[HN Gopher] Hacker's Guide to Numerical Analysis
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Hacker's Guide to Numerical Analysis
Author : bollu
Score : 223 points
Date : 2021-03-19 17:34 UTC (5 hours ago)
(HTM) web link (bollu.github.io)
(TXT) w3m dump (bollu.github.io)
| apbytes wrote:
| In the derivation for conditioning what is `O(dx^2)` and why is
| it there? Looks something like error margin?
| sampo wrote:
| https://en.wikipedia.org/wiki/Big_O_notation#Infinitesimal_a...
| ubavic wrote:
| Digression: for all authors that want minimalistic website,
| please don't push it too hard, as author of this website did.
| Color scheme is nice, as is font selection, but it lacks
| navigation links. Putting simple link to website home would do so
| much for explorers of site.
| leephillips wrote:
| Since I dumped on the author in another comment, let me defend
| him here: going up one level in the URL is really not a
| problem. It's even easier if you use something like Vimperator,
| where you just need to type "gu". And there is a nice table of
| contents.
| SavantIdiot wrote:
| "It is not transparent to me why this definition is sensible."
|
| Because if I have a six digit meter and I measure 5.0000 volts I
| know that it is 5.0000 volts. I don't see why that is confusing
| at all.
|
| "Here, yyy has correct one and three significant digits relative
| to xxx, but incorrect 2 significant digits, since the truncation
| at x2x_2x2 and y2y_2y2 do not agree even to the first significant
| digit. "
|
| This doesn't make sense. Y has the correct one?
|
| I think you might be confused about the purpose of significant
| digits. SD are intended to be paired with a tolerance. If there
| is no tolerence quote with the number, it is meaningless to talk
| about significant digits. You arbitrarily defined the tolerance
| as +/0.5 (half a unit). That is not always true, almost never
| true. Consider test equipment, not every multimeter is going to
| be +/0.5mV, some might be +/-.3uV. It varies.
| leephillips wrote:
| Yes, I think all we've learned from this article is that the
| author never learned about significant digits.
| PhaseLockk wrote:
| Exactly this. It may be useful for OP to consider what the
| hypothetical opposite of significant digits would be:
| insignificant digits, ie. digits that tell you nothing. A
| leading zero can be omitted from a number without any loss of
| information. A trailing zero, when correctly used in a position
| that is not below the tolerance, gives you significant
| information. On the other hand, if you start combine numbers
| with different tolerances, but do not truncate the result, you
| will have a bunch of trailing digits that are meaningless in
| practice because they are below the combined tolerance.
| taeric wrote:
| I think you missed what they found weird. The subject for that
| was that 0.0491 only has three significant digits. It is not
| obvious why, if you don't normalize the number.
| Trombone12 wrote:
| No, what is weird is that the original author seems to think
| you can look at a number and know how many significant digits
| it has.
|
| This is almost completely wrong: the number of significant
| digits is something given by metadata, you cannot look at a
| number and _know_ that it has been properly rounded to the
| correct number of significant digits. You always need to be
| told this special circumstance because most numbers you
| encounter have in fact not been properly rounded.
| jhallenworld wrote:
| In the world of general chemistry, the significant figures
| rule is assumed. It's not a great rule because in this
| case: .000300 you have 3 significant
| figures.
|
| But in this case: 300000 you could
| have 1 to 6 significant figures.
|
| The convention to deal with this is always to use
| scientific notation: 3.00x10^5 has three
| significant figures.
| taeric wrote:
| You lost me on this one. :(
|
| There is the odd case of trailing zeroes in non decimal
| numbers, but otherwise if a number is written, when is that
| not assumed to be significant?
|
| That is, there are fairly standard rules for conveying
| significant digits, I thought?
| alisonkisk wrote:
| It's assumed that you shouldn't write down insignificant
| digits.
| wsh wrote:
| I agree that it's better to state the uncertainty [1] together
| with the measurement, but when this isn't done, there is a
| standard convention for significant digits, explained in ISO
| 80000-1:2009 [2], subclause 7.3.4: When a
| number is given without any further information, it is
| generally interpreted so that the last digit is rounded
| with a rounding range equal to 1 in the last digit (see
| Annex B). Thus, for example, the number 401 008 is
| generally assumed to represent a value between 401 007,5
| and 401 008,5. In this case, the maximum magnitude of the
| error in the number 401 008 is 0,5.
