TweetFollow Us on Twitter

Curve Fitting 3
Volume Number:1
Issue Number:13
Column Tag:Forth Forum

Curve Fitting, Part III

By Jörg Langowski, Chemical Engineer, Grenoble, France, MacTutor Editorial Board

Last time I promised you to add some input/output to the curve fitter program so that it becomes a useful application. We'll do this here today.

A lot of basic graphic routines which you would have to write yourself on other machines are included in the Quickdraw ROM. This makes life very easy; we can use the screen like we would use a plotter (of course, we can do much better than that, but this is what we are dealing with today). What is not included in the ROM, of course, are utilities that plot coordinate axes, draw lines through pairs of x/y points, add symbols etc.

Most computer systems that are used for numerical calculation purposes contain a set of standard plotting routines, mostly callable from FORTRAN, that will help you generating a 'nice' graphical output with not too much effort. This column gives you some similar routines in FORTH; I have tried to stick closely to the Calcomp plotting routines, since they are something of a standard.

Our objective is to produce something like the curve in Fig. 1. The data points that the model is fitted to will be displayed using some symbols, while the fitted curve is plotted as a smooth line. X- and Y-axes are added to the plot.

The numbers that are to be plotted will be given in floating point format (in this case, single precision arrays as defined previously). So first of all we have to know how to scale the data to the integer values of the bit map that we are going to plot to.

The Calcomp convention is, for an array of N floating point numbers, to store the minimum of these numbers in cell N+1 of the array and the difference between the minimum and the maximum in cell N+2. These two numbers are then rounded to one significant figure. Given this information, one can write routines that automatically draw an axis that spans the range of the data values.

The rounding to one significant figure is done by the word next.int, which given the address of an extended number on the stack returns with the address of the rounded number (which is a local variable of next.int).

fscale finds the minimum and maximum values of an array and stores the scaling numbers above the last data point after rounding them to one significant digit.

xaxis and yaxis draw coordinate axes in x- and y-direction. Input parameters are (#ticks\length\array\npts), #ticks gives the number of ticks to be drawn (to the bottom of the x- and to the left of the y-axis), length is the total axis length in points, array the address of the scaled data array and npts the number of points in array.

Given two scaled arrays, line draws a line through the (x y) pairs defined by the two arrays. The symbol defined by the global symbol is plotted at each of the data points. Symbols could be e.g. '+' or '*'. Setting symbol to zero will draw no symbols (how could you have guessed). Depending on the setting of the global flag connecting, the symbols are either connected by a line or not.

For all the plotting routines cartesian has to be switched on and the origin to be defined within the output window by xyoffset. init.plot is used to do this job.

The main curve fitter program now consists of the following:

- the data arrays xdat and ydat are initialized with the simulated 'experimental data' (init). In this month's example, we use a sum of two exponentials as our model; this gives us five parameters to fit.

- a routine is called in which one can deliberately change the values of the parameters (for this we need floating point input, see below), so that one can see how the correct curve is fitted through the data starting from wrong parameters. One may also change the total number of parameters to be fitted; for instance, fitting three parameters only will force a single exponential fit to the data points. In this case, parameters 4 and 5 will have to be set to zero.

- the main loop calculates theoretical function values from xdat and the parameters and stores them in zdat. The data arrays are scaled and the plot displayed (like in Fig. 1); then one iteration of the fitting routine is taken, the parameters changed accordingly, displayed and the process repeated until parameter changes are below 1 part in 104.

Floating point input

As I mentioned, the program would not be very useful without floating point input routines. In order to be independent of any particular Forth implementation, I am including a simple floating point input routine here which accepts a string and converts it to the decimal string format used by the SANE conversion routines.

Any such routine would mainly consist of taking successive characters from the input string, generating a decimal mantissa and keeping track of the number of digits behind the decimal point, then looking for an 'e' or 'E' to indicate an exponent and converting the exponent finally. In that aspect, the routine written here is very similar to the MacForth floating input routine. There is one difference, in that numbers without an exponent will be accepted and converted to floating point numbers, too. Therefore, you won't be able to use this routine as an extension of the Forth interpreter, as MacForth level 2 does. But in entering a lot of data, it can be very tedious always having to type an exponent.

fnumber takes a string as its input parameter and leaves on the stack:

( addr of float\true ) if a valid floating point number was read from 
the string,
( false ) if the conversion was not successful.

