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
$97.67
Apple Inc.
+2.95
MSFT
$45.38
Microsoft Corpora
+0.55
GOOG
$597.41
Google Inc.
+2.67

MacTech Search:
Community Search:

Software Updates via MacUpdate

Firefox 31.0 - Fast, safe Web browser. (...
Firefox for Mac offers a fast, safe Web browsing experience. Browse quickly, securely, and effortlessly. With its industry-leading features, Firefox is the choice of Web development professionals... Read more
Little Snitch 3.3.3 - Alerts you to outg...
Little Snitch gives you control over your private outgoing data. Track background activityAs soon as your computer connects to the Internet, applications often have permission to send any... Read more
Thunderbird 31.0 - Email client from Moz...
As of July 2012, Thunderbird has transitioned to a new governance model, with new features being developed by the broader free software and open source community, and security fixes and improvements... Read more
Together 3.2 - Store and organize all of...
Together helps you organize your Mac, giving you the ability to store, edit and preview your files in a single clean, uncluttered interface. Smart storage. With simple drag-and-drop functionality,... Read more
Cyberduck 4.5 - FTP and SFTP browser. (F...
Cyberduck is a robust FTP/FTP-TLS/SFTP browser for the Mac whose lack of visual clutter and cleverly intuitive features make it easy to use. Support for external editors and system technologies such... Read more
Airmail 1.4 - Powerful, minimal email cl...
Airmail is a powerful, minimal mail client.It was designed to retain the same experience with a single or multiple accounts and provide a quick, modern and easy-to-use user experience. Airmail... Read more
Macs Fan Control 1.1.12 - Monitor and co...
Macs Fan Control allows you to monitor and control almost any aspect of your computer's fans, with support for controlling fan speed, temperature sensors pane, menu-bar icon, and autostart with... Read more
A Better Finder Rename 9.37 - File, phot...
A Better Finder Rename is the most complete renaming solution available on the market today. That's why, since 1996, tens of thousands of hobbyists, professionals and businesses depend on A Better... Read more
MacBook Air EFI Firmware Update 2.9 - Fo...
MacBook Air EFI Firmware Update is recommended for MacBook Air (Mid 2011) models. This update addresses an issue where systems may take longer to wake from sleep than expected and fixes a rare issue... Read more
FileZilla 3.9.0.1 - Fast and reliable FT...
FileZilla (ported from Windows) is a fast and reliable FTP client and server with lots of useful features and an intuitive interface.Version 3.9.0.1: MSW: Fix installation issue with locked DLLs... Read more

Latest Forum Discussions

See All

Stronghold 3: The Campaigns Review
Stronghold 3: The Campaigns Review By Jennifer Allen on July 23rd, 2014 Our Rating: :: DULL STRATEGIZINGiPad Only App - Designed for the iPad A cumbersome strategy game, Stronghold 3: The Campaigns has a few too many issues to... | Read more »
Table Tennis Touch on Sale for a Limited...
Table Tennis Touch on Sale for a Limited Time Posted by Jessica Fisher on July 23rd, 2014 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Secret Files Tunguska Review
Secret Files Tunguska Review By Jennifer Allen on July 23rd, 2014 Our Rating: :: CONSPIRACY-LITTERED ADVENTURINGUniversal App - Designed for iPhone and iPad Offering traditional adventuring with no fear of in-app purchases, Secret... | Read more »
Celebrate Summer With a Cat in the Hat L...
Celebrate Summer With a Cat in the Hat Learning Library Sale Posted by Ellis Spice on July 22nd, 2014 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Dragon Raiders Review
Dragon Raiders Review By Nadia Oxford on July 22nd, 2014 Our Rating: :: RUN, DRAGON, RUNUniversal App - Designed for iPhone and iPad Dragon Raiders is rough and scaly in some parts, but overall it’s an enjoyable level-based running... | Read more »
MyTaskList Review
MyTaskList Review By Jennifer Allen on July 22nd, 2014 Our Rating: :: EFFECTIVE IF PLAINUniversal App - Designed for iPhone and iPad It’s not the most stylish of task management apps, but MyTaskList has all the features you could... | Read more »
FlyCraft Herbie: Crazy Machines Review
FlyCraft Herbie: Crazy Machines Review By Jennifer Allen on July 22nd, 2014 Our Rating: :: TRICKY FLYINGUniversal App - Designed for iPhone and iPad A tough game of careful thrusting and navigation, FlyCraft Herbie: Crazy Machines... | Read more »
MTN Review
MTN Review By Jessica Fisher on July 22nd, 2014 Our Rating: :: ADORABLE, SERENE, AND AMUSINGUniversal App - Designed for iPhone and iPad MTN is an adorable, talking pet mountain that is less game and more zen garden.   | Read more »
Fly High with Ninja UP! Now Available o...
Fly High with Ninja UP! Now Available on the App Store Posted by Jessica Fisher on July 22nd, 2014 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Bio Inc. Review
Bio Inc. Review By Nadia Oxford on July 22nd, 2014 Our Rating: :: SICKENING - IN A COMPELLING WAYUniversal App - Designed for iPhone and iPad Bio Inc is about orchestrating the medical destruction of a single person. If that doesn’... | Read more »

