TweetFollow Us on Twitter

Linear Equations
Volume Number:1
Issue Number:11
Column Tag:Forth Forum

Solving Systems of Linear Equations

By Jörg Langowski, Chemical Engineer, Fed. Rep. of Germany, MacTutor Editorial Board

This is the first of a series of columns that will deal with the general problem of doing numerical calculations in MacForth. Forth's philosophy is to use integer arithmetic in many cases that would be handled with floating point in other languages. The reason for this has to be seen historically in the development of Forth, which first was used almost exclusively as a language to do process control. It was desirable not to have the ballast of a floating point package in implementations that used 8-bit processors with a limited amount of memory, and there is, of course, a great speed advantage in using integer arithmetic.

When used in 'custom-designed' routines for one particular problem, integer arithmetic can do as well as floating point. However, one has to scale all the numbers involved so that they fit into the range that is given by the 4 bytes of the Mac's integer arithmetic (or the 2 bytes of some other system). On the other hand, numbers shouldn't get too small, either, because accuracy is lost very quickly. The constant need to haul scaling factors around between parts of the program then makes the code rather hard to read and bug-prone.

Again this is the old tradeoff between speed and low memory requirement on one side and flexibility and readability on the other. If we want to write a set of mathematical routines in Forth that will be useful no matter what the particular problem is (whether distances are in nanometers or lightyears, weights in tons or micrograms) the easiest way to do this is to use floating point arithmetic. This is especially true on the Macintosh, since we have an excellent floating point package with 80-bit accuracy built in.

This FP package, also called SANE (Standard Apple Numeric Environment) conforms to the proposed IEEE standard on floating point arithmetic (see my article in MacTutor V1#1). MacForth 2.0 offers Forth code for a very slick interface to the SANE routines, using its own floating point stack and even modifying the interpreter so that real numbers are accepted as input. There are two problems with this code, though: First, we cannot print it here for obvious reasons and therefore our program would run only under MacForth 2.0, which would be a little too restricted for this Forum. Second, according to my own tests the floating point interface adds so much overhead that actual calculations are slowed down by a factor of 2 to 3.

The code that we write here uses a more 'direct' approach to FP arithmetic, which is variable-oriented rather than stack-oriented (see V1#1). It looks a little more clumsy and is definitely harder to read, but since we want to generate a set of Forth words for general use which should be fast more than anything, this is justified.

Definition of the problem - fitting experimental data to a theoretical equation

Enough of the preliminaries, I should tell you now what exactly we want to do. One of the bread-and-butter problems in experimental science is to extract theoretical parameters from a set of experimental data points, given a theoretical equation that can predict those data points from the parameters.

Example: You measure the time response of a physical system, for instance the voltage across a capacitor C as it is discharged through a resistor R. The time behavior of the voltage versus time looks like:

U(t) = Uo exp(-t/RC)

or, if the voltage does not drop all the way down to zero (e.g. some bias applied),

U(t) = Uo exp(-t/RC) + U1 .

In practice, we may have measured a series of points Ui at times ti. Our problem is to get Uo, U1 and RC from that data. Fig. 1 shows how the data and the 'exact' theoretical curve might look like.

Fig. 1: Fitting a theoretical curve to experimental data

Of course, for all U(t) curves with different U0, U1 and RC, there is only one that fits the data points best. The quality of the fit is usually checked by summing the squared differences (the 'residuals') between the data points and the theoretical curve. We have to vary the parameters U0, U1 and RC in such a way that this sum-of-squares becomes a minimum.

Iterative least-squares fitting

Let's state the problem in a more general way. We have a function y = f(t,a1,a2,a3...am) that, given certain values for the parameters a1,a2,a3...am, tells us the time dependence of some quantity y that can be measured. Furthermore we have a set of n data points (ti,yi), the y-values that are actually measured at times ti. The residual for data point i is then

There exists a variety of techniques that one can use to minimize the sum of the squared residuals in such a case. All of them require that one first estimates initial values for the parameters that are not too far away from reality; this is usually possible. From these initial values one can then compute a better estimate of the parameters, and iterate this process until the fit does not improve anymore.

