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 ;
 
AAPL
$96.13
Apple Inc.
+0.53
MSFT
$42.86
Microsoft Corpora
-0.30
GOOG
$566.07
Google Inc.
-5.53

MacTech Search:
Community Search:

Software Updates via MacUpdate

Data Rescue 3.2.4 - Recover lost data on...
Data Rescue is a robust and reliable hard-drive recovery solution for your Mac. Recover lost or deleted files, mount corrupted drives, and more -- Data Rescue offers complete relief from crippling... Read more
Adobe Lightroom 5.6 - Import, develop, a...
Adobe Lightroom software helps you bring out the best in your photographs, whether you're perfecting one image, searching for ten, processing hundreds, or organizing thousands. Create incredible... Read more
OneNote 15.2 - Free digital notebook fro...
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
iStat Menus 4.22 - Monitor your system r...
iStat Menus lets you monitor your system right from the menubar. Included are 8 menu extras that let you monitor every aspect of your system. Some features: CPU -- Monitor cpu usage. 7 display... Read more
Ember 1.8 - Versatile digital scrapbook....
Ember (formerly LittleSnapper) is your digital scrapbook of things that inspire you: websites, photos, apps or other things. Just drag in images that you want to keep, organize them into relevant... Read more
OmniPlan 2.3.6 - Robust project manageme...
With OmniPlan, you can create logical, manageable project plans with Gantt charts, schedules, summaries, milestones, and critical paths. Break down the tasks needed to make your project a success,... Read more
Command-C 1.1.1 - Clipboard sharing tool...
Command-C is a revolutionary app which makes easy to share your clipboard between iOS and OS X using your local WiFi network, even if the app is not currently opened. Copy anything (text, pictures,... Read more
Knock 1.1.7 - Unlock your Mac by knockin...
Knock is a faster, safer way to sign in. You keep your iPhone with you all the time. Now you can use it as a password. You never have to open the app -- just knock on your phone twice, even when it's... Read more
Mellel 3.3.6 - Powerful word processor w...
Mellel is the leading word processor for OS X and has been widely considered the industry standard since its inception. Mellel focuses on writers and scholars for technical writing and multilingual... Read more
LibreOffice 4.3.0.4 - Free Open Source o...
LibreOffice is an office suite (word processor, spreadsheet, presentations, drawing tool) compatible with other major office suites. The Document Foundation is coordinating development and... Read more

Latest Forum Discussions

See All

Bio Inc. is $0.99 for the Weekend, Recei...
Bio Inc. is $0.99 for the Weekend, Receives Small Update Posted by Ellis Spice on August 1st, 2014 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Happy 7th Birthday Readdle! Thank You fo...
Happy 7th Birthday Readdle! | Read more »
Sharknado: The Video Game Review
Sharknado: The Video Game Review By Lee Hamlet on August 1st, 2014 Our Rating: :: SHARKNA-DON'TUniversal App - Designed for iPhone and iPad Sharknado: The Video Game brings the craziness of the movies to iOS, though it quickly... | Read more »
Clima (Weather)
Clima 1.0 Device: iOS iPhone Category: Weather Price: $.99, Version: 1.0 (iTunes) Description: Clima show you all weather information, just beautifully simple. A series of color bars can tell you at a glance exactly current... | Read more »
Sticky Soccer Review
Sticky Soccer Review By Andrew Fisher on August 1st, 2014 Our Rating: :: STICK THIS GAMEUniversal App - Designed for iPhone and iPad Sticky Soccer puts too much emphasis on the ‘sticky’ and not enough on the ‘Soccer’ or ‘Fun’.   | Read more »
Graphics-less Apocalyptic Adventure A Da...
Graphics-less Apocalyptic Adventure A Dark Room Goes Free for a Limited Time Posted by Rob Rich on August 1st, 2014 [ permalink ] | Read more »
Fraud Tycoon Review
Fraud Tycoon Review By Rob Thomas on August 1st, 2014 Our Rating: :: UNHEALTHY CREDITUniversal App - Designed for iPhone and iPad Fraud Tycoon is a half-baked, messy, promotional tie-in that does their sponsor no favors whatsoever... | Read more »
Guardians of the Galaxy: The Universal W...
Guardians of the Galaxy: The Universal Weapon is on Sale for the Weekend Posted by Rob Rich on August 1st, 2014 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Mister Beam Review
Mister Beam Review By Jordan Minor on August 1st, 2014 Our Rating: :: ILLUMINATINGUniversal App - Designed for iPhone and iPad Mister Beam’s puzzles are great. But its platforming? Not so much.   | Read more »
Hook Some More Fun With MapHook’s New Up...
Hook Some More Fun With MapHook’s New Update Posted by Jessica Fisher on August 1st, 2014 [ permalink ] iPhone App - Designed for the iPhone, compatible with the iPad | Read more »

