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COSC 4364 Numerical Methods - Lecture01_Floating-Point_Number_representation
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00:04 Okay, figured it out. So to this class. So I'm Leonard
00:12 , the instructor structure is a student and that helped me ta of
00:20 So first I'll talk to you a bit about the syllabus or the course
00:26 and talkative sort of start the but the power, So this first
00:37 of slides is basically thrown. The tried to make it actually work
00:47 So, so I also sent out email, um two zoom in terms
00:57 remote taxes and then I will also the classes and post the video after
01:06 at one point or that said somewhere . Um so mhm that's, and
01:19 for the first two weeks because of and soft opening, then obviously this
01:25 is face to face as well as taxes and so will be next
01:31 but monday next week and the following for the sort of zoom. So
01:39 one lecture a week in the first weeks because the current rules and we
01:46 back to normal then from then I will be here for every lecture
01:54 will still remain criminals access to all nutrition as well. Yes questions.
02:08 it will be the same link. a moment. Um, it's posted
02:13 blackboard. Yeah, so the syllabus all this stuff from this first,
02:19 tennis slides, I will show you it comes to the overview that is
02:23 the syllabus posted on blackboard as well the official, one of the core
02:28 that you get that so hard today we upload syllabus and stuff.
02:38 Okay, and the slides are posted left for performance and the video is
02:48 this other website. Video points for to see the best fights somewhere.
02:58 . Okay, so, so a bit to guns, what is the
03:07 about is an introductory course? So the first several lectures, I'm sure
03:16 will recognize different amounts of it from I've taken to court. So let's
03:25 that just to kind of remind you little bit what american methods do mind
03:34 , that kind of deals with numerical representation and approximations in different
03:42 So and that's sort of the whole . And we talk about the consequences
03:47 these approximations as well as some, know, standard american methods that they
03:53 for problem solving some floating point numbers be kind of the basis for everything
04:02 do. And they are kind it is for the pain because there
04:08 because of the approximations and there are as easy to understand us in figures
04:15 have to work with, but but for most practical things you
04:23 So, and part of it is approximation what happens is because of but
04:32 position in computers, another approximations to as that the methods themselves, you
04:40 the problem formulation. So there are proximal approximation, not just the number
04:43 representation. So we'll talk both about things. Um So this is a
04:53 bit so wise useful. Well, I mentioned pretty much anything. One
04:58 up doing one and seven needing to broken point numbers. Now the difference
05:06 floating point numbers and integral postal know that. That's not perfect over.
05:12 it's kind of a trade off or you used when you have limited number
05:20 states or digits to represent something. a tradeoff between position and dynamic range
05:27 you don't have but the same So um, part of it talk
05:37 this in a few contexts and for matters but also work. We tried
05:43 teach you somewhat tools and protector if don't already know it. They were
05:49 my flat putting the assignments. The for math lab is that would claim
05:58 most commonly used tools after not only academia but industry everywhere there are other
06:06 , but the method is very It's used and it's kind of a
06:09 language designed for linear algebra and most , the particular topics is the laundry
06:22 . So it corresponds to chapters in book that we will be using is
06:28 very commonly used books. It's a bit old at this point, but
06:33 kind of fairly well written. It a few bucks unfortunately is still,
06:41 there's still a book that is tailored , I was a computer scientist,
06:50 totally focused on the mat, but is sort of programming exercises as well
06:56 the boat and if, so that's picture of the front cover of the
07:04 and it will get the books or there's the rest in one way or
07:10 sections of the book as you pleased Yes, that's for the books.
07:17 terms of math lab, there's also versions provided by the companies or one
07:23 not necessarily need to download and install doctor. But you can download and
07:29 the software from you always software website should have access. Thank you.