One rather simple algorithm that solves the fitting problem in such an interative way is given by T.R.McCalla in his book 'Introduction to Numerical Methods and FORTRAN Programming' (Wiley & Sons, New York 1967). I won't give the details here, since we are mainly interested in how to program such an algorithm in Forth. The only thing we need to know is the final result: a set of linear equations whose solution gives correction terms ak. These ak have to be added to the initial ak to get the new estimate.

The linear equations that one gets look like this:

Fig.2: System of linear equations

where the cij are coefficients that one calculates from the set of n data points (ti,yi) and the derivatives of the function f(ti, a1,a2,a3...am ) at each data point with respect to the parameters ak. The bi contain the residuals.

So the first problem that we have to solve - and this will be plenty for this column - is to solve a system of linear equations like the one given above. In later columns we will build on the basics of floating-point arithmetic that we develop here and end up with a functional curve-fitting program.

The Gauss Algorithm

A linear equation system like the one above is often solved using the Gauss algorithm. One starts writing the coefficients on the left and right hand sides of the equations as a m*m+1 matrix:

(3 by 4 in this example).

The algorithm then converts this matrix into a triangular matrix:

where the bottom left 'triangle' is equal to zero: multiples of the first row are subtracted from the rows below it until the first column is all zeroes except for the first row, then multiples of the second row are subtracted from the rows below it until the second column is all zeroes except for the first two rows, and so on.

After that procedure is completed, the bottom row has become a simple equation of one variable:

from which a3 can easily be calculated. a3 is then substituted into the equation above it and a2 obtained, and from a3 and a2 finally a1. This procedure can, of course, be expanded to be used on any number of equations.

The Gauss algorithm is given as a Pascal program (to improve readability) in Listing 1. To code it in Forth we first have to give the problem of data representation a little thought, namely: how are we going to store a matrix?

Data representation for arrays of floating point numbers

The SANE routines work on 80-bit numbers. This is ideal for accurate calculations, but a little expensive as far as storage goes; a 100 * 100 matrix would already occupy 80K. If high precision is not needed, large arrays may be stored as lower precision FP numbers. Single precision uses only 32 bits, less than half of the standard SANE length. Therefore we are going to store matrices as two-dimensional arrays of 32-bit long words that contain single precision real numbers. The MATRIX definition (in the example program in listing 2) is modified from the example released by Creative Solution on the Forth disk. We have separated the DOES> part that calculates the address of a matrix element from its indices and defined it as a separate word, CALC.OFFSET. This was done so that our routine works with any matrix variable whose address is passed on the stack.

You define a matrix with r rows and c columns by

r c MATRIX X    .

When you later execute

i  j  X   ,

the address of the element in row i and column j of matrix x will be on the stack. When you execute 0 0 X (all rows and columns start with 0), the address of the first element in the matrix will be on the stack. If we want to write a Gauss algorithm routine that works with any matrix of any size, we have to be able to calculate the offset into the matrix from the row and column indices just as the DOES> part of the MATRIX defining word does. In our definitions, i j addr CALC.OFFSET leaves on the stack the address of the element at row i and column j of the matrix whose first element is at addr.

The solution of the linear equation system will be stored in an array z. For this array we do not need a DOES> part because it is one-dimensional, no need to keep track of row and column lengths here.

Strategy for floating point calculations using the SANE package

The SANE routines expect addresses of floating point numbers on the stack as their parameters (see V1#1). All arithmetic operators are two-address operators, where the first parameter is added to, subtracted from, divided or multiplied into the second parameter. The second parameter is always 80-bit extended precision, while the first one may be any precision. So for any calculation we will transfer the numbers out of the 32-bit variables into 80-bit variables (or add them in etc., if it is convenient), then do all intermediate calculations in 80-bit precision and at the end store the 80-bit result into a 32-bit single precision variable again.

