another matter). For electronic grading, I'd either have to correct a student's LaTeX or (worse) fiddle around with the tedious Word equation editor. I'm sure this would double or triple the time I spend grading.
I sometimes have students wondering if I can put stuff up on the Web via WebCT. This would be a lot of work. This is something people don't think about, but at least for math all the electronic alternatives are a pain in ass.
You might want to check your math on your metabolic rate. A lb. of fat is about 3000 calories. A 150 calorie daily deficit is 1050 per week or about 1/3 lb. of fat. You say you are losing 1-2 lbs. per week. Sounds to me like your daily deficit is closer to 500 calories. This is a good amount.
Also, exercise is good. Most people want to lose fat, not muscle. If you don't exercise 30%-50% of your weight loss will be lean body mass, not fat. This will reduce your metabolism, something the body is inclined to do anyway when you are restricting calories. Lifting weights is good to help off-set this. Most people who keep weight off take up some sort of sustainable exercise program.
On acid-free paper.
Algorithms for such matrices which usually have Toeplitz or Toeplitz-like structure (depending on whether it is an actual autocorrelation matrix or computed from samples) is my field of research. Is this matrix structured? The fastest algorithms are then O(nlog^2(n)) using O(n) storage. O(n^2) Schur or generalized Schur algorithms are well known and easy to code. Using general system solvers is not the way to go. If you want references, just let me know here. Otherwise do a search on the "generalized Schur algorithm". Even inefficient Matlab implementations of these algorithms should be fine. 1700x1700 is really small if the matrix is structured.
I hacked it up in C, used gsl to solve the matrix, and had results MUCH faster than Matlab can provide them. And, the kicker in this case was that the memory usage of C was trivial compared to that of matlab - gsl used about 25 MBytes, Matlab threw the 256 M RAM machine into swap.
It's easy to make poor use of memory with Matlab. If you don't put some thought into memory management, you can easily write code that allocates lots of unecessary temporary storage. Without seeing your Matlab code, I can't really comment. If you are just using the "\" operator for solution, it should go to LAPACK for Gaussian elimination. And I stand by my claim that you probably aren't going to do better than that in C. I will concede that Matlab won't do an in-place factorization, but 1700x1700 isn't particularly huge. So one extra 1700x1700 matrix shouldn't be your problem with swapping. In fact, I just used "\" to solve a 1700x1700 autocorrelation system (without exploiting Toeplitz structure) and it took my system less than 45 seconds. There was no swapping and I also have 256M. Whatever is making your code run with problems it isn't solution of a 1700x1700 system.
Matlab is a toy scripting language that is good for small data sets, where memory dwarfs the problem at hand, and when the speed of prototyping the algorithm is the limiting factor (which is really really often).
Matlab is a scripting language for putting together quite good matrix solvers. If you use it for anything else, I agree it will not be efficient. But the mistake made is usually using it incorrectly or for the wrong problem. Other than the fact that you might be not exploiting matrix structure, it doesn't sound like your problem is something Matlab couldn't handle. But perhaps there is more to your problem than the autocorrelation matrix.
But still, get down to serious crunching, and people are rewriting algorithms in C or ForTran to save time.
Sure, especially if they have an even somewhat novel algorithm that isn't based on basic matrix operations. But not if they value their time and their code is just glueing together some system solvers or FFTs or the like.
Matlab has evolved considerably from a Matrix Laboratory to a general all purpose numerical programming and plotting scripting language. Kinda like a blunt swiss army knife.
People do use it like that, perhaps when they shouldn't. But strictly speaking if you use it like a scripting language it is fine (i.e. if you use it for glueing together LAPACK solvers, or other optimized internal Matlab functions that do most of the work). What it sucks at is being a general purpose programming language; that I agree with.
As the poster said, this is based on the assumption that matrix operations are used. If you make heavy use of matrix operations on large matrices via the builtin functions, which is what Matlab is intended for, then the speed will be largely determined by the code that Matlab uses. (LAPACK I believe for matrix operations.) Unless you are an expert at this sort of thing and have a lot of free time you aren't going to write C code to beat LAPACK.
On the other hand, if you write a lot of your own loops, work with lots of small problems or force Matlab to do lots of stupid allocations for matrices to contain temporary results, then it will be very slow. However this shouldn't be a problem if your problem is at all suited to MATLAB; if your code doesn't require pretty heavy use of matrices, a MATrix LABoratory might not be the right tool for the job.