In the latter case, an error message is printed.

input.float reads a string from the keyboard into pad, then calls fnumber to convert it.

With the additions from this column, the curve fitter should finally be a utility that is useful to you (in case you have any curves to fit). Changing the function to be fitted is easy, and you might even install a make switch for vectored execution (V1#7) so that you can easily switch between different functions within one program.

Data input still has to be done manually, number by number. However, input.float may easily be extended to read input from a file. To transfer data to/from other applications through the scrap requires some more work; I'll deal with that problem soon.

Feedback dept.

Re: finding object code of unnamed tokens

The procedure to decompile the object code of an axed token is the following (V1#2):

- convert the token to an absolute address, using token>addr. If the word contained there is $4e4f (TRAP $F), the next words will be Forth code. Start decompiling at the following word. Of course, different versions of MacForth will give different addresses for the individual words due to different dictionary arrangements; but this procedure should work for any version of Level 1 or Level 2.

Re: MacModula floating point

Shortly after my comment on errors in MacModula 2's 32-bit floating point arithmetic, I received a letter from the author of those routines, Daan Strebe:

" A few weeks ago I followed several bug reports to find two fundamental errors in the floating-point routines, one a conceptual error in the rounding and the other a misunderstanding about the 68000 processor instruction set. These two errors affected the multiply routine most, although the others were somewhat affected also. I revamped those routines and then ran a complex set of comprehensive tests, and the results allow me to state with a margin of confidence that the floating point in the latest rev is now not only slightly faster than it was, but also that the only errors in the basic routines (not necessarily including those in the math library) are from rounding. The rounding as it is now implemented yields 3 downward, 4 upward, and 1 non-round per 8 random floating-point operations. This should be satisfactory for most users; full IEEE rounding would considerably decrease the speed. I believe the compromise was successful. "

So everything should be fixed now. Let's hope that the new update will be distributed soon (it probably is when this goes to the printer's).

Listing 1: Plotting routines and floating point input for the curve fitter 
program

( Additions to the curve fitting program) 
( for graphical output and floating point input.)
( © 1985 J. Langowski for MacTutor)
( Again, only the parts that have been changed) 
( or added with respect to the last two columns )
( are printed here)

: s> 1008 fp68k @sr 10 and ;

: numstring create 24 allot ;  
numstring zzs1 ( internal conversion string )

: dec. ( float\format# -- )
       zzformat ! zzformat swap zzs1 b2d
       zzs1 dup w@ 255 > if ." -" else ."  " then
       dup 4+ count over 1 type ." ."
       swap 1+ swap 1- type ( mantissa )
       2+ w@ ( get exponent )
            1 w* zzformat @ + 1- 
            ." E" 0 .r ;

create xdat 400 allot   create ydat 400 allot
create zdat 400 allot   create residues 400 allot
100 10 matrix derivmat    10 11 matrix resmat
     5 constant npars        80 constant npts
create par 5 10 * allot   
create delta 5 4*  allot

float logten  ten logten x2x  logten lnx
: log10x dup lnx logten swap f/ ;

( define function )                               ( 090485 jl )
                                     float temp ( local to func)

 func ( x -- f[x] = par[1] + par[2] * exp[par[3]*x]
                                       + par[4] * exp[par[5]*x] )
  dup temp x2x
  par 40 + temp f* temp expx  par 30 + temp f*
  par 20 + over f* dup expx  par 10 + over f*
  par over f+  temp over f+ ;
                                     axe temp
: >fa1  fa1 s2x ;
: init_pars   one par x2x  two par 10+ x2x 
                 -one par 20 + x2x  two par 20 + f/
                   one par 30 + x2x  
                 -one par 40 + x2x ten par 40 + f/ ;
init_pars

: one_iter
  make_derivmat  residuals  
  make_resmat     delta 0 0 resmat npars gauss ;

: new_pars 16 ( true if no significant changes)
  npars 0 do par i 10 * +
    delta i 4* +  over  s+
    delta i 4* + fa1 s2x  fa1 f/
    fa1 fabs  eps fa1 f> and loop ;

80 ' npts !    5 ' npars !