The Gauss Algorithm Routine

Listing 2 shows the example program containing the GAUSS routine for solution of linear equation systems of any size. The routine expects on the stack, from bottom to top: the address of a solution vector z, which for n unknowns has n 32-bit words allocated; the address of the n (rows) by n+1 (columns) matrix X that contains the coefficients of the linear equation system; and n, the number of equations (or unknowns, respectively). The routine first converts the X matrix into its triangular form (so X is changed upon exit), then proceeds to calculate the values of the unknowns, starting in the bottom row of the matrix and working its way up.

The K function: extracting the loop index 2 levels up

The first part of the algorithm has DO..LOOP constructs nested 3 levels deep. The inner loop needs the outermost loop index, and there is no standard word in MacForth that handles this. Therefore we define : k rp@ 20 + @ ; which does this job. (There is also a k defined in machine code; see V1#9).

The example program

Our example calculates the solution of the system of equations

The solution is x1 = 1.2308, x2 = -1.0769, x3 = -0.1538. The word gbm calculates and prints this solution (it actually calculates n times, with n on top of the stack, for benchmark purposes).

Listing 1: Gaussian algorithm - Pascal example

program LinEqu;
 type  matrix = array[1..10, 1..11] of real;
          column = array[1..10] of real;
 var  x : matrix;    z : column;   n, i : integer;

 procedure gaussalg (var x : matrix;
         var z : column;  n : integer);
  var   dg, fk, ee : real;   i, j, k : integer;
 begin
  for i := 1 to n - 1 do
   begin  dg := x[i, i];
    for j := i + 1 to n do
     begin  fk := x[j, i] / dg;
      for k := i to n + 1 do
       x[j, k] := x[j, k] - fk * x[i, k]
     end
   end;
  for i := 1 to n do   z[i] := x[i, n + 1];
  for i := n downto 2 do
   begin   dg := x[i, i];    ee := z[i];
    for j := i - 1 downto 1 do  
        z[j] := z[j] - ee * x[j, i] / dg
   end;
  for i := 1 to n do   z[i] := z[i] / x[i, i]
 end;

begin  n := 3;
 x[1, 1] := 1; x[1, 2] := 1; x[1, 3] := 1; x[1, 4] := 0;
 x[2, 1] := 1; x[2, 2] := -1; x[2, 3] := 2; x[2, 4] := 2;
 x[3, 1] := 4; x[3, 2] := 1; x[3, 3] := -1; x[3, 4] := 4;

 gaussalg(x, z, n);
 for i := 1 to 3 do  writeln('z[', i : 1, ']= ', z[i] : 7 : 4)
end.
Listing 2: Gaussian algorithm, FORTH example

( Floating point primitives )
( This is part of the SANE interface given in MT V1#1; not all of it 
is needed here)
hex a9eb w>mt fp68k     ( package 4 )
    a9ec w>mt elems68k  ( package 5 )
( extended precision operations )
: f+ 0 fp68k ; : f- 2 fp68k ; : f* 4 fp68k ; : f/ 6 fp68k ;
: x2x e fp68k ;  : fneg d fp68k ;
( single to extended operations )
: s+ 1000 fp68k ; : s- 1002 fp68k ; : s2x 100e fp68k ;
: s* 1004 fp68k ; : s/ 1006 fp68k ; : x2s 1010 fp68k ;
( long integer to extended operations )
: in+ 2800 fp68k ; : in- 2802 fp68k ; 
: in2x 280e fp68k ; : in* 2804 fp68k ; 
: in/ 2806 fp68k ; : x2in 2810 fp68k ;
: d2b 9 fp68k ; : b2d b fp68k ;
   ( decimal <--> binary conversions )
: float create 10 allot ; : integer create 4 allot ;
: wvar create 2 allot ;    ( type declarations )
( floating point i/o )
decimal
: numstring create 24 allot ;  ( decimal display string )
hex 1000000 constant fixdec decimal 
( format style control )
variable zzformat 
( internal format for conversion routine )
numstring zzs1 ( internal conversion string )
: dec. ( float\format# -- )
       zzformat ! zzformat swap zzs1 b2d
       zzs1 dup w@ 255 > if ." -" else ."  " then
       dup 4+ count type ( mantissa )
       2+ w@ ( get exponent )
            1 w* ( convert to 32 bit integer )
            ." E" . ;

