via “Some disasters caused by numerical errors”:

On February 25, 1991, during the Gulf War, an American Patriot Missile battery in Dharan, Saudi Arabia, failed to track and intercept an incoming Iraqi Scud missile. The Scud struck an American Army barracks, killing 28 soldiers and injuring around 100 other people. A report of the General Accounting office, GAO/IMTEC-92-26, entitled Patriot Missile Defense: Software Problem Led to System Failure at Dhahran, Saudi Arabia reported on the cause of the failure. It turns out that the cause was an inaccurate calculation of the time since boot due to computer arithmetic errors. Specifically, the time in tenths of second as measured by the system's internal clock was multiplied by 1/10 to produce the time in seconds. This calculation was performed using a 24 bit fixed point register. In particular, the value 1/10, which has a non-terminating binary expansion, was chopped at 24 bits after the radix point. The small chopping error, when multiplied by the large number giving the time in tenths of a second, led to a significant error. Indeed, the Patriot battery had been up around 100 hours, and an easy calculation shows that the resulting time error due to the magnified chopping error was about 0.34 seconds.

(The number 1/10 equals

1/2⁴+1/2⁵+1/2⁸+1/2⁹+1/2¹²+1/2¹³+....

In other words, the binary expansion of 1/10 is

0.0001100110011001100110011001100....

Now the 24 bit register in the Patriot stored instead 0.00011001100110011001100 introducing an error of 0.0000000000000000000000011001100... binary, or about 0.000000095 decimal. Multiplying by the number of tenths of a second in 100 hours gives 0.000000095×100×60×60×10=0.34.)

A Scud travels at about 1,676 meters per second, and so travels more than half a kilometer in this time. This was far enough that the incoming Scud was outside the “range gate” that the Patriot tracked. Ironically, the fact that the bad time calculation had been improved in some parts of the code, but not all, contributed to the problem, since it meant that the inaccuracies did not cancel, as discussed here.

The following paragraph is excerpted from the GAO report. The range gate's prediction of where the Scud will next appear is a function of the Scud's known velocity and the time of the last radar detection. Velocity is a real number that can be expressed as a whole number and a decimal (e.g., 3750.2563...miles per hour). Time is kept continuously by the system's internal clock in tenths of seconds but is expressed as an integer or whole number (e.g., 32, 33, 34...). The longer the system has been running, the larger the number representing time. To predict where the Scud will next appear, both time and velocity must be expressed as real numbers. Because of the way the Patriot computer performs its calculations and the fact that its registers are only 24 bits long, the conversion of time from an integer to a real number cannot be any more precise than 24 bits. This conversion results in a loss of precision causing a less accurate time calculation. The effect of this inaccuracy on the range gate's calculation is directly proportional to the target's velocity and the length of the the system has been running. Consequently, performing the conversion after the Patriot has been running continuously for extended periods causes the range gate to shift away from the center of the target, making it less likely that the target, in this case a Scud, will be successfully intercepted.

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The March 13 issue of Science carried an article claiming, on the basis of a report from the General Accounting Office (GAO), that a “minute mathematical error … allowed an Iraqi Scud missile to slip through Patriot missile defenses a year ago and hit U.S. Army barracks in Dhahran, Saudi Arabia, killing 28 servicemen.” The article continues with a readable account of what happened. The article says that the computer doing the tracking calculations had an internal clock whose values were slightly truncated when converted to floating-point arithmetic. The errors were proportional to the time on the clock: 0.0275 seconds after eight hours and 0.3433 seconds after 100 hours. A calculation shows each of these relative errors to be both very nearly 2⁻²⁰, which is approximately 0.0001%. The GAO report contains some additional information. The internal clock kept time as an integer value in units of tenths of a second, and the computer's registers were only 24 bits long. This and the consistency in the time lags suggested that the error was caused by a fixed-point 24-bit representation of 0.1 in base 2. The base 2 representation of 0.1 is nonterminating; for the first 23 binary digits after the binary point, the value is 0.1 × (1 - 2⁻²⁰). The use of 0.1 × (1 - 2⁻²⁰) in obtaining a floating-point value of time in seconds would cause all times to be reduced by 0.0001%. This does not really explain the tracking errors, however, because the tracking of a missile should depend not on the absolute clock-time but rather on the time that elapsed between two different radar pulses. And because of the consistency of the errors, this time difference should be in error by only 0.0001%, a truly insignificant amount. Further inquiries cleared up the mystery. It turns out that the hypothesis concerning the truncated binary representation of 0.1 was essentially correct. A 24-bit representation of 0.1 was used to multiply the clock-time, yielding a result in a pair of 24-bit registers. This was transformed into a 48-bit floating-point number. The software used had been written in assembly language 20 years ago. When Patriot systems were brought into the Gulf conflict, the software was modified (several times) to cope with the high speed of ballistic missiles, for which the system was not originally designed. At least one of these software modifications was the introduction of a subroutine for converting clock-time more accurately into floating-point. This calculation was needed in about half a dozen places in the program, but the call to the subroutine was not inserted at every point where it was needed. Hence, with a less accurate truncated time of one radar pulse being subtracted from a more accurate time of another radar pulse, the error no longer cancelled. In the case of the Dhahran Scud, the clock had run up a time of 100 hours, so the calculated elapsed time was too long by 2⁻²⁰ × 100 hours = 0.3433 seconds, during which time a Scud would be expected to travel more than half a kilometer. The roundoff error, of course, is not the only problem that has been identified: serious doubts have been expressed about the ability of Patriot missiles to hit Scuds.