07:35 downloaded tomorrow. Mm hmm. So idea is to have sort of one
07:46 for chapter of the book and the will be some combination of some simple
07:56 that program or some pencil and paper extends to just do some variation by
08:05 . And also has said slides and be on blackboard. The leader recordings
08:13 be constant work. We also just for interaction students discussions, You
08:22 what else? one midterm and final and the grading policy. Yes,
08:27 associates and this usual thing that you in every class. Academic honesty
08:37 We tried to check that it's not much copying going on. See then
08:47 probably have seen this in most Well, we'll see if you haven't
08:52 that yet. But you signed up the corvids. Some of the coverage
08:57 used and probably have the address. , then I made this preliminary schedule
09:09 so you can see what's expected to . Hopefully it will managed to keep
09:17 it's been a really small one. it's just it's a little bit,
09:22 of the things follows the chapter order the book but not everything does something
09:27 will just junk a little bit and I cover anything that is not in
09:32 books you will get something else about . But most of the things that
09:37 will be, oh The next one sort of this assignment schedules want to
09:49 things one way sometimes and as I be somewhat more expensive and I get
09:57 time but pretty much something every week spring break and probably a little bit
10:07 I think that was what I had terms of that general kind of
10:14 So additional questions, the process of to cover From or they could be
10:28 or 100 um mostly 100 tests them some very simple. That's that
10:37 Right. Mhm. The commitments yes, I remember that, it's
10:44 two years before for this past so think in general, so that's not
10:48 programming class per six. So it's that it will be expected to write
10:54 the code so that so the emphasis understanding the american procedures or the approximations
11:04 what gets so got some practice they use most likely for the kind of
11:11 to do in this class to reduce or some other tool that provides you
11:16 the procedures that gives you do solving or something and all approximations or
11:23 They wouldn't write the code. So point is we live in this class
11:29 to understand how can you trust your ? It will give you some
11:33 but how accurate is it? So lot of it is the emphasis to
11:37 you aware what accuracy you can A number of things can go
11:43 So it makes you sort of conscientious not blindly trust whatever comes up of
11:48 . Such a move. So sometimes not, you shouldn't expect neither of
11:56 or assignment that things only have to pages and pages of code. You
12:02 , there should be many in the , just a few lines of cold
12:07 . I was saying morning, maybe few tens of them. Thanks want
12:16 comment on that. Yeah, I say the same thing as for each
12:20 , it's going to be mostly lines of quotes. And the reason
12:24 mostly as for you to understand what's on in the code for more on
12:31 . You will get a little bit of providing reasons for what you understood
12:36 the whole exercise. There's only partial for writing the code and getting,
12:42 more about your understanding what's going on the, if you're getting high rollers
12:46 lawyers, you know the reason when take courses, you know, we
12:54 you when you start working, there's greeting. So you have to figure
13:01 the self test or self assess, know, I can really trust what
13:05 think what I do. That's sort the new point I'm taking for the
13:10 and that's why we also tried to to understand what happened to a way
13:16 reasoning about double check this out. this makes sense. Okay. Any
13:27 questions? Okay. So we'll switch to do the other and to slides
13:36 should be here somewhere. Mhm Yeah. Mm hmm. No.
14:03 , good. But mhm wow, hmm, mm hmm. Of
14:22 Mhm. No. Yeah. And that don't know. All right.
14:33 . Right. Yeah. Okay. , alright. So okay. So
14:54 topic, so the topic today is floating point number representation that you may
15:01 familiar with already but in case you're , I was trying to cover
15:07 Mhm. So most of the concepts . But it's important to the effect
15:17 which is this notion of significant And then we talked about some of
15:25 different positions and over and on the . That is important. Some of
15:35 . So typical the way things are is what is usually known as place
15:42 notation. So as we know where numbers receiving decimal numbers is positioned in
15:48 string of digits as a particular ah or value associated with it. That's
15:57 they did it. Um, so right and the strength uh is the
16:03 that since the list manager to the . So Yes um take 1
16:10 3. That's a 123 But three best for that Brand zero Tonight and
16:17 next month is you know the number factors, General powers of 10 and
16:25 next one is the number of Of of the factor to the power of
16:31 or 100. So each position in string corresponds to some power of
16:37 Let me look at the decimal number . And so they give you try
16:41 get 0-9 for each position. So the standard things that I used to
16:46 and I'm sure you're also familiar in other sides when it comes to my
16:52 instead of The small digiscents by heritage of zero but it's the same idea
16:57 the position in the string tells That's particularly. Yeah. And the
17:08 . This is a position that there's still the number of digits that you
17:13 trust in northern leading zeros. I don't rent out for leading zeros but
17:21 that can happen sometimes substitue happens. so now a little bit on the
17:32 . Oh the number of the just that they do have. It's a
17:38 trivial example of this from the So you can see it. So
17:43 it's a little system of equation two to equations. And somewhere in your
17:51 school math. You learn how to the system of equations um because you
17:56 your skills and determination successive elimination procedure the way that works is to multiply
18:04 the first equation with someone factors so multiplication of the first equation for that
18:10 and you subjected in this case from second one and what you end up
18:14 there's something where you only have one left and then we can solve for
18:20 in this case of it. So example says well let's start it up
18:26 four days just perception four digits in case started to the right of it
18:31 small point. Now I suppose in somebody only gave you to work for
18:39 so acquisition so this is a related and in this case it was correctly
18:48 and we'll talk about rounding a little but this is a typical way you
18:52 if it's same higher than the midpoint you round up and it's it's slower
18:59 the midpoint turn you round down and it's at the midpoint. Yeah we
19:04 procedures to deal with. So in case you can take this like before
19:11 numbers and sentiments of three with the rounding. You got this set of
19:16 that donation you get this thing and got this one nothing that mhm nah
19:27 you in fact we're allowed to use four digits and take them as they
19:33 then I get a little bit messy terms of the numbers, but the
19:37 procedure and you know, figure out to what the multiplier is and do
19:40 subtraction now you get to public sold , I think you have three digits
19:47 ended up being -547. And if have four digits It's just that plus
19:55 affordable. So why do you kind Drop down from four days? Just
20:03 give it to you And just use ? Not a single digit. And
20:08 result was perhaps so that's sort of point of being aware of the significance
20:17 the number of digital work and not trying to plug something yet depending upon
20:23 you may declare something in your program cold. But then it's a question
20:28 happens underneath. So something can go wrong. We talked much later in
20:36 course about how you can get a for if this is likely to
20:43 So it has to deal with the to the numbers and what's called.