( floating point initialization )
: fclear 0 over ! 0 over 4+ ! 0 over 8+ w! drop ;
: sclear 0 swap ! ;

( Matrix Operators )                               
: calc.offset  ( row\col\addr -- addr )
           dup>r  4+ @  ( #cols)  4*        ( 32-bit )
           rot *  ( offset to row)  swap 4*    ( 32-bit )
           +  ( offset to element ) r> 8+  + ( add base addr) ;

: matrix  ( #rows\#cols -- )
    create over ,  ( #rows )  dup ,  ( #cols )
            *  4* allot  ( allot the space for the matrix )
    does>  calc.offset ;

( Gauss algorithm for linear equations, definitions)
: k rp@ 20 + @ ;
variable nv   variable coeff variable solution
( addresses for storing actual parameters)
float one  float -one  float zero  float two  float four
1 one !  -1 -one !  0 zero !  2 two !  4 four !
one one in2x  two two in2x  -one -one in2x  
zero zero in2x four four in2x
float fa1   float fa2   float fa3   float fa4
( define some floating accumulators)
float dg    float fk    float ee
create z 12 allot   3 4 matrix x
: ztest 
      3 0 do i 4* solution @ + fa1 s2x fa1 5 dec. loop cr ;
( setup coefficient matrix for example)
one 0 0 x x2s  one 0 1 x x2s  one 0 2 x x2s  
                                                           zero 0 3 x 
x2s
one 1 0 x x2s -one 1 1 x x2s  two 1 2 x x2s   
                                                           two 1 3 x 
x2s
four 2 0 x x2s  one 2 1 x x2s -one 2 2 x x2s  
                                                           four 2 3 x 
x2s
( Gauss algorithm for linear equations) 
: gauss ( z\x\n | --)  nv !  8- coeff !  solution !
  nv @ 1- 0 do  ( i-loop)
     i dup coeff @ calc.offset dg s2x ( diag elem)
     nv @ i 1+ do  ( j-loop)
        i j coeff @ calc.offset fk s2x   dg fk f/
        nv @ 1+ j do  ( k-loop)
            k i coeff @ calc.offset fa1 s2x
                      fk fa1 f*  fa1 fneg  ( -fk*x[i,k])
            j i coeff @ calc.offset dup fa1 s+
                      fa1 swap x2s
                  loop
              loop
           loop
nv @ dup 0 do i over coeff @ calc.offset  fa1 s2x
                       fa1 solution @ i 4* + x2s loop
1 nv @ 1- do
     i dup coeff @ calc.offset dg s2x
     solution @ i 4* + ee s2x  dg ee f/
     0 i 1- do i j coeff @ calc.offset fa1 s2x
                         ee fa1 f* fa1 fneg
               solution @ i 4* + dup fa1 s+ fa1 swap x2s
            -1 +loop
       -1 +loop
nv @ 0 do  solution @ i 4* +  fa1 s2x
           i dup coeff @ calc.offset  fa1 s/
           fa1 solution @ i 4* + x2s
       loop ;

: soln ." The solution is: " ztest ; 

: gbm 0 do z 0 0 x 3 gauss loop soln ;
 

Community Search:
MacTech Search:

Software Updates via MacUpdate

VirtualBox 5.1.10 - x86 virtualization s...
VirtualBox is a family of powerful x86 virtualization products for enterprise as well as home use. Not only is VirtualBox an extremely feature rich, high performance product for enterprise customers... Read more
Pixa 1.1.9 - Quickly and easily organize...
Pixa is an image-organizing application. The new app functions well, is easy to use, and helps people organize their images quickly and easily on their computers. For those who prefer not to use the... Read more
Civilization VI 1.0.1 - Next iteration o...
Sid Meier’s Civilization VI is the next entry in the popular Civilization franchise. Originally created by legendary game designer Sid Meier, Civilization is a strategy game in which you attempt to... Read more
Google Chrome 55.0.2883.75 - Modern and...
Google Chrome is a Web browser by Google, created to be a modern platform for Web pages and applications. It utilizes very fast loading of Web pages and has a V8 engine, which is a custom built... Read more
Chromium 55.0.2883.75 - Fast and stable...
Chromium is an open-source browser project that aims to build a safer, faster, and more stable way for all Internet users to experience the web. Version 55.0.2883.75: Security fixes: High CVE-2016... Read more
Luminar 1.0.2 - Powerful, adaptive, conf...
Luminar is the new full-featured image editor that adapts to the way you edit photos. Over 300 essential tools to fix, edit, and enhance your photos with comfort. The future of photo editing is here... Read more
Slack 2.3.3 - Collaborative communicatio...
Slack is a collaborative communication app that simplifies real-time messaging, archiving, and search for modern working teams. Version 2.3.3: Fixed window zoom jumping back-and-forth OS X 10.9... Read more
WhatRoute 2.0.10 - Geographically trace...
WhatRoute is designed to find the names of all the routers an IP packet passes through on its way from your Mac to a destination host. It also measures the round-trip time from your Mac to the router... Read more
Luminar 1.0.2 - Powerful, adaptive, conf...
Luminar is the new full-featured image editor that adapts to the way you edit photos. Over 300 essential tools to fix, edit, and enhance your photos with comfort. The future of photo editing is here... Read more
WhatRoute 2.0.10 - Geographically trace...
WhatRoute is designed to find the names of all the routers an IP packet passes through on its way from your Mac to a destination host. It also measures the round-trip time from your Mac to the router... Read more

Latest Forum Discussions

See All

Amateur Surgeon 4 Guide: Become the worl...
It's time to wield your trusty pizza cutter again, as Amateur Surgeon has returned with a whole fresh set of challenges (and some old, familiar ones, too). Starting anew isn't easy, especially when all you have at your disposal is a lighter, the... | Read more »
Le Parker: Sous Chef Extraordinaire (Ga...
Le Parker: Sous Chef Extraordinaire 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: | Read more »
Telltale Games really is working on a Gu...
Telltale Games' next episodic adventure is indeed Guardians of the Galaxy. A document tied to the voice actors strike suggested that the project was in the work, but now we have direct confirmation following an announcement at the Game Awards that... | Read more »
Amateur Surgeon returns to iOS and Andro...
Amateur Surgeon and its two sequels disappeared from the App Store some time and it was sad days for all. But now, just in time for the holidays, the Adult Swim favorite makes its joyous return in the shape of Amateur Surgeon 4, a remake with... | Read more »
The best board games on mobile
Sometimes you need to ditch all of the high speed, high action games in favor of something a little more traditional. If you don't feel like parting ways from your mobile device, though, there are still plenty of ways to get that old-school fix.... | Read more »
The best Facebook Messenger Instant Game...
Facebook's new Instant Games is now here, meaning you can play games with your friends directly via Facebook. It's a fun new way to connect with friends, of course, but it's also proving to be a solid gaming experience in its own right, with a... | Read more »
You can now play game's on Facebook...
Facebook launched its new Instant Games platform in an exciting new attempt to engage its user base. As a result, you can now play a number of different games directly through Facebook Messenger. All of these games run with HTML5, meaning you play... | Read more »
Apollo Justice Ace Attorney (Games)
Apollo Justice Ace Attorney 1.00.00 Device: iOS Universal Category: Games Price: $.99, Version: 1.00.00 (iTunes) Description: Court Is Back In Session Star as rookie defense attorney, Apollo Justice, as he visits crime scenes,... | Read more »
KORG iWAVESTATION (Music)
KORG iWAVESTATION 1.0 Device: iOS Universal Category: Music Price: $19.99, Version: 1.0 (iTunes) Description: A revolutionary new world of sound.The Wave Sequence Synthesizer for iPad - KORG iWAVESTATION | Read more »
Don't Grind Guide: Tips for becomin...
Don’t Grind is a surprising, derpy little one touch game with fun hand-drawn graphics. The goal is simple -- get the high score without being chopped to bits. That can be tough when you’re not used to the game, and that’s compounded by the fact... | Read more »