Robert Skeel is a professor of computer science at the University of Illinois at Urbana-Champaign. From SIAM News, July 1992, Volume 25, Number 4, page 11

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lmao

This week I decided to write something more directly in my wheelhouse, and talk about an unusual way to prove the fundamental theorem of calculus. This doesn't have any direct action items for you take away in your daily life, but it's a fun bit of math!

Weekend To Do List 🌺

to do list for this weekend is ready let's see how much i will be able to cover it 😅

#just trying to be more active on tumblr ....🥹

#Pov : Exam season ahead 🫠

-R.W. Hamming 

9.26.22

First week of Fall Quarter is underway!!

I spent my gap between classes today in the library working on some readings for Thermodynamics . I have a homework assignment for Thermodynamics and a research proposal for Energy due this Friday, so I’m trying to stay on top of it to avoid overwhelming myself.I took a look at the Thermo homework and I could answer the first half of the questions. I’m planning on writing out the solutions tomorrow.

I haven’t actually had a meeting for Energy, so I’m hoping the professor will explain the proposal a little better in class tomorrow. I haven’t started it for two reasons. One, the prompt is a little vague, so I don’t know where to start. And two, I’m super intimidated by research projects and proposals. Hopefully tomorrow will clear up some doubts in my mind. Based on how the Energy class goes, I’ll decide whether to do some homework or work on the proposal.

Overall, I feel excited for my classes this quarter. Thermodynamics and Fluid Mechanics seem pretty straightforward. I’m a little nervous for Energy and Numerical Analysis, but I think that’s just first day jitters.

Ive mentioned this before on my other blog, but everytime I think about it, it astonishes me how much Arcane feels like it successfully pulled off what Hazbin Hotel was trying to do. You got:

Lesbian lead couple that have amazing chemistry despite their different backgrounds and personalities? Arcane did it better than Hazbin

Male characters that are fleshed out enough to have their own narratives but don't overshadow the female leads and instead support their plotlines? Arcane did that better than Hazbin

Diverse character designs that tell you so much about each character and helps them stand out from one another? Arcane did that better than Hazbin

An accurate portrayal of social hierarchy and how even if those in elite positions genuinely care about every person, they still may be blinded by their own privilege? Hazbin barely had that but Arcane did that better than Helluva Boss

Building conflict between two nations where one abuses their power over the other and the lower nation is ready to rise up and start a war for their equality? Arcane did that better than Hazbin

Fast paced plot that tells multiple nuanced stories in a single digit amount of episodes but everything still feels comprehensive and realistic? Arcane did that better than Hazbin

Overarching themes of parenthood without feeling like the show is telling the same story again and again? Arcane did that better than Hazbin and Helluva Boss

Both have gorgeous animation and wonderful musical segments and this one is more debatable than the others, but I feel like Arcane uses their animation and soundtrack much better as a way to tell their story while Hazbin uses it more for style than for adding to or progressing the plot (but obviously both shows use it for both plot and style simultaneously)

I don't feel qualified to talk about their portrayals of trauma, especially when it comes to Angel Dust and Jinx, but I'd love to hear from someone with more experience than me how these two shows compare in that department and in regards to making fictional trauma feel realistic without glamorizing it or only showing it for the sake of being edgy

Good god ep18 s6. House dealing (god-awfully .Full disclosure. The episode opens with house drinking himself half to death with zero regard for himself to the point of unintentionally breaking into his neighbours house and passing out there) with wilson getting back together with his ex-wife while the patient of the episode grapples with his possibly reciprocated love for the woman he loves but cannot Have because she's engaged to be wed with his friend.......... Thirteen egging the patient on to tell her how he really feels but he genuinely Cant Bring Himself To because "my friend's a great guy . He's rich, smart, and he'll treat her well. Me-- I'd do everything I could for her. But it wouldn't be much. And she... she deserves the best." And the episode ends with house resignedly throwing away the sealed envelope that held all the dirt he hired lucas to get on wilson's ex wife to try and break them upOhhhhhhhh oh. OHHHHHHHHHHHHH Ouhhhhhhh. Ouhhhhh the acknowledgement that house would do everything & anything & more for wilson Give .all of himself To Wilson and it still wouldn't be enough because house is House. Because even at his best-- house would still be house and because of that house would never be able to give wilson what he wants What he Needs. Wilson doesn't. Need the misanthropic bastard he's been in love with for 20 years who bites and spits at any threat or sign of emotional intimacy or vulnerability. What wilson. Needs is a woman . A wife and two kids and a white picket fence because that's all wilson is