20:52 , ill uh ill conditioned systems, condition systems or systems where these things
21:00 have to Okay, so, and is if you actually have more digits
21:07 this case, assuming that everything. in this case, you know,
21:14 . So in that case you don't know what follows after the six in
21:18 case, assume that there is all . But for the doing the application
21:23 additions, you would actually work Mm hmm. Let the slightly different
21:29 compared to three. The difference is technology. Yes. For this boss
21:40 difference. So this is the service posted email post or systems depending upon
21:47 properties of the conditions in the system the questions. So any questions on
21:57 , How you find out this complicated . So you know that way towards
22:03 end of the course, understanding when systems also enough. Mhm some fairly
22:18 straightforward things about absolute and relative So those are important to know.
22:25 in particular is very clear whether relative or the absolute terrors and most of
22:35 time it's the relative areas that maybe . But sometimes the absolute terrorists are
22:44 the relative embarrassing. You know, it you know the right to answer
22:48 a few percentage points of what's the you get or is it somewhere absolute
22:58 . So this is just to the of in my mother in some
23:04 Absolutely. They're so small relative. um, no. So yes,
23:15 just china straightforward example. Something I guess that's stress the difference.
23:25 , absolute terrorists. Very large. hmm. So I am mhm.
23:39 different physical and in terms of the between decimal place is gorgeous digits isn't
23:48 in this case means the number of just after the just another binary points
23:54 this binary number systems. Whereas in of the they just themselves, you
24:00 look at the full number you know the just to the left to
24:05 Yes my report and it's just the track on which ones when we're talking
24:11 . Many times as you can see the next several slides I want to
24:15 a normalized representation in which case pretty either you have zero or one to
24:24 left of the decimal or binary points then they have a bunch of days
24:30 the decimal about many points on It's not a big difference between what's
24:38 so talking about accuracy in terms of smell diggers or the total number of
24:44 pickups. Again it's normalized such that for a while. So there's a
24:56 of concept that you should be aware exactly what they mean because it's important
25:02 particular activity papers and stuff like that that's what typically ask some of these
25:09 midterms and you know what the concepts and what they need. Ah so
25:17 number representation assisted place failures have talked already and In that case one has
25:24 . That's more points. So they leading sort of intricate numbers and then
25:28 fraction that follows the decimal point now I mentioned, typically once tried to
25:35 some normalized representation in which case specified in the terms of the decimal number
25:43 that have uh leading zero to the of the decimal point and then have
25:49 bunch of digits and forget the number want then you have not provided by
25:55 The power of 10 with the shipment course left or right of the decimal
26:03 . Mm hmm. So and or binary number system. That is what's
26:16 used in most computers today. There other ways of dealing with them
26:21 But pretty much anything you have used will be using uselessly by the system
26:29 the computer systems from that place The are zero and 1 numbers and
26:39 this notion that is normalized a while following this notion that to the left
26:45 the decimal or by her point in case it's zero and then there are
26:50 . So once following mm hmm by points in this case buy cheap believe
26:57 point standard about covering next two It's slightly different than how it normalizes
27:05 . The other things in all. the concept of anticipation? Exponents Samantha
27:12 part of the number that is for right of the decimal or binary point
27:18 the exponents is the value to which base and the number system is going
27:23 be raised to get the correct numbers they want to represent. So it
27:30 the base 10 number system. Dr number system is Whatever the power of
27:36 that's a month's supply. The number the Manchester with in order to get
27:41 correct if it's a binary number systems too. And the power can be
27:49 positive or negative depending on how you to shift the number make it so
27:54 left to make it a larger number shifted right to make this morning right
28:05 questions. And that's what Yeah. as I'm sure we all know,
28:20 , and it is, you data type to use them in the
28:22 language corresponds to a certain number of in the computer system. So it's
28:28 string of zeros. Someone and they to use it integers are long integers
28:33 floats or upper floors double some such . That is, it defines so
28:39 bits you are going to use or be used border of sensational the numbers
28:46 you have. And because of that means you can't represent every number
28:57 computer system. Just a limited set numbers. And depending upon of your
29:02 floating point or different sets of numbers can be represented in the system.
