8-Bit Software Online Conversion

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.