Inversion is always a bad idea due to the effect of round-off error. What you really want to do is to solve the linear system Ax=b using a stable method without actually computing the inverse of A. For solving Ax=b it is clearer to speak of parallelizable algorithms instead whether solving a general system is parallelizable; there are too many different types of matrices A. Some algorithms work well on some A and not on others.
Guassian elimination is the one method that works on any system but it does not exploit any structure in A (presence of zero elements, etc.). Also, assuming that is what you mean by "matrix inversion", you are correct. It is not terribly parallelizable: the amount of communication required increases pretty quickly as you increase the number of processors until you don't really gain anything at all. And since it doesn't exploit the structure of A it is typically not very fast. Nevertheless, it's the slow steady method that will eventually give you a solution to a problem on which other methods fail.
Iterative methods can easily exploit the structure of matrices A that come from partial differential equations. (This covers fluid flow and many other physics/engineering simulations). In some cases there are methods (e.g. domain decomposition) which can make the methods highly parallel. But whether iterative methods work well in parallel, or even whether they work at all, depends on the problem. These methods can be efficient when they work, but can sometimes require an expert to make them to work at all.
In practice if you pick the right algorithm and have a bit of luck you might get a good speedup solving a linear system on a parallel computer. More often than not, unless you put a lot of work into matching your algorithm to a very specific matrix A, the gains from parallel solution will not be amazingly impressive. Having a faster computer that doesn't depend on parallelism is better. This was true even in the old days when supercomputers had fewer processors; in my experience the gains from vectorization were easier to get and more dramatic than gains from parallelization. Now that we try to divide up the problems on supercomputers with many more processors, the limits imposed by communication are even more significant.
But I have no problem paying for something that works well.
There are plenty of people with specialized needs for whom it doesn't work well. Last time I used the Word equation editor it crashed on me, in addition to being a not-very-convenient interface. Everything I have to do with Word is mercifully short, but I still hear people complaining about what a mess it is for large documents. But since it's a closed format and a de facto standard, we're forced to use it anyway (in my case only occasionally). That sucks.
Body analysis devices all have a certain degree of error. Some methods have so high a degree of error as to be almost worthless (those scales that claim to electrically measure bodyfat for example) Calipers are OK for body fat, but don't even pretend to measure water weight. They also have to be used by someone who knows what they are doing. Hydrostatic testing is the most accurate, but even it relies on assumptions on average body composition. (Note that the specific gravity of muscle is 1.08 which means that it is not easy to tell the difference between muscle and water weight loss by hydrostatic testing.)
Weigh that against that numbers: The figure of 3000 calories/lb. of fat is correct. If you really lost 20lbs. of fat in two weeks then you had to be burning 4286 calories per day. If you were running a lot and not eating anything at all and your metabolism didn't crash, that might be possible. Otherwise, something doesn't add up.
Then consider that losing water weight upon entering ketosis is normal and think about what this all suggests about the likely accuracy of your body composition test.
Those "exceptions" are the most important lifts. IMO a proper weight lifting routine should be mostly lower body and back work. The real benefits from weight lifting are more likely to come from squats, deadlifts and cleans. Just ask any power lifter or olympic lifter. Just because gyms are filled with guys doing 20 sets of bicep curls doesn't mean that is the smart way to lift weights.
Perhaps not, but you said that Newton and Leibniz were great for "understanding what calculus really is", but very poor for practical calculus. Considering that from Newton and Leibniz up to the 19th century calculus had very little theoretical basis and was in large part a method of formal manipulation (with many practical applications in physics), it seems to me that you have things backward. You did not say Newton, Leibniz, Cauchy, Riemann and Weirstrass are good for understanding what calculus really is. I would have agreed with that statement.
You are giving short shrift to Newton and Leibniz. The "incorrect" or "incomplete" ideas of the past are what informed the "correct" and "complete" ideas of the present.
And you are giving short shrift to Bishop Berkeley and his "ghosts of departed quantities". Newton and Leibniz did not have a rigorous basis for the calculus. I'm not saying they were anything other than amazingly original geniuses. However we've built on their ideas, extending and strengthening them. Ideas advance and this can't be ignored.
What you and others are reflexively attempting to say to me in reponse is that your method of study of mathematics is the only valid method, and the approach I am recommending is clearly inferior to yours. Given that I am not making an apparently chauvinistic argument about my own preference, and you are, I suspect that the bias lies with you.
I'm not sure how you make the determination of what constitutes a chauvanistic argument. I took your implication that Newton and Leibniz were sufficient without Cauchy, etc. to be a chauvanistic dismissal of the importance of studying later work. It looks to me like you have done nothing more than assert that my position is chauvanistic without any basis.