29:08 irrational numbers can often represented correctly. without this because their best have an
29:16 string of religious or whatever space you . So we never terminates. So
29:21 can't represent irrational numbers and sometimes the are too large. So you don't
29:29 enough, let's say in the computer to represent them. And sometimes because
29:35 numbers are also too small. So they get overloaded And computers that do
29:42 ugly standards. Yeah. Um, system will report to you whether it's
29:47 the pillow off. Oh, that's or some other 2nd No. That
29:53 work. That's not the number. overflow. Underflow And knowing that just
30:01 finite number but our numbers that you represent is important as well as multiple
30:09 the machine. Absalon, basically the between The two smallest numbers.
30:19 repair smallest number seven you can represent won the number and then to try
30:25 get the next number up whatever it , the distance between these two numbers
30:30 the machine. So if you have come back to that as an
30:39 So, mm hmm. So here's little bit of it perhaps and somewhat
30:48 behavior of floating point numbers if you thought about it. So let's take
30:53 look. That's a trivial example where have three blackberry digits right over
31:01 I might have a point and then have just one bit for the
31:06 But it can be So it can -1 there or for what? It's
31:10 car long each one of the potential Values -1, 01. And then
31:18 have all the different permutations of do want me to be three district?
31:23 a possible numbers from the contessa and they have three different varies depending upon
31:31 fact. So, so one thing notice is um, that there is
31:42 overlap between what is different columns Um, so for instance, this
31:53 you know In reality is sort of . This is also one of the
31:59 . So depending on the some numbers representative in more than one way In
32:05 case. So even though the 24 numbers here, It doesn't know that
32:11 distinct numbers. So and the other , depending upon what day exponents is
32:21 distance between successive members. When you the bits in the contessa, it's
32:27 the same. So you can see ceremony for the first numbers here.
32:33 must be one of the 16 Difference the numbers and next one is basically
32:38 . So there are further separated and I have this that's part of ah
32:53 properties of the floating point numbers that kind of the trade off. Unlike
32:58 integration you just have bigger numbers, don't have anything with the master fire
33:04 spawn it. So in that case time you fit the last bit it's
33:09 incremental but they fixed them up which not the case of the campus because
33:14 exponents kind of tells you what the is. And the number starts.
33:23 other ones that can put us here that You cannot get arbitrary imposed to
33:28 . So we sometimes call it the and zero Because that's the smallest number
33:36 . Yes. So what's what's the a little bit about the stone in
33:42 numbers. Thanks. So I guess are again I said we're talking about
33:52 numbers before and then the normalized Gentlemen with the binary representation. That
33:59 that the leading after the decimal points be a one. Which is not
34:05 case for this first important numbers would think. Also it's numbers according to
34:12 notion that believing did you So you explain in that case they should be
34:18 and then just the experiment in order get there. I'm very naughty.
34:24 as a normal X number. So what computers most computers do but there's
34:37 guarantee. So sometimes if you don't something processor chip from intel in the
34:45 and they ah then you may have check whether it's actually support that.
34:53 should be looking for justification but most the GPS were late in supporting electrically
35:01 from standard people. So then depending the number of it's that you use
35:15 the representation. So in terms of actual refloating with standard, the notion
35:20 have binary because the final numbers and there is a number that follows that
35:27 the number of bits that's being used the representation. So The binary of
35:32 is corresponds to float in sea or single position whatever you want to call
35:38 . and by no 64 is double in recent years. Ah half precision
35:48 also become quite popular and mostly because machine government. So I thought there
35:55 a comment more about that soon um the machine learning sometimes position is not
36:06 important at least for certain stages. in that case you don't want to
36:11 the memory space computer effort. Also energy required to do the computation.