Price Scanner via MacPrices.net

13-inch MacBook Airs on sale for $100 off MSR...
B&H Photo has the new 2014 13″ MacBook Airs on sale $100 off MSRP. Shipping is free, and B&H charges NY sales tax only. They also include free copies of Parallels Desktop and LoJack for... Read more
16GB iPad Air on sale for $399, save $100
Best Buy is offering the 16GB WiFi iPad Air for $399.99 on their online store for a limited time. Their price is $100 off MSRP. Choose free shipping or free store pickup (if available). Price is for... Read more
All Over For Tablets Or Just A Maturing, Evol...
CNN’s David Goldman weighs in on tablet sector doom and gloom, asking rhetorically: “Is this the beginning of the end for the tablet?” Answering that, he contends that hysteria and panic are... Read more
Letterspace 1.0.1 – New Free iOS Text Editor...
Bangkok, Thailand based independent developer Sittipon Simasanti has released Letterspace, a new text editor for iPhone, iPad, and iPod touch devices. Letterspace is a note taking app with an... Read more
Save up to $130 on an iPad mini with Apple re...
The Apple Store has Certified Refurbished 2nd generation iPad minis with Retina Displays available for up to $130 off the cost of new models, starting at $339. Apple’s one-year warranty is included... Read more
iPad Cannibalization Threat “Overblown”
Seeking Alpha’s Kevin Greenhalgh observes that while many commentators think Apple’s forthcoming 5.5-inch panel iPhone 6 will cannibalize iPad sales, in his estimation, these concerns are being... Read more
Primate Labs Releases July 2014 MacBook Pro P...
Primate Labs’ John Poole has posted Geekbench 3 results for most of the new MacBook Pro models that Apple released on Tuesday. Poole observes that overall performance improvements for the new MacBook... Read more
Apple Re-Releases Bugfixed MacBook Air EFI Fi...
Apple has posted a bugfixed version EFI Firmware Update 2.9 a for MacBook Air (Mid 2011) models. The update addresses an issue where systems may take longer to wake from sleep than expected, and... Read more
Save $50 on the 2.5GHz Mac mini, plus free sh...
B&H Photo has the 2.5GHz Mac mini on sale for $549.99 including free shipping. That’s $50 off MSRP, and B&H will also include a free copy of Parallels Desktop software. NY sales tax only. Read more
Save up to $140 on an iPad Air with Apple ref...
Apple is offering Certified Refurbished iPad Airs for up to $140 off MSRP. Apple’s one-year warranty is included with each model, and shipping is free. Stock tends to come and go with some of these... Read more

Jobs Board

Position Opening at *Apple* - Apple (United...
**Job Summary** The Apple Store is a retail environment like no other - uniquely focused on delivering amazing customer experiences. As an Expert, you introduce people Read more
Position Opening at *Apple* - Apple (United...
**Job Summary** At the Apple Store, you connect business professionals and entrepreneurs with the tools they need in order to put Apple solutions to work in their Read more
*Apple* Retail - Multiple Positions (US) - A...
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
Sr. Product Leader, *Apple* Store Apps - Ap...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring Read more
Sr Software Lead Engineer, *Apple* Online S...
Sr Software Lead Engineer, Apple Online Store Publishing Systems Keywords: Company: Apple Job Code: E3PCAK8MgYYkw Location (City or ZIP): Santa Clara Status: Full Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.