|
| This is consistent with the author's "less than half a unit."
|
| [1] JCGM 100:2008, _Evaluation of measurement data -- Guide to
| the expression of uncertainty in measurement_ ,
| https://www.iso.org/sites/JCGM/GUM-JCGM100.htm
|
| [2] ISO 80000-1:2009, _Quantities and units -- Part 1: General_
| SavantIdiot wrote:
| Thanks for the ISO link, that is incredibly useful.
| ticklemyelmo wrote:
| I'm not entirely clear on the author's thinking. But consider
| the case of watching a real world voltmeter read 1.02v, 1.01v,
| 1.00v, 0.99v. Did we really lose a digit of precision there?
| The leading digit transition between 1 and 9 is a boundary
| condition that messes up that rule.
| taeric wrote:
| I think the catch is that you aren't keeping the number
| normalized. Significant digits somewhat assume scientific
| notation. So that last number is 9.9e-1. Two digits.
| ska wrote:
| You didn't lose a digit, you wrote it that way. If your
| instrument has 3 digits, you should have written 0.990v
| alisonkisk wrote:
| You can't make your meter add an extra digit (unless you
| can turn the dial to a different scale). The digit is lost.
|
| But of course it isn't a big deal to lose that dgiti
| because a trailing digit is 0.1 as much error fraction when
| the leading digit is 9 vs 1.
| ska wrote:
| I'm not sure what you mean here, probably because I
| worded it poorly. Of course your instrument will have
| different scales, and you should use the appropriate one.
|
| But the measurement was done with a number of significant
| figures, this is independent of how you write it. Nothing
| is "lost", if measurement had 2 sig figs it is 0.99, if
| it had 3 it was 0.990. Or 0.991 or whatever.
|
| Am I missing something?
| taeric wrote:
| You skipped the example.
|
| They said if your device measures 1.00, it is three
| significant digits. If it measures 0.01, it is one
| significant digit. From the same measuring device.
|
| So it is counter intuitive that the measurement just
| going down below 1 somehow causes it to "lose"
| significant digits. It didn't lost precision, though.
| Just the number of digits that are significant.
| yudlejoza wrote:
| When it comes to measurement, 1.7320 is not the same as 1.732
|
| The trailing zero was part of the measurement and, hence, is
| included in the significant digit count.
|
| That also means you can't just measure 1.7320, and report
| 1.73200 instead (or 1.732). You have to report only, and
| exactly, what you measured.
|
| (I'm ignoring rounding in what I said).
| singhrac wrote:
| Since it was linked, Herbie will help you find numerically stable
| versions of your expression: https://herbie.uwplse.org/
| jphoward wrote:
| This is such a great example of how people that know math(s)
| really struggle to explain things to people who don't, because
| they struggle to realise how much lingo they are using.
|
| The first point explains something as basic as absolute and
| relative errors. Great.
|
| 1 page down (s)he's casually thrown in function notation with R
| representing the real numbers with no explanation. Anyone who
| isn't familiar with this notation is immediately intimidated. But
| I'd wager almost anyone who is familiar with it didn't need to
| read the page before it. But also, I think it's just unnecessary
| to be this specific in the first place, if you're trying to
| quickly explain concepts.
|
| It's actually a huge problem in maths and even Wikipedia (I feel)
| is pretty guilty of this.
|
| If you take a (relatively) key mathematical concept, such as the
| Taylor series, then the opening gambit is completely
| incomprehensible to anyone who isn't already knowledgeable in
| maths: https://en.wikipedia.org/wiki/Taylor_series
|
| This, to me, seems in huge contrast with other scientific fields.
| Medicine wikipedia, for example, caters towards the "average
| reader" first, and doctors later. This is entirely appropriate
| for an encyclopedia. For example:
| https://en.wikipedia.org/wiki/AV_nodal_reentrant_tachycardia...
| epsilonclose wrote:
| Is encountering function notation and the symbol for the set of
| real numbers all that different from encountering unfamiliar
| vocabulary? It is slightly harder to Google for notation, but
| otherwise I do not see a major difference.