36:19 you don't really want to use more that we need community countries. So
36:28 below here is sort of the format by Tripoli and other bits are being
36:36 . All formats have designed it with numbers but then the rest of the
36:42 and they stayed for that 16 sort wonders for the sign and then you're
36:46 15 And the standard decided to use for the exponents and then for the
36:54 of socks and that sort of Ah The minimum and maximum values you
37:02 represent ah is in terms of the numbers will come back. So so
37:15 is in terms of there is also standard. We'll talk about that next
37:23 for two. That is known as flow succeed just question Is the number
37:32 qualified by either two or 10. . So the Manchester is the digits
37:41 the right though sort of decimal And the decimal number system was a
37:44 point. And and then it's multiply Based on the number system 10 or
37:53 to some power that is determined by expo, how is determined by the
37:58 covered yesterday because it still it gets when it comes to attribute but that's
38:06 of the card, you just have figure out that interpret with it in
38:10 experiment. Ah So I started to I'll talk about that. So some
38:19 you may get too many or There's something called the flow 16.
38:27 is the format introduced by for their . So for that they decided that
38:38 experiment it kind of is predominantly defining range because that's amount about how much
38:47 want to buy the number out Now. That's for machine learning.
38:53 precision which is determined by the mantis that anticipates if you have more or
39:00 the more bids you have, the is sort of the distance between success
39:04 numbers that you can represent. So decided that the position was less important
39:11 the dynamic bridge. I'll come back that. So they Still 16 bits
39:18 they have a different division between the and marty um As you can see
39:26 I went to single position the double , the dynamic range did not in
39:34 activity standard wasn't increased all that much the position was so I was a
39:44 off that was typically and driven by was needed in scientific and engineering
39:53 So that was ended up to you standards people decide and then there are
40:00 in this position whether for a position terms of how the bits are used
40:07 between uh what's the range of the you can do And the precision of
40:12 # seven. So now we'll come to the question in terms of my
40:26 , that explanation there's a little bit intricate I guess that's interprets. So
40:35 one doesn't take the bits directly and what the experiment is. So in
40:42 to represent both negative values in the conservative values, €1 sports known as
40:52 called excess representations or thanks whatever the are and they feel then there's the
41:01 , the number that is kind of midpoint rate To get something for
41:06 something two plus the gap in front the actual despondent. So here's the
41:12 of the bit string and the exponents then he computes the value by subtracting
41:18 particular number. That is, that's the number of its point respondents.
41:23 midpoint -1. So in this case in a position. Yes. The
41:31 is true to this. There were bits in the ah His positive yes
41:43 32 bits Let's say eight bits. 8.1 and seven. and so that's
41:50 is the shift next year the number miners want server. And to
41:59 And uh so the other point is carbon normalization is done in this
42:09 The idea of electric police standard is normal such that the legion leading ah
42:19 digit to the left of them by point. It's a what? So
42:26 example in scientific notation is normalized based the first digit to the right.
42:33 the binary adjustable parts of this it's the first digit to the
42:38 So that's what they're standing departments Since the standard defines that that's normal
42:47 number hasn't won in this place but decided to do it and I can't
42:57 represented. We know it's because it's . It's the rules for the normalized
43:03 they get shifted them so that they this one. So in that case
43:08 have But the three bits for single but the fraction class Jack should have
43:14 Vietnam. But it also means since is the one here. Is that
43:23 part. That means the first lady ? But the decimal point may not
43:30 , it depends on the life. . And then there's some things you
43:37 want to Be able to represent zero . So that means they there's certain
43:46 values That was used for special purposes zero, zero. Does not mean
43:54 . Then subtract 27 and gets The by the -1. That's not.
44:07 As well. So zero. And . What a stupid divide. So
44:14 a conspiracy. You can represent things first the minus infinity Sarah.
44:26 So and the corresponding things you figuring out the rain. Sorry.
44:35 now if you have mm hmm. since you're not allowed to do
44:41 So the smallest number of exponents have used for a number of observation other
44:45 zero Would be the one Based the , it was 127 so one was
44:53 gives you the smallest His pockets I -136. And similarly to 55 specials
45:01 their faces. Look it's before when 1 27. So you just want
45:05 itself, it's the largest export and you have just a bit this is
45:14 zero. It's okay as long as number digital someone. So and then
45:21 other end is all gonna taste the star one. So you may still
45:24 the right from 50 years of minus to the 23 which is the step
45:33 can figure out. So eventually you all these things out. Start to
45:37 forward. That means essentially if you single position numbers, we got in
45:46 decimal number system that we're used to about what values are and what they
45:51 , This is basically seven significant decimal . So if one needs more,
46:03 position is not good. So that's of one of the points of having
46:09 discussion to understand and things were used thinking about that tends to be that
46:14 small number system. How many decimal to regard for the different levels of
46:21 and your data. Thank you when comes to the floor. Mhm.
46:34 so a double positions as the much the solution of position and some of
46:44 direct So in this case we've got . Yeah but 15 significant. The
46:51 is actually the number of decimal Yeah but increases the position more.