: init ( initialize data arrays)
          npts 0 do i sp@ fa1 in2x 4*
          xdat over + fa1 swap x2s
          ydat over + fa1 func  ranf fa2 x2x
          ten fa2 f/  fa2 over f+  swap x2s
          drop  loop ;

( plotting routines)                        ( 092385 jl )
float 1/2  1 sp@ 1/2 in2x drop  2 sp@ 1/2 in/ drop
float small  ten small x2x  -200 sp@ fa1 in2x drop
fa1 small x^y  
( anything smaller than small will be zero)
float sc.aux  float sc.exp
: next.int ( float -- rounded to 1 dec. place )
  dup small f>
  if dup sc.aux x2x  sc.aux  dup fabs  dup log10x
     dup 1/2 swap f- frti
     ten sc.exp x2x  sc.aux sc.exp x^y   sc.aux x2x
     sc.exp sc.aux f/  sc.aux frti  sc.exp sc.aux f*
     sc.aux
  else drop zero then  ;

float xlow  float xhi  float sc.factor
: fscale 
    ( array \ n -- | start, scale -> array[n+1..n+2] )
  over xlow s2x  over xhi s2x
  over over 1 do
    dup i 4* + dup xlow s> not if dup xlow s2x then
               dup xhi  s> if xhi s2x  else drop then  loop
  xlow xhi f-
  over 4*  +  xlow next.int swap x2s
     1+ 4* +  xhi  next.int swap x2s  ;

: .fscale ( array \ n -- )
  4* + dup fa1 s2x ." min = " fa1 7 dec.
  4+ fa1 s2x ." , scale = " fa1 7 dec. cr ;

( xtick, ytick )                                  ( 092385 jl )

: xtick ( rx x -- ) dup 0 move.to
  dup -5 draw.to  dup 15 - -16 move.to
  get.textsize >r 9 textsize swap 2 dec. r> textsize
  0 move.to ;

: ytick ( ry y -- ) -45 over 4- move.to
  get.textsize >r 9 textsize swap 2 dec. r> textsize
  0 over move.to    -5 over draw.to
  0 over 6- move.to  0 swap draw.to ;

( xaxis)                                           ( 092385 jl )
float start  float del
: xaxis 
   ( #ticks\length\array\npts -- | starts at origin )
  0 0 move.to
  4* + dup  start s2x  4+ delta s2x
  over /  ( #ticks\length/tick -- )
  over 0 do dup i 1+ * dup 0 draw.to
         delta fa1 x2x i 1+ sp@ fa1 in* drop
                    3 pick sp@ fa1 in/ drop
         start fa1 f+    fa1 swap xtick loop
  drop drop ;

( yaxis)                                          ( 092385 jl )
: yaxis 
  ( #ticks\length\array\npts -- | starts at origin )
  0 0 move.to
  4* + dup  start s2x  4+ delta s2x
  over /  ( #ticks\length/tick -- )
  over 0 do dup i 1+ * 0 over draw.to
            delta fa1 x2x i 1+ sp@ fa1 in* drop
                    3 pick sp@ fa1 in/ drop
         start fa1 f+    fa1 swap ytick loop
  drop drop ;

( line, variables )                               ( 092585 jl )
variable xline.length variable yline.length
variable xpos  variable ypos  variable symbol
variable connecting  connecting off
float xstart float ystart float xdel float ydel

( line)                                           ( 092585 jl )
: line ( xarray\yarray\npts\xlength\ylength -- )
  yline.length !  xline.length !   0 0 move.to
    over over 4* + dup ystart s2x 4+ ydel s2x
  3 pick over 4* + dup xstart s2x 4+ xdel s2x
  0 do over i 4* + fa1 s2x  xstart fa1 f-  xdel fa1 f/
       dup  i 4* + fa2 s2x  ystart fa2 f-  ydel fa2 f/
       xline.length fa1 in*  fa1 xpos x2in
       yline.length fa2 in*  fa2 ypos x2in
       xpos @  ypos @  over over
       connecting @ if draw.to else move.to then
       -4 -4 rmove symbol @ emit move.to
    loop   drop drop ;

