After some poking around, a Program library (PDP-7) and a Floating point reference manual for a PDP-8 turned up and is now slowly dawning on me that libraries could exist back then too, and that complex polynomials could also be written into routines albeit a bit tedious. http://www.bitsavers.org/pdf/dec/pdp7/DIGITAL-7-30-A_FltPtPkg.pdf http://bitsavers.trailing-edge.com/pdf/dec/pdp8/software/DEC-08-YQYB-D_PDP-8_Floating-Point_System_Programmers_Reference_Manual_Sep69.pdf On Sun, Oct 20, 2019 at 12:02 AM Abhinav Rajagopalan < abhinavrajagopalan@gmail.com> wrote: > Forgive me for both hijacking this thread, and to address my amateurish > gnawing concern, but how was it be possible to write differential/integral > equations at an assembly/machine level at the time, especially in machines > such as the PDP-7 and such which had IIRC just 16 instructions and operated > on the basis of mere words, especially the floating point math being done. > Surmising from some personal experience that writing mathematical programs > is hard even now, although there exist certain functional paradigms, and > specialised environments such as MATLAB or Mathematica. The > complexity seems to remain the same if not more now, due to the vast oodles > of data to handle stemming from the nature of the world. > > Were they loaded as just words as any other instruction or were there > separate coprocessors that did the number crunching? I'm guessing > Fortran-ish kind of implementations were done, but the hardware level > computation itself I just can't process. > > It just blows my mind now thinking backwards in terms of those > monster machines being loaded with trails of paper tape instructions to > play Space Travel. Being born in the late 90's doesn't help me too. > > Also, on a related note, don't know if you've watched the interview > of Ken done by Brian at the Vintage > Comptuer Federation 2019, there might be a few surprises lurking around the > middle of that when they discuss pipes and grep. > > Thank you! > > On Sat, Oct 19, 2019 at 8:11 PM Doug McIlroy > wrote: > >> I was about to add a footnote to history about >> how the broad interests and collegiality of >> Bell Labs staff made Space Travel work, when >> I saw that Ken beat me to telling how he got >> help from another Turing Award winner. >> >> > while writing "space travel," >> > i could not get the space ship integration >> > around a planet to keep from either gaining or >> > losing energy due to floating point errors. >> > i asked dick hamming if he could help. after >> > a couple hours, he came back with a formula. >> > i tried it and it worked perfectly. it was some >> > weird simple double integration that self >> > corrected for fp round off. as near as i can >> > ascertain, the formula was never published >> > and no one i have asked (including me) has >> > been able to recreate it. >> >> If I remember correctly, the cause of Ken's >> difficulty was not roundoff error. It >> was discretization error in the integration >> formula--probably f(t+dt)=f(t)+f'(t)dt. >> Dick saw that the formula did not conserve >> energy and found an alternative that did. >> > > > -- > > Abhinav Rajagopalan > > > -- Abhinav Rajagopalan