In any event, I am no longer sure what you mean by a "historical" and "general" approach. I'm certainly not going to argue against the need for comprehension or intuition. However I should point out that conceptual intuition is not sufficient by itself for doing mathematics. As a researcher I've seen too many conceptual ideas for a proof vanish when confronted with details. If the point of studying mathematics is not to do mathematics or to use mathematics in practice, then I'm not sure what it is. Since your approach seems to be dismissive of practical applications and hostile to the idea that mathematics should always be taught at a level of detail sufficient for research, perhaps you could expound on the use of learning mathematics your way?
I won't say much about the other stuff conerning "revelatory visions". I have never supported such a view and I would have thought my call for detailed study would in fact suggest that I hold the opposite view. IMO understanding is hard won. It also seems to me that the highly formal approach commonly used to teach math is contrary to this "revelatory" approach that you claim is in conflict with your "conceptual" but surprisingly not "revelatory" approach.
It's also not a very good basis for understanding the theory behind calculus. The theoretical background for calculus came much later than Newton or Leibniz: think instead of Cauchy and Riemann. If you've studied only Newton and Leibniz, you've studied a small part of the history and origin of calculus---not its theory, not its practical use and not even its full history.
The concept of a limit was fuzzy at best and Leibniz worked with infinimetesimals. They didn't really understand "what the calculs really is." This was something that took a couple of centuries to figure out. The idea that these later technical refinements are not relevant to "a deep comprehension of the subject matter in general" is nonsense IMO.
Which is not to say that the larger point of studying generalities first is bad. But ultimately math is about details. Dismissing these details as being irrelevant to a deep understanding is misleading.
The faculty and courses don't remain nearly the same if they expect the upgrade to work
credibly. This change usually involves a plan
for new courses. Subsequent hiring will also
take the change into account. College and University are not arbitrary labels.
I am a math professor at a four year university. I'll make the ad hominem note that a lot of people most critical of math teaching are flunking math because: 1. they aren't doing the work 2. they aren't coming to class 3. they get caught cheating (the most common way to fail my course last semester). Which is not to say that teaching isn't bad sometimes; but there are also a lot of people who just won't do the work and then act angry and blame the teacher when the natural results follow.
Obviously teaching varies from place to place. The comment that at major universities the professors are mostly concerned with grad students and research is probably true for a significant number of professors---but not all professors for all courses. This is worse the lower the course: it's a cold truth, but very few people with a PhD relish teaching high-school level math. In fact, even at a major university I'd wager very few people teaching such courses have a PhD.
On the other hand, although I've known some pretty dedicated community college teachers, the chances of getting a decent course in multivariable calculus or differential equations much less abstract algebra or analysis at a community college is not good. I'd say make up any pre-college level math at a community college and then switch to a university if you want to follow it through or past calculus.
For someone who is self motivated and just needs a refresher on rusty math skills in algebra, self-study with any decent text-book should also work. Just make yourself work problems with the discipline you would if you had to turn them in.
Lot's of people don't want to (or can't) work. In my experience policies that allow students to get help from other students end up being abused. I'm currently teaching a numerical analysis class in which the assignments are half programming and half math. Students think nothing of copying another student's code or downloading some code found via a web search. My stated policy was that they could discuss general methods of solution but not duplicate another person's code. Unfortunately this leaves a grey area that makes it harder to enforce a consistent policy. ("But Professor, I didn't copy. My code is similar to Steve's because he explained to me how Romberg integration works.") In the future I will require that they work completely independently.
On the other hand, if the policy at Georgia Tech really does prevent students from referencing anything outside the course materials, that is ridiculous. Students who take the initiative to do a little outside research should be encouraged.
I don't believe it is quite that simple. Adaptive noise cancellation uses a reference microphone (e.g. a microphone on the headband, or someplace away from the ear), two microphones in the ear and an adaptive filter. The filter processes the signal from the reference microphone and adds it to the signal going out through the speakers. The filter is tuned (e.g. using the LMS algorithm) to try to drive the signal at the ear microphones to zero.
Without the reference signal and the adaptive filter, this won't work too well. You need something to cancel the signal with (the reference signal) and a way to adjust the cancellation (to take into account changes in the phase of the signal.)
This might also be a senior level undergrad project if the undergrad in question was really strong on DSP. But it's not just a lowpass filter.
Except that the industry standard rats nest in engineering and science is Fortran. Compatability with people writing GUI programs is less important than the fact that F90 compiles the enormous base of F77 numerical code. Also F90 has features of special interest to those writing numerical code. With the features for handling arrays, F90 matrix code is very easy to write.