Price Scanner via MacPrices.net

MacBook Airs on sale starting at $799, free s...
B&H Photo has the new 2014 MacBook Airs on sale for up to $100 off MSRP for a limited time. Shipping is free, and B&H charges NY sales tax only. They also include free copies of Parallels... Read more
Apple 27″ Thunderbolt Display (refurbished) a...
The Apple Store has Apple Certified Refurbished 27″ Thunderbolt Displays available for $799 including free shipping. That’s $200 off the cost of new models. Read more
WaterField Designs Unveils Cycling Ride Pouch...
High end computer case and bag maker WaterField Designs of San Francisco now enters the cycling market with the introduction of the Cycling Ride Pouch – an upscale toolkit with a scratch-free iPhone... Read more
Kingston Digital Ships Large Capacity Near 1T...
Kingston Digital, Inc., the Flash memory affiliate of Kingston Technology Company, Inc.,has announced its latest addition to the SSDNow V300 series, the V310. The Kingston SSDNow V310 solid-state... Read more
Apple’s Fiscal Third Quarter Results; Record...
Apple has announced financial results for its fiscal 2014 third quarter ended June 28, 2014, racking up quarterly revenue of $37.4 billion and quarterly net profit of $7.7 billion, or $1.28 per... Read more
15-inch 2.0GHz MacBook Pro Retina on sale for...
B&H Photo has the 15″ 2.0GHz Retina MacBook Pro on sale for $1829 including free shipping plus NY sales tax only. Their price is $170 off MSRP. B&H will also include free copies of Parallels... Read more
Apple restocks refurbished Mac minis for up t...
The Apple Store has restocked Apple Certified Refurbished Mac minis for up to $150 off the cost of new models. Apple’s one-year warranty is included with each mini, and shipping is free: - 2.5GHz Mac... Read more
Twelve South HiRise For MacBook – Height-Adju...
If you use your MacBook as a workhorse desktop substitute, as many of us do, a laptop stand combined with an external keyboard and pointing device are pretty much obligatory if you want to avoid... Read more
Why The Mac Was Not Included In The Apple/IBM...
TUAW’s Yoni Heisler cites Fredrick Paul of Network World whoi blogged last week that the Mac’s conspicuous absence from Apple and IBM’s landmark partnership agreement represents a huge squandered... Read more
Save $100 on 13-inch Retina MacBook Pros, plu...
Adorama has 13″ Retina MacBook Pros on sale for $100 off MSRP. Shipping is free, and Adorama charges sales tax in NY & NJ only: - 13″ 2.4GHz/128GB MacBook Pro with Retina Display: $1199 - 13″ 2.... Read more

Jobs Board

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
*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
*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 *Apple* Engineer - IT - Requisition #: -...
For more information about TIAA-CREF, visit our website . The Apple Engineer will provide engineering and third-level incident support for 300- 500 MacOS desktop/laptop Read more
*Apple* Systems Administrator - DISH (United...
…satellite service provider, and Dish is currently looking for an experienced Apple /Mac Systems Administrator. Apple systems administrator will be responsible for Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.