PROGRAM: Non-linear Equations. 
     
    BASIC DEMO FILENAME = $.NLinEq - Side 0 
    ASCII TEXT FILENAME = T.NLinEq - Side 0 
     
    The program consists of two main modules: 'graph plotter' that depicts 
    any function y=f(x) on the screen and 'equation solver' which 
    calculates roots of an equation f(x)=0 using one of the three 
    iteration methods.  The former module has been included to obtain some 
    idea of a range in which a solution is likely to appear (if a solution 
    exists at all).  When running the program you are asked to enter the 
    formula for f(x).  It must be given in terms of 'X' (or 'x') and must 
    be a valid BBC BASIC expression, except that lower-case letters and 
    capitals can be used interchangeably.  Although an attempt has been 
    made to check the formula as soon as it is entered, some mistakes are 
    detected a little bit later and the program returns you to re-enter 
    the formula.  Therefore, do not forget to use '*' whenever 
    multiplication occurs and make sure that the number of opening 
    brackets is equal to the number of closing brackets. 
     
    When you have defined the function you will be asked to specify the 
    range of X and Y to appear on the plot.  If you do not choose these 
    suitably you may lose most or all of the points by plotting them way 
    off screen somewhere.  It is advisable, however, to use rather broad 
    X and Y ranges initially, say [-5,+5] and [-10,+10], respectively.  
    You can always change the range later and re-plot the graph with the 
    refined range.   
     
    The plotting routine is programmed so that it works even though the 
    function is not continuous at some points or it does not exist at all 
    within some intervals.  However, as the X-axis must always appear on 
    the screen, so the user is forced to enter a negative number as the Y 
    minimal value and a positive number as the Y maximum value.  While 
    plotting is in progress, X-values, at which the function value is 
    zero, are displayed in the text window at the bottom of the screen.  
    These values should be treated as rough estimates and not as a final 
    solution.  In some circumstances (in particular when the equation 
    y=f(x) has two or more coincident roots) those X-values do not appear 
    in the text window even though a solution of the equation exists 
    within the given X-range.  If the curve does not cross (or does not 
    touch) the X-axis you will have to try a different X-range, unless you 
    have decided there is no solution at all, in which case either exit or 
    change the formula. 
     
    When the graph is drawn you can use the second module, 'equation 
    solver'.  Again, you will be asked to answer a number of questions and 
    the first of them deals with selection of the method of calculation. 
    There are three method available: a tangent method of Newton as 
    modified by Raphson, a bisection method, and the so called 'regula 
    falsi' in its most efficient version ('variant Illinois').  None of 
    them is universal and each fails under some circumstances.  However, 
    if one method will fail you can use other method and it may work 
    perfectly.  Whatever method you choose you will be asked for the 
    accuracy required.  It is reasonable to start with low accuracy value, 
    for example 1E-4 (ie. 0.0001) and then increase the accuracy by 
    decreasing the 'epsylon' value, even to its limit (ie. 1E-9). 
     
    With the Newton-Raphson method you have to enter an initial guess 
    value.  This value should be close to the expected root value. With 
    two other methods you have to enter an initial range of X values 
    straddling the expected root value so that the function must change 
    sign within this range (ie. if the function value is positive at the 
    lower (ie. left) bound, it must be negative at the upper (ie. right) 
    bound, and vice versa). 
     
    If the method will work for a given case, a result is displayed on the 
    screen and you are offered to a number of options. So you can alter 
    single parameters, change a method, return to the graph plotting 
    section or re-run the program.  As each method allows to find only one 
    solution at a time so if you want to find another root just press 'P' 
    key and change initial values.  The program is highly subject to user 
    errors. However, it is believed that all possible errors are trapped. 
    The program simply displays various messages and returns you to the 
    point, where you can change everything and try again. 
     
    At last a short comment on the output of results. 
    Using the program you will discover very soon that solving the 
    equation x^2-4=0 you may obtain as a solution '2.00000247' or 
    '1.99999036' instead of '2'. There is no error at all as long the 
    accuracy value declared by you is 1E-5 (ie. 0.00001) or greater. 
    Unfortunately, similar 'strange' results often appear on the screen 
    when the highest permissible accuracy of 1E-9 has been declared and in 
    such cases inaccuracy is due to the round-off errors generated by the 
    computer during floating point calculations. 
     
    As you probably know, when decimal numbers are entered into the 
    computer they are converted into binary floating point notation and 
    stored as the 32-bit 'mantissa' and the 8-bit 'exponent' (40 bits or 5 
    bytes in total).  This allows to store the first 9 digit of a decimal 
    number without error.  However, if two numbers, each with 32 binary 
    digits mantissa, are multiplied together, their true product will have 
    64 digits but only 32 of them can be stored, thus causing rounding 
    error. With a large number of multiplications or other floating point 
    operations some of these rounding errors compensate but the others can 
    accumulate, contaminating the final result. 
     
    Miroslaw Bobrowski L1J.