( calc.theor, disp.theor, disp.data, scale.all)   
( 092585 jl )
: calc.theor
          npts 0 do xdat i 4* + fa1 s2x fa1 func
                    zdat i 4* + x2s  loop ;
: disp.theor
    0 symbol !  connecting on
    xdat zdat npts 400 250 line ;
: disp.data
    43 symbol !  connecting off
    xdat ydat npts 400 250 line ;
: scale.all
    xdat npts fscale  ydat npts fscale
    ydat npts 4* + @  zdat npts 4* + !
    ydat npts 1+ 4* + @  zdat npts 1+ 4* + !  ;


( disp.axes)                                      ( 092585 jl )
: disp.axes
    5 400 xdat npts xaxis  5 250 ydat npts yaxis ;
: init.plot
    page 50 255 xyoffset cartesian on ;

( floating point input)                           ( 092385 jl )
: fsign zzs1 ;  : fexpo zzs1 2+ ;
: fmant zzs1 4+ ;  variable frac   float float.out
variable decimals
: input.sign 0 fsign !
  dup c@ case 45 of -1 fsign w! 1+ endof
              43 of             1+ endof endcase ;
: input.mantissa   0 decimals !   0 frac !
  fmant 21 +  fmant 1+ do  i fmant 1+ -  fmant c!
          dup c@ dup
             case 48 57 range.of ic! frac @ decimals +!
                                              1+ 1 endof
                     46 of   drop 1 frac !    1+ 0 endof
             20 swap endcase  +loop drop ;
: input.exponent
  dup c@ dup bl = not
      if dup 69 =  swap 101 =  or not
         if  drop 0
         else dup 1+ c@ 43 = 1 and +
              number decimals @ - fexpo w! -1
         then
      else  decimals @ -1 * fexpo w! -1  2drop
      then  ;

: fnumber zzs1 24 blanks
   input.sign input.mantissa input.exponent
   if fsign float.out d2b float.out -1
   else ." floating input error!" cr 0 then ;

: input.float pad 22 blanks pad 22 expect pad fnumber ;

: get.pars
  npars 0 do
          begin cr ." par[" i 1+ . ." ] = " input.float until
          par i 10 * + x2x loop
  cr begin cr ." total # of parameters to be fitted "
           5 input.number until 
  ' npars !  cr ;

: .pars npars 0 do
      ." par[" i 1+ . ." ] = " i 10 * par + 7 dec. cr loop ;

( main curve fitter program)                 ( 092685 jl )
: fit.curve  page
  init_pars ." initializing data arrays..." cr init
  5 ' npars ! init.plot get.pars .pars
  ." fitting " npars . ." parameters to "
               npts . ." data points "
  calc.theor scale.all page
  disp.data disp.axes disp.theor
  begin  one_iter new_pars not while
    page .pars
    calc.theor page disp.data disp.axes disp.theor
  repeat
  300 300 move.to ;