46:58 it's kind of rule of thumb. is good. I mean you have
47:02 to do things single position roughly. decimal digits. Double position.
47:16 Alright. So this was the big . Yes. I already said that
47:22 decided. Mostly because of the machine , that position was less important than
47:28 range. So that kind of uh this partitioning of the Justin this imperative
47:41 . Listen, it's for the exporter uh Tripoli does for a single position
47:49 six and 3 That you Billy 65 here. But the new talk has
47:55 like the central position And that means seven bits left for the marchesa or
48:02 position. So, so that's kind sense. So the shifted the balance
48:10 economic branch conversation. Okay. Perhaps serve them well. Machine compensations
48:23 So the default 16 is now being also I mean right architecture for their
48:32 , what they call the tensor product . Tpu that the use of their
48:38 centers. It was so successful in learning. So now ain't that A
48:46 B. And NVIDIA also support 50 . So now it's super supportive on
48:55 platforms. Mm hmm. And at this is that's what the information.
49:03 has an IBM I think still at uncertain of their processes. Support
49:09 They had with more bits. for the manatees are very cheaply double
49:18 format. So there has been other there are still some formats but it's
49:24 to make you aware that everything is particularly standing up. Now. This
49:34 kind of just conference set over a in terms of Couple things are one
49:41 memory. Um, so, and you have logical problems that even the
49:53 between single and double position becomes important the doubling the amount of memory that
50:00 need, they won't consider the So if you don't really need
50:03 don't use it. And it also for energy consumption and more silicon has
50:11 this more memory. Um, for thematic units. So yes. And
50:23 questions are not from the formats. it's a gun. This format
50:33 And the position of range that we , the different forms of things attempted
50:40 either on assignments or it's important that have a gut feeling, uh,
50:46 the right code and what position you there are men can also have significant
50:59 the results you get sometimes. So not typically specified several rounding votes,
51:11 , you can often control um, type of rounding is being used.
51:21 the typical thing is to do is you around to kind of nearest and
51:26 make sure the midpoint right in between there's different things you can swap,
51:32 that Tripoli has, but you're fired other things, I think there's even
51:38 running boards, but these are the common one. So round to nearest
51:43 you can Either taste or even or away from zero was not uh directed
51:50 the goal. And this old table . Yes, for example, what
51:56 in the different grounding worlds values here this kind of the point, with
52:02 positive numbers and negative numbers, so this is all of this. So
52:09 you're not surprised to even if you from 8.11.5, while the closest even
52:16 is 12 soldiers round up. Um It's at 10.5, The closest even
52:25 is 12, so that is run . So it's not always that you
52:30 up or down, I think that's it is that taste, if the
52:35 is negative, then you know, even is mine as well. And
52:41 that place is also might as Now if you're on the way from
52:48 , then we'll run this up and the sun Because that's the way it
52:52 zero and in this case you're acting . So it has consequences for what
52:59 rounding is and eventually if they do lot of computation that affects the final
53:05 around 20, that means instead of up away from zero, store
53:11 so that the best for the trump the number six. The positive
53:15 And we have the negative number of in the Bronx. So I'll just
53:21 you examples of what happens depending on . Mm hmm. All right.
53:38 yes. So it's the other thing do it with a tiebreaking randomly select
53:44 one you are around things and will him as well. Okay. Mm
53:54 . Okay. So I requested on notion of arming and they're wearing all
54:02 arming and configured for it. You choose to play with it.
54:12 Another thing that is quite tricky unfortunately class of significance. There is a
54:22 examples of what happenings that means significant digits are the ones that you
54:27 trust and are correct. and if in this case of subtract two numbers
54:33 are private close to end up losing the greatest opposition. So in this
54:44 , but I guess that's significant digits . And cynics cynics. What's more
54:51 ideas of ex uh it is pretty access. So that's nice. These
55:00 cynics separately close to each other as can see in this particular case.
55:05 the first thing we did this year the same in both X and Sine
55:08 . And then there's the defense. to subtract these two numbers.
55:16 there's three leading zeros to the right that. That's my point. And
55:23 you normalize it then the best way shift this numbers you get something like
55:29 and then it's just the exponents of to get the correct number. But
55:35 means that this tree training, you're mhm They don't don't know what they
55:43 Doing is putting up to zero but don't really know what they are.