Its the foundation of his entire self that would bring everything else down with it if it were to crumble. The face that wilson has spent his entire Life honing and sculpting and perfecting to present as the Perfect Hardworking American Man and Son. The perfect husband and treasured son with his own big shiny department and a stainless-white doctor's coat and the Exact Man a woman would Need

What is wilson. who is he What is left of him if he is not needed by a woman; not needed nor wanted any longer by the world he's lived his entire life by to please. What is James Wilson if not what everyone else expects him to be

House is the antithesis to all of that. A man rough and abrasive as sandpaper who makes wilson selfish makes him emotional and stupid. Who encourages wilson to lash out and fight and get angry and stand up for himself and be the exact opposite of what he's worked his entire life to be. Instead of accepting and taking wilson's painstakingly pedantically constructed facade at face value House fucking. Crashes through the walls with a bulldozer. Snatches the mask right off of wilson's face and dangles it over his head goading wilson to go ahead; try and get it back

Wilson is so deathly terrified at the idea of breaking out of the norms he himself has walled himself into-- he can't Bear to think of any other future for himself that is anything other than wholly and completely unnoticeable average monotonous unextraordinary

and House is the exact opposite of unnoticeable average monotonous unextraordinary. House is the apple of Eden that rests on the other side of wilson's pristine-white picket fence. The object of Wilson's every true desire that simultaneously threatens to doom and tear down everything wilson regards protects worships as the one untouchable unquestionable unchangeable truth of his life

House loses before it even starts. No one can compete with that; not even house. By nature, house can never be what wilson needs. What wilson truly desires or wants or needs is another subject entirely, something im genuinely not sure wilson could even grapple with, let alone come to terms with canonically. I fully believe wilson and house are the loves of each other's lives but house will never be what wilson "needs" or "wants" no matter what he does or changes about himself. They love each other more than anything and they want each other and they cant live without each other but House-- intrensically, by nature-- cannot be what Wilson wants.

last exam of my undergrad career is shaping up to be long & annoying contour integral hell times

(granted the entire course has been long and annoying integral hell, so... I suppose that's fitting for the final? still doesn't make me want to do it any more)

Seeing people write off Viviana as mid when she's at SL7... She gains 47% more damage with her masteries. How many times must we do this?

One of the many cool things about language is that we will commonly absorb words and expressions from people around us, people we talk to, live with, love...

How peculiar, too, that both a miracle and a betrayal "of the highest order" have taken place. Ava is a "miracle", but so is Adriel still being alive inside that tomb, in a way; Areala "betrayed" Adriel (or vice-versa...), but so did Vincent betray the OCS.

Spotify recommended Bari Weiss's podcast to me so I spent some time today hate-reading the episode descriptions. And obviously this doesn't matter at all in the larger picture, but the phrasing of this one is killing me. Why did they write it as though 14 is bigger than 33. It feels so transparent that they had to do this because if they phrased it in a natural way they might possibly have appeared to be ceding ground to the "girls have it worse" crowd.

well.. amazing period!!! yay boyos!!

learning python rn and nothing boosts your confidence like finding the bug while the person teaching you is coding live

math, art, or both

I will not apologize for the person I'll become while reading this book

((One of the more subtle things that I hope is brought up in more detail is the efforts that Flynn and Bo go to in order to restore Kariom's memories. In no particular order we have: conversation, exposure to places and important objects, keeping multiple journals/diaries, playing recordings of his voice back to him, legitimately hypnotizing him, and so on.

A few of these things? Fine, normal even. But the escalation is stark and all it really shows is that neither Bo nor Flynn know what to do or how to help and they make it incredibly clear that the things they've tried aren't foolproof. Even Bo's reasoning for giving Kariom a camera and having him take Flynn and Bo's picture directly references it:

((The gesture and gift itself is very sweet (and I'm sure Kariom would love to have a camera, that's very fitting for him) but the underlying reason? The adding on to the efforts/measures they can rely on to restore his memories without nary a mention of anything else on top of knowing that it's not going to work long term if at all? That just makes it.....sad. No, not just sad: it's desperate. It's a reality made worse by how unbothered Bo is and by how bothered Flynn is---Flynn even makes a sarcastic remark and then reminds Bo that the journals didn't work and so why would photos---and how more and more they're (seemingly) on opposite sides. They're fighting a losing battle and only one of them is actively pushing for it to be stopped.))