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

BusyCal 3.7.2 - Powerful calendar app wi...
BusyCal is the powerful, flexible, reliable calendar app for macOS. It's packed with time-saving features and compatible with all leading cloud services including iCloud, Google, Exchange and more.... Read more
BusyContacts 1.4.2 - Fast, efficient con...
BusyContacts is a contact manager for OS X that makes creating, finding, and managing contacts faster and more efficient. It brings to contact management the same power, flexibility, and sharing... Read more
iShowU Instant 1.3.0 - Full-featured scr...
iShowU Instant gives you real-time screen recording like you've never seen before! It is the fastest, most feature-filled real-time screen capture tool from shinywhitebox yet. All of the features you... Read more
Capo 3.8 - Slow down and learn to play y...
Capo lets you slow down your favorite songs so you can hear the notes and learn how they are played. With Capo, you can quickly tab out your songs atop a highly-detailed OpenCL-powered spectrogram... Read more
Viber 11.7.0 - Send messages and make fr...
Viber lets you send free messages and make free calls to other Viber users, on any device and network, in any country! Viber syncs your contacts, messages and call history with your mobile device, so... Read more
Drive Genius 5.3.1 - $79.00
Drive Genius features a comprehensive Malware Scan. Automate your malware protection. Protect your investment from any threat. The Malware Scan is part of the automated DrivePulse utility. DrivePulse... Read more
CorelDRAW 21.2.0.708 - Graphic design so...
CorelDRAW - professional graphic design software for vector illustration, layout, and so much more. Get started quickly and easily with a wealth of intuitive tools, built-in learning materials,... Read more
Navicat Premium Essentials 12.1.27 - Pro...
Navicat Premium Essentials is a compact version of Navicat which provides basic and necessary features you will need to perform simple administration on a database. It supports the latest features... Read more
OmniFocus 3.4.3 - GTD task manager with...
OmniFocus is an organizer app. It uses projects to organize tasks naturally, and then add tags to organize across projects. Easily enter tasks when you’re on the go, and process them when you have... Read more
Microsoft OneNote 16.30 - Free digital n...
OneNote is your very own digital notebook. With OneNote, you can capture that flash of genius, that moment of inspiration, or that list of errands that's too important to forget. Whether you're at... Read more

Latest Forum Discussions

See All

Breakout: Dark Prison is a fast-paced ac...
Breakout: Dark Prison is an action RPG from LaterSoft. Set in the aftermath of a deadly virus outbreak your daughter has been taken from you because she has an immunity to the illness in her DNA. Not being a fan of experimentation on children –... | Read more »
Apple Arcade in review
This weekend, Apple Arcade will officially be one month old. That means anyone who signed up for the free trial on day one has a decision to make: Stick with the service and shell out $5 a month, or cancel and go about your merry way. | Read more »
Alluris is a choose-your-own adventure g...
Alluris is an RPG that the developer's are calling a swipe-your-own adventure game. This is because the game incorporates a Reigns-style - swiping left or right - selection mechanic to make all the decisions you'd usually expect to make across... | Read more »
Hello Hero All Stars receives update wit...
The first Hello Hero game hit global platforms in 2013 and proved a huge success, with developer Fincon adding two more entries to this popular series of casual RPG games since. Released in June this year, Hello Hero All Stars brings many of the... | Read more »
Zombieland: Double Tapper, a cartoon idl...
Zombieland: Double Tapper is the idle RPG tie-in to the upcoming Zombieland: Double Tap. Oddly, it's one of two different Zombieland games launching today, with the other being the Switch title Zombieland: Double Tap - Road Trip. [Read more] | Read more »
Rusty Lake's The White Door launche...
Rusty Lake and Second Maze's intriguing point-and-click adventure game, The White Door, is now up for pre-order on the App Store. This one sees you playing as Robert Hill, a mental health patient who is suffering from severe memory loss. The game... | Read more »
Hellrule is an auto-runner inspired by G...
Hellrule is an upcoming auto-runner game from independent developer Pedrocorp where players will take control of a dapperly dressed gentlemen who comes equipped with a razor-sharp umbrella for slicing up his foes. The game will be available for... | Read more »
Grobo is a gravity bending puzzle platfo...
Grobo is a 2D puzzle platformer that marks the first release from developers Hot Chocolate Games. You'll find yourself manipulating gravity as you make your through this title that's available now for iOS and Android. [Read more] | Read more »
Adrenaline, Compulsive Entertainment’s h...
Compulsive Entertainment’s high-octane arcade racer, Adrenaline, has now made its way to the App Store following a successful launch on Google Play. It’s a ton of challenging, fast-paced fun, boasting easy-to-learn controls and a varied selection... | Read more »
Mario Kart Tour is adding Super Mario Ga...
Earlier today on Twitter, Nintendo announced that Mario Kart Tour is getting a new racer and track. Fans of Super Mario Galaxy will be pleased to hear that Rosalina is the first post-launch character being added, while the iconic Rainbow Road is... | Read more »