55:49 in this case the last three digits and this subtraction process. So that's
55:59 that sometimes can the result of some the bad results. Not done in
56:05 simple example model. What else would ? So sometimes there are unfortunately no
56:16 method. It takes good ideas to to figure out the limit the loss
56:19 position and your computation and I think the next side uh not quite ah
56:30 it will be a subsequent size. this is just a little exercise show
56:35 you to get away for you to how many this opposition you might do
56:43 get the bond that lower bond and bomb how many latest opposition to the
56:51 . And just manipulating this expression again subtraction and fraction of Pisa in this
56:59 the sooner the mantis to provide as of this and then it's normalized and
57:07 in this case binary. So that these two values that leading Um digits
57:13 the final value to the right of decimal point is the one. So
57:18 means our is at least 0.5% f it's no more than one. So
57:25 have arranged and then they can manipulate . That's for this type of expression
57:30 is on the right, you have the values are and the relative values
57:35 that shifted by ours on this one evaluate this expression. Let's see.
57:43 I think that that's on the next . So in this case we have
57:47 out what things are and in that it's yeah, I can see that
57:52 place is off. That was the position. So so ah so here
58:08 another example of mm hmm this is you lose position of subjection again.
58:22 if X is really small, so with the one plus for the
58:28 this prison what you want. So order to get a little bit better
58:35 not lose as much decision. One manipulate this. That's what everyone has
58:43 use. Kind of whatever tricks one remember in terms of expression to manipulate
58:49 . Two try to capture uh the value sector concerned about by itself some
59:00 this place. If you have this and multiply it by putting a plus
59:09 instead of the -1, then the is basically the wow ! This expression
59:17 minus. Oh the one so you basically expert comes up by yourself.
59:26 you multiply with this thing and then also divided by the same expression.
59:30 it's coming also since the first Here this is basically the one we read
59:37 . But if you evaluate this impression what you've got is X squared comes
59:42 by itself. So now I have but it's totally equivalent to this expression
59:49 now X squared does not get messed compared to one. It's all by
59:54 . And this is the reason why would rather because it's dropped it too
59:59 or less 1 1st 1. So that case you actually have X by
60:06 . So this is a while trying not lose so much prestige but it's
60:13 to this particular expression so they can't this general rule whatever expression I
60:20 But this doesn't work. Mhm. So yeah, so this is
60:33 Example to coming back to this one example I had before. So when
60:42 of those, a good suggestion how would do that to try to salvage
60:49 instead of doing it directly by the Synnex. Thank you. And it's
60:58 . Yeah. What Is that expression 1st? Mhm. I don't know
61:13 it takes place. It is. a serious inflation six. Mhm.
61:22 , it's ah Thank you. I . I think questions. Okay.
61:30 have another. Okay. Yes, , that's more than this.
61:40 exactly. That's what I want to that one where it is. So
61:44 this case they used the table series of science X. That has these
61:52 . Right? And keep going. that means in the case what you
61:57 now is but x minus this for access disappears on the other leading term
62:05 something directly in accent is divided. is my number so it does a
62:09 good position. So again you have be creative depending upon what the expression
62:17 . Yes sir, more or less numbers and it's a subtraction and try
62:22 figure some way of its closing the numbers so trying to get it.
62:34 hmm. What's that? Another This is because no one and that's
62:45 happened. Um not too uncommonly that you're cold you may end up having
62:54 by the argument of the trig functions not within sort of the first period
63:00 that's something sort of a large number then you can basically do an argument
63:05 production and basically evaluated in the first of function. So this kind of
63:11 to subtract out. If you're thinking of it, get the smaller and
63:16 accurate the trim function and so is , we won't play with it much
63:29 at all. But I wanted you be aware of it and it's,
63:34 can't remember if it's mentioned in the or not, but it's something that
63:39 wait software and computer systems for a time but not so much anymore.
63:49 there's been anyone known about this notion big and little indians. Yeah.
63:58 what it's all about is ah except computer systems interpret sort of the
64:10 other bits from right to left. means the right words, this is
64:14 one that is least significant and others it in the offices order so that
64:19 left most businesses is the least So that means if you take a
64:28 representation generated in a system that has little endian, take that bit strength
64:37 interpreted as if it was the big their number, you get their own
64:44 . So if you run cool and the vote the result of colds and
64:53 use that on a different computer Thanks. Family offered it off.
65:03 particularly important in these days, I say when crowds is the norm for
65:08 of users. There's one called the system and the other system and you
65:11 necessarily tell what's the computer system representation on that particular platform. But to
65:19 the data in between, it's a of the software that's to figure out
65:26 to hide and make up for this . So make sure you always get
65:30 correct results and some of these middleware is between whatever the compiler generates the
65:36 is and their execution and what happens the other hand, us too.
65:41 just for this nowadays, most systems now, the little indian system.
65:53 , so at this point it's not much of as the standard user using
65:58 computer system. It's not a big . But again, you should know
66:01 between your computer systems nowadays. It's such a big deal. Again because
66:07 systems used have agreed on the same the R. S. System.
