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

Together 3.6.1 - Store and organize all...
Together helps you organize your Mac, giving you the ability to store, edit and preview your files in a single clean, uncluttered interface. Features Smart storage. With simple drag-and-drop... Read more
Cloud 4.1.1 - File sharing from your men...
Cloud is simple file sharing for the Mac. Drag a file from your Mac to the CloudApp icon in the menubar and we take care of the rest. A link to the file will automatically be copied to your clipboard... Read more
OmniFocus 2.7.1 - GTD task manager with...
OmniFocus helps you manage your tasks the way that you want, freeing you to focus your attention on the things that matter to you most. Capturing tasks and ideas is always a keyboard shortcut away in... Read more
CleanApp 5.1.1 - Application deinstaller...
CleanApp is an application deinstaller and archiver.... Your hard drive gets fuller day by day, but do you know why? CleanApp 5 provides you with insights how to reclaim disk space. There are... Read more
ForkLift 3.0 Beta 2 - Powerful file mana...
ForkLift is a powerful file manager and ferociously fast FTP client clothed in a clean and versatile UI that offers the combination of absolute simplicity and raw power expected from a well-executed... Read more
Sublime Text 3126 - Sophisticated text e...
Sublime Text is a sophisticated text editor for code, markup, and prose. You'll love the slick user interface, extraordinary features, and amazing performance. Features Goto Anything. Use Goto... Read more
1Password 6.3.3 - Powerful password mana...
1Password is a password manager that uniquely brings you both security and convenience. It is the only program that provides anti-phishing protection and goes beyond password management by adding Web... Read more
WhatsApp 0.2.1880 - Desktop client for W...
WhatsApp is the desktop client for WhatsApp Messenger, a cross-platform mobile messaging app which allows you to exchange messages without having to pay for SMS. WhatsApp Messenger is available for... Read more
NeoFinder 6.9.3 - Catalog your external...
NeoFinder (formerly CDFinder) rapidly organizes your data, either on external or internal disks, or any other volumes. It catalogs all your data, so you stay in control of your data archive or disk... Read more
Amadeus Pro 2.3.1 - Multitrack sound rec...
Amadeus Pro lets you use your Mac computer for any audio-related task, such as live audio recording, digitizing tapes and records, converting between a variety of sound formats, etc. Thanks to its... Read more

Paperback: The Game (Games)
Paperback: The Game 1.0 Device: iOS Universal Category: Games Price: $3.99, Version: 1.0 (iTunes) Description: You are an author trying to finish kitschy paperback novels. Complete Westerns, Science Fiction, Romance or even a Crime... | Read more »
How to Rule With a Firm Hand in My Majes...
My Majesty is a kingdom management sim not unlike August’s magisterial hit, Reigns. It’s essentially a reskin of developer Tigrido’s previous management sim, Dictator. As supreme ruler of the land, you must consult with a number of subjects to... | Read more »
Our 5 Favorite iMessage Sticker Packs
At long last, iMessage joins the ranks of messaging apps the likes of LINE and Whatsapp, adding an impressive collection of stickers. They’re a great way to add a little something extra to your daily conversations. [Read more] | Read more »
How to get past Vulture Island's tr...
Vulture Island is a colorful and quirky mish-mash of platforming and puzzles. It’s creative and fresh, but sometimes the game can throw a curveball at you, leaving you stuck as to how you should progress. These tips will help you explore smoothly... | Read more »
The new Clash of Kings is just for Weste...
If you’ve played the original Clash of Kings, you’ll probably recognise the city building, alliance forging and strategic battles in Clash of Kings: The West. What sets this version apart is that it’s tailor made for a Western audience and the... | Read more »
Frost - Survival card game (Games)
Frost - Survival card game 1.12.1 Device: iOS Universal Category: Games Price: $3.99, Version: 1.12.1 (iTunes) Description: *Warning: the game will work on iPhone 5C and above and iPad Pro / 4. Other devices are not supported* | Read more »
How to build and care for your team in D...
Before you hit the trail and become a dog sledding legend, there’s actually a fair bit of prep work to be done. In Dog Sled Saga, you’re not only racing, you’re also building and caring for a team of furry friends. There’s a lot to consider—... | Read more »
How to win every race in Dog Sled Saga
If I had to guess, I’d say Dog Sled Saga is the most adorable racing game on the App Store right now. It’s a dog sled racing sim full of adorable, loyal puppies. Just look at those fluffy little tails wagging. Behind that cute, pixelated facade is... | Read more »
Let the war games commence in Gunship Ba...
Buzz Lightyear famously said, “This isn’t flying, this is falling – with style!” In the case of Gunship Battle: Second War, though, this really is flying - with style! The flight simulator app from Joycity puts you in control of 20 faithfully... | Read more »
How to get a high score in Fired Up
Fired Up is Noodlecake Games’ high score chasing, firefighting adventure. You take control of a wayward firefighter who propels himself up the side of a highrise with blasts of water. Sound silly? It is. It’s also pretty difficult. You can’t... | Read more »