Price Scanner via MacPrices.net

13-inch Silver Touch Bar MacBook Pro in stock...
Amazon has the new 2016 13″ 2.9GHz/256GB Silver Touch Bar MacBook Pro (MLVP2LL/A) in stock today and on sale for $1749 including free shipping. That’s $50 off MSRP, and it’s the lowest price... Read more
Parallels Toolbox 1.3 for Mac Offers 25 Singl...
Parallels has launched Parallels Toolbox 1.3 for Mac, an upgrade that adds five new utilities to the stand-alone application which was released in August and is available exclusively online at http... Read more
OWC Mercury Elite Pro Dual mini Ultra-Portabl...
OWC has introduced the new OWC Mercury Elite Pro Dual mini, a powerful yet ultra-portable dual-drive RAID solution. The new Mercury Elite Pro Dual mini packs phenomenal performance into a small... Read more
Clearance 13-inch Retina MacBook Pros availab...
B&H Photo has clearance 2015 13″ Retina Apple MacBook Pros available for up to $200 off original MSRP. Shipping is free, and B&H charges NY tax only: - 13″ 2.7GHz/128GB Retina MacBook Pro: $... Read more
Roundup of 2016 13-inch 2.0GHz MacBook Pro sa...
B&H has the non-Touch Bar 13″ MacBook Pros in stock today for $50-$100 off MSRP. Shipping is free, and B&H charges NY sales tax only: - 13″ 2.0GHz MacBook Pro Space Gray (MLL42LL/A): $1449 $... Read more
New 13-inch 2.0GHz Space Gray MacBook Pro in...
Adorama has the new 13″ 2.0GHz Space Gray MacBook Pro (non-Touch Bar, MLL42LL/A) in stock for $1499 including a free 3-year AppleCare Protection Plan. Shipping is free, and Adorama charges sales tax... Read more
Finnair Adopts iOS Enterprise iPad Apps from...
Finnair and IBM have announced a first-of-its-kind agreement to utilize iOS enterprise apps from IBM to support the airline’s overall digital transformation. Finnair is focused on Asia-Europe traffic... Read more
Tech21 Launches Evo Go iPhone 7 Case Availabl...
Tech21 has announced the launch of the Evo Go case for Apple iPhone 7 and iPhone 7 Plus, exclusively at T-Mobile. Available online and at participating T-Mobile stores nationwide, Evo Go cases start... Read more
Apple Turns (RED) with More Ways to Join the...
In recognition of World AIDS Day, Apple is offering more ways than ever for customers to join (RED) in its mission to create an AIDS-free generation. Apple is the worlds largest corporate contributor... Read more
Deals on new 15-inch Touch Bar MacBook Pros,...
B&H Photo has new 2016 Apple 15″ Touch Bar MacBook Pro models in stock today with some available for $50 off MSRP, each including free shipping plus NY sales tax only: - 15″ 2.7GHz Touch Bar... Read more

Jobs Board

*Apple* Brand Ambassador (Macy's) - The...
…(T-ROC), is proud of its unprecedented relationship with our partner and client, APPLE ,in bringing amazing" APPLE ADVOCATES"to "non" Apple store locations. Read more
US- *Apple* Store Leader Program - Apple (Un...
…Summary Learn and grow as you explore the art of leadership at the Apple Store. You'll master our retail business inside and out through training, hands-on Read more
*Apple* Retail - Multiple Positions- White P...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
Automotive Detailer - *Apple* Used Autos -...
We are currently conductinginterviews and will be accepting applications for a part-time detailer. Apple Used Autos is a great place to work andstart a career. We Read more
*Apple* Retail - Multiple Positions - Apple,...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.