|
| In the Wikipedia page you linked, I immediately encounter
| several terms that I do not know the meaning of. I must either
| click through or search separately to fill in my own gaps.
|
| Your criticism is not totally disagreeable, but we have to
| wonder where to draw the line. Is a "Hacker's Guide to
| Numerical Analysis" obligated to hold your hand all the way
| through basic notation? The notation that you mention in your
| comment are the "Hello world" of mathematics.
| elmomle wrote:
| I have found undefined notation to be the bane of my
| existence when trying to teach myself a field of math. How do
| you Google mathematical notation? It can be a giant and
| opaque barrier to deepening one's knowledge when studying the
| subject on one's own.
| jaykru wrote:
| Maybe this will help:
| https://detexify.kirelabs.org/classify.html Then you can
| e.g. use the name of the TeX command for the symbol to
| Google its meaning (Stackexchanges are often full of
| questions looking for the TeX command for a symbol, so you
| could use these to infer in the other direction)
| ZephyrBlu wrote:
| Strongly agree. It's very difficult to work out what a
| symbol means if you've never seen it before.
|
| In saying that, in general I've found it's not too bad.
| Mostly greeks and the occasional symbol like sigma along
| with some set notation.
| marzullo wrote:
| Off-topic, but interesting to see my former heart condition
| randomly linked on HN!
| howenterprisey wrote:
| I run into that same problem a lot while trying to find
| tutorials that target beginners. Absolute beginners, the kind
| showing up to my organization's Google Summer of Code chat
| rooms. It's rare to find tutorials that just keep it simple
| enough.
|
| I wanted a regex one recently. First off, they all presumed
| mastery of the concept of a string, but more seriously, none of
| them had a really good definition of a regex! I still can't
| think of one myself, but I know it's probably not "pattern to
| match specific strings", because it uses an uncommon meaning of
| "pattern" and a jargon meaning of "match". And of course they
| all immediately used terms like "formal languages".
|
| The perldoc page 'perlretut' did pretty well with "a template
| that is used to determine if a string has certain
| characteristics" - ok, a bit abstract for my taste, but let's
| keep going - then a bit further down they use "modifier" (flag)
| with no definition. But I really had to look hard for flaws in
| that one. I guess the only way to actually nail this is to
| supervise successive groups of students as they try to learn
| using your tutorial-in-progress...
| moehm wrote:
| I explained regexp to my mum as a "single character string,
| which has the power to describe an amount of various,
| different character strings".
|
| And then an example like "Look, you have your Word document
| and you are looking for ..."
| 1ceaham wrote:
| One comment regarding Wikipedia: you may be aware of the
| "Simple English" language option, which can help with making
| some pages more accessible. It's in the language menu, or you
| can just change the language prefix in the URL, such that your
| example above becomes
| https://simple.m.wikipedia.org/wiki/Taylor_series. Granted,
| it's not immediately obvious that it's there, but perhaps it's
| one reason that the main article can be more precise off the
| bat rather than explanatory. Furthermore, I wonder if Wikipedia
| math editors are less concerned with the notion of an
| encyclopedia as learning device and treat an article as a
| canonical overview of a topic, with all of the baggage that
| comes with that.
| pixel_fcker wrote:
| I've been using wikipedia for years and never knew about
| this. I wish there was a prominent link on the page "Switch
| to simplified explanation" or something. As someone who knows
| barely enough maths to get by this would have helped me
| immensely over the years.
| _hl_ wrote:
| On the other hand, as someone who knows a bit of math, I love
| those Wikipedia articles that get straight to the point, and
| hate those where I first have to dig through a pile of (to me)
| useless "intuition".
|
| I'm not sure if it is really possible to learn/teach math (like
| actual math, not just some loose intuitions, because those
| don't really help you with anything) without going through
| notation and prior knowledge. Notation exists for a reason.
| ZephyrBlu wrote:
| I've found Wikipedia quite hit or miss. Sometimes it's
| straight to the point, other times it's impenetrable and I've
| had to do more research.
| rarefied_tomato wrote:
| For a deeper look, Trefethen's Numerical Linear Algebra starts
| with these ideas and applies them to matrix algorithms.
| bollu wrote:
| Thank you for the reference!
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(page generated 2021-03-19 23:00 UTC)