Price Scanner via MacPrices.net

11″ WiFi iPad Pros on sale today for up to $2...
Amazon has new 2018 Apple 11″ WiFi iPad Pros in stock today and on sale for up to $250 off Apple’s MSRP. These are the same iPad Pros sold by Apple in its retail and online stores. Be sure to select... Read more
Apple offers clearance 2018 15″ 6-Core MacBoo...
Apple has clearance 2018 15″ 6-Core Touch Bar MacBook Pros, Certified Refurbished, available starting at only $1999. Each model features a new outer case, shipping is free, and an Apple 1-year... Read more
Apple continues to offer refurbished 2019 21″...
Apple has Certified Refurbished 2019 21″ & 27″ iMacs available starting at $929 and up to $350 off the cost of new models. Apple’s one-year warranty is standard, shipping is free, and each iMac... Read more
Verizon offers free iPhone 7 for customers wh...
Verizon is offering a free 32GB iPhone 7 for new or existing customers who open a new line of service, no trade-in required. Cost of the phone is credited to your account monthly for 24 months. The... Read more
13″ 2.4GHz 4-Core MacBook Pros remain availab...
Apple has a full line of Certified Refurbished 2019 13″ 2.4GHz 4-Core Touch Bar MacBook Pros available starting at $1529 and up to $300 off MSRP. Apple’s one-year warranty is included, shipping is... Read more
Lease one iPhone Xs, Xr, or X at Sprint and g...
Purchase an Apple iPhone X, Xr, Xs, or Xs Max at Sprint, and get a 64GB iPhone Xr for free. Requires 2 new lines or 1 upgrade-eligible line and 1 new line. The fine print: “iPhone Xs (64 GB) $37.50/... Read more
How to use your Apple Education discount to s...
Purchase a new MacBook Pro or MacBook Air using Apple’s Education discount, and take up to $200 off MSRP. All teachers, students, and staff of any educational institution with a .edu email address... Read more
Base 2019 13″ 1.4GHz 4-Core MacBook Pro on sa...
Amazon has the new 2019 13″ 1.4GHz/128GB 4-Core Space Gray Touch Bar MacBook Pro on sale for $100 off Apple’s MSRP. This is the same MacBook Pro sold by Apple in its retail and online stores: – 2019... Read more
Save up to $30 on Apple’s AirPods at these re...
Amazon is offering discounts on new 2019 Apple AirPods ranging up to $30 off MSRP. Shipping is free: – AirPods with Charging Case: $144 $15 off MSRP – AirPods with Wireless Charging Case: $169 $30... Read more
Save $15 on Apple Watch Series 5 models on Wa...
Walmart has new Apple Watch Series 5 models on sale for $15 off Apple’s MSRP on their online store. Choose free shipping or free local store pickup (if available). Sale prices for online orders only... Read more

Jobs Board

*Apple* Mobile App Developer - eiWorkflow So...
…eiWorkflow Solutions, LLC is currently looking for a consultant for the following role. Apple Mobile App Developer Tasks the role will be performing: ? Mobile App Read more
Essbase Developer - *Apple* - Theorem, LLC...
Job Summary Apple is seeking an experienced, detail-minded Essbase developer to join our worldwide business development and strategy team. If you are someone who Read more
Best Buy *Apple* Computing Master - Best Bu...
**738230BR** **Job Title:** Best Buy Apple Computing Master **Job Category:** Store Associates **Location Number:** 000233-Almeda-Store **Job Description:** The Core Read more
Best Buy *Apple* Computing Master - Best Bu...
**738182BR** **Job Title:** Best Buy Apple Computing Master **Job Category:** Sales **Location Number:** 000036-Independence MO-Store **Job Description:** **What Read more
*Apple* Mobility Pro - Best Buy (United Stat...
**741450BR** **Job Title:** Apple Mobility Pro **Job Category:** Store Associates **Location Number:** 000168-Mentor-Store **Job Description:** At Best Buy, our Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.