Price Scanner via MacPrices.net

15-inch 2.2GHz Retina MacBook Pro on sale for...
B&H Photo has the 2015 15″ 2.2GHz Retina MacBook Pro (MJLQ2LL/A) on sale for $1799, including free shipping plus NY sales tax only. Amazon also has the 2015 15″ 2.2GHz Retina MacBook Pro (... Read more
Toughbook Celebrates 20 Years of Ruggedized M...
Panasonic System Communications Company of North America, Division of Panasonic Corporation of North America (Panasonic) today celebrates the 20th anniversary of its industry-leading Toughbook mobile... Read more
12-inch 1.1GHz Gray Retina MacBook on sale fo...
B&H Photo has the 2016 12″ 1.1GHz Gray Retina MacBook on sale for $1199.99 including free shipping plus NY sales tax only. Their price is $100 off MSRP. Read more
13-inch 2.5GHz MacBook Pro (Apple refurbished...
Apple has Certified Refurbished 13″ 2.5GHz MacBook Pros available for $829, or $270 off the cost of new models. Apple’s one-year warranty is standard, and shipping is free: - 13″ 2.5GHz MacBook Pros... Read more
Save 30% on Camera Guard’s Secure Protection...
To celebrate the release of macOS Sierra, Miami-based security solutions company, ProtectStar has announced a special 30% discount on Camera Guard Professional for Mac 2016. This innovative security... Read more
DVDFab Special Deal – Get a 1-Year Free Licen...
Beijing, China based specialist in the field of DVD, Blu-ray and video backup solutions, Fengtao Software has launched its Autumn Special Deals 2016, giving a 1-year free license of a randomly picked... Read more
21-inch iMacs on sale for up to $120 off MSRP
B&H Photo has 21″ iMacs on sale for up to $120 off MSRP including free shipping plus NY sales tax only: - 21″ 3.1GHz iMac 4K: $1379 $120 off MSRP - 21″ 2.8GHz iMac: $1199.99 $100 off MSRP - 21″ 1... Read more
13-inch 2.7GHz/256GB Retina MacBook Pro on sa...
Amazon.com has the 13″ 2.7GHz/256GB Retina Apple MacBook Pro on sale for $151 off MSRP including free shipping: - 13″ 2.7GHz/256GB Retina MacBook Pro (sku MF840LL/A): $1348 $151 off MSRP Read more
Apple TVs on sale for up to $50 off MSRP
Best Buy has 32GB and 64GB Apple TVs on sale for $40-$50 off MSRP on their online store. Choose free shipping or free local store pickup (if available). Sale prices for online orders only, in-store... Read more
Apple refurbished 13-inch Retina MacBook Pros...
Apple has Certified Refurbished 13″ Retina MacBook Pros available for up to $270 off the cost of new models. An Apple one-year warranty is included with each model, and shipping is free: - 13″ 2.7GHz... Read more

Jobs Board

Restaurant Manager (Neighborhood Captain) - A...
…in every aspect of daily operation. WHY YOU'LL LIKE IT: You'll be the Big Apple . You'll solve problems. You'll get to show your ability to handle the stress and Read more
Sr. *Apple* Mac Engineer - Net2Source Inc....
…staffing, training and technology. We have following position open with our client. Sr. Apple Mac Engineer6+ Months CTH Start date : 19th Sept Travelling Job If Read more
*Apple* Retail - Multiple Positions-Norfolk,...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
Restaurant Manager (Neighborhood Captain) - A...
…in every aspect of daily operation. WHY YOU'LL LIKE IT: You'll be the Big Apple . You'll solve problems. You'll get to show your ability to handle the stress and Read more
Lead *Apple* Solutions Consultant - Apple (...
# Lead Apple Solutions Consultant Job Number: 51829230 Detroit, Michigan, United States Posted: Sep. 19, 2016 Weekly Hours: 40.00 **Job Summary** The Lead ASC is an Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.