66:15 what makes? Mm hmm. That's . Uh huh. So for
66:26 what's that bad clams some absolutely has number of representation that uh regardless of
66:36 of the computer system, you will the number to move data between different
66:41 . So that's the challenge. And would you read evaporate? Mm
66:47 That's an internal representation. Okay. hmm. Yes. So this is
66:53 this so um but okay. It's the past, you know, things
67:10 the fact that the interpretation of the and then data for different different computer
67:19 . Her heart was thank you. examples of things that happened and part
67:27 the and then there's a part of other issues and floating point representation that
67:36 go wrong. So thanks. So just an example. Pretty much ah
67:50 don't believe so just pointing out. . There's a number of issues,
67:59 parts as well as the number of . It's important to you understand the
68:07 of the range of numbers you can and the different formats that's available.
68:15 , mm hmm. So, so the next lecture we'll talk about
68:30 So how many have used that? spinners. Okay. Sorry, what
68:40 talking about is also not helpfully I'll you enough to the comfortable using that
68:48 up. Ah It's already the last of the brightest informer. Ah so
68:59 that's all I have in addition to is also very well validated tested,
69:08 , routines for doing all kinds of and I get all your problems and
69:14 of that. They're very, very . But it also has also
69:18 um, I think graphs and reports so it's pretty nice too. That
69:25 of the reason that will simplify also things, you know, report for
69:31 science and so and it's fine and breakfast you can comment on what for
69:47 rest of students, you know I think syntax is pretty similar to
69:57 enjoyable. Mostly confused with makeup. I spent playing along with matrix and
70:03 so they don't multiply major says you have the right to know differentness,
70:09 lobes of the, of the launch , symbolic to answer the matrices and
70:16 assignments and figures, helicopters doing the thing. So it's kind of often
70:22 as an array syntax. So you work with the race the same way
70:26 you work the scale hours and the language. This story Overloaded said something
70:34 front of the operators. I think much more correct. Okay.
70:46 Last question, last measurements and answer question about lots of significance.
70:54 So I noticed that a lot of lots and lots of significant stuff.
70:57 were just trying to get the X of some function in or just on
71:01 own. But when you did that was there was always like X squared
71:05 execute something like that. When x close to zero does not make it
71:08 smaller number. Wouldn't Yes. So , you can still end up there
71:15 in position that in this case you the smaller numbers but in the
71:22 the leading zero sports and they still , they add something squared. So
71:27 usually have enough digits in the representation the exponents. It's uh different.
71:34 it's less than one than this. higher negative exponents. Uh when the
71:40 we still have a number of but subjects. So to start with say
71:50 , months is a different single uh and when you square it you
71:55 have the 23 minutes. Um, then your shifts its form.
72:01 But yes, that's a good So it's, it has to be
72:05 . It's not. So let us us. I love you. Mm
72:16 . Yes. Would you like us have that lab in stone by the
72:20 lecture? Yeah, I think that be good. So there is then
72:26 just I had to fix. That's good point. There is also I
72:35 productive version that has happened. The of students have felt them not for
72:43 know like structure but for some assignments laptop maybe a little bit to show
72:49 not having enough memory or something. there are also online versions that you
72:54 need to do some things you can into the website and do things interact
73:00 bit. Mhm. Do we need license together in that lab?
73:08 there is the university has a You need to go through the steps
73:14 , if you look downloaded it from USa website shouldn't be history. Remember
73:20 are some steps we need to take get the license key and so
73:24 But it's right here in the virtual license if they go directly from,
73:32 . Mhm. Access to it and software illness. Right. I
73:38 yeah, one of the, you , icons is software and down there
73:42 someone that I think on the It wasn't obvious to me the first
73:47 we did just no, I don't about it sometimes. And the
73:55 Mhm. Yeah. Mhm. Any questions till monday for his own
74:08 The cost of the Lakers ah and next week back here. Mm
74:18 Yeah. Excuse me. Yes. the syllabus on blackboard, it's not
74:34 . It's mm hmm. I hope . Maybe just a couple of the
74:44 . That's it. Okay. Okay. No one's ended up
74:54 you know the test that it works . Can we know? It's
75:03 Can we email you to remind you that. Oh yeah, absolutely.
75:07 , absolutely. No problem. They should make sure that they find
75:14 photo. Yeah. So yes, they are to set up the I
75:23 something elder, you're also you can and we also post things on the
75:29 . Right? Why you can also with each other? Yes, we'll
75:37 that. I don't know if it's right now, but probably not.
75:41 I will make sure. Okay. huh, mm
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