The Learning Curve

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Documentation to Comgraph """"""""""""""""""""""""""" """"""""""""""""""""""" """"""""""""""""""" Configuration needed : Kickstart 1.2 or higher 1 MB Memory Libraries needed in the systemdirectory LIBS: MathIEEEDoubBas.library MathIEEEDoubTrans.library Devices needed in the systemdirectory DEVS: printer.device parallel.device serial.device Comgraph is a 2D-functionplotter, handling up to 10 functions and 5 of their differentations at the same time. Further it has the possibility to create 10 bands per function (including their differentations!). The screen used has always the same resolution as the Workbenchscreen, so also the new resolutions can be used, whereas the printer output is not simply a hardcopy (is also possible), but a to the resolution of the printer fitted result. The size (in cm) and the density of the print can be choosen freely. A build-in curvediskussion calculates the first and second symbolic differentiation and determine zero digits, extrema, turning points (saddle points), pole digits and definition gaps and symmetry. Tables of values can be printed, for those who like more to draw the functions by themselves. Furthermore the program has a calculator, prime number calculations and prime factor reduction. The programm contains a german- and also an english version. It uses routines working with LONGREAL numbers (64 Bit floating point) and so they are a bit slow, but they work in smallest areas exact. The loss of speed against FFP (Fast Floating Point) is less bad than inexact values. Start of the program : From the CLI : Stack 100000 (can also work with less memory) Comgraph [-gb] [+gb] From the WorkBench : If there are no options to start with, just doubleclick the icon of Comgraph, else click that icon once, press SHIFT and click all wished options, the last option have to be doubleclicked. The options : +g install german version -g install english version +b show all bitplanes -b kill one bitplane => memoryprofit (loss of colors) Standard the options are set to : english version with all bitplanes Furthermore there is the possibility to set the options by the Environmentsystem : SetEnv Comgraph [-gb] [+gb] When the environment is used, all options are set to the given values. Given options at the start of the program have a higher priority than the environment options. After the start an empty screen should appear with a titlebar containing the name of the program and the copyright. The Menus : Project : Info <AMIGA>+A shows informations about the program, the copyright and the free Chip- and FastMemory. Workbench <AMIGA>+B close/open whe WorkBench to spare ca. 45 KB Memory (If there is less than 200 KB ChipMem at the start of the program, it tries to close the WorkBench automatically). In order to close the WorkBench there may only be WorkBenchWindows on it! Quit <AMIGA>+Q leaves the program after a security check. Plotter : Coordinate system <AMIGA>+S asks the x- and y-axis interval to be displayed, on which values marks should be set (e.g. an interval [0;100] could have marks lying 10 units apart from each other, so 11 marks would be created. If a mark would be set at each unit, there would be 101 marks that can not clearly be distuingished!). The number of points between the marks are displayed in a grid lying over the coordinate system. Each part of that grid has a certain number of points, normaly 10 points in each direction (dividing the marks in 10 parts). Logarithmic scales can be installed in X-, Y- or in both directions by selecting the respective gadgets. Because negative logarithm do not exist, all logarithmic scales start at 1. The marks would in higher areas accumulate, so these marks will not be set in equal distances. Each mark has twice the value of the former (e.g. 1,2,4,8,16...). Zoom <AMIGA>+Z creates a cross on the screen after selecting, that can be moved with the mouse over the screen. By a simple click on the left mousebutton one corner of the zoomwindow is choosen. After moving the pointer to the opposite corner of that zoomwindow and clicking the left mousebutton the second time the area will be zoomed. This function will not work if something was changed in the functions or in the coordinate system between the drawing of the curves and the selection of 'Zoom'. In that case the curves have to be drawn again (at least a part of them). Enter functions <AMIGA>+E asks at first what function should be edited. To do that a requester is opened showing all functions in memory. After a function was choosen an other requester is opened where the function terms can be edited. The format of the terms will be given later on. Curve bands will be used with a constant in the function term that can be replaced by given values. So functions are possible that have no concrete value for a variable (e.g. f(x)=k*x^2 ). It is possible to draw the graph for different values of 'k' at the same time. The number of bands (curves) of one function that should be drawn must be set on values over 1, but only if there is really a constant 'k' in the term. Otherwise the same curve will be drawn several times over itself and that will make the program lots slower. Every (!) function can have up to ten bands that can be displayed at the same time on the screen. Next the start value of the constant have to be choosen and its step that will be added on the constant for the next band. So pseudo-3D functions are possible. At least the number of differentiations is asked that will be drawn on the screen. The given number of differentiations will be drawn for all bands of the actual function! The maximal number of derive functions are five. As higher the degree of the derive as more time will be needed to be drawn! So unneeded differentiations should not be drawn. The syntax of the functions terms : Small and capital letters (also mixed) are allowed, as well as Spaces (not in names of functions or of constants). Brackets can be set everywhere as long as they maintain the mathematical order. The following commands are allowed : x : X-axis value +,-,*,/,^ : The five basic kinds of calculation () : Brackets (to set priorities) sin(),cos() : Sine and cosine tan(),cot() : Tangens and cotangens asin(),acos() : Arcussine and arcuscosine atan(),acot() : Arcustangens and Arcuscotangens sinh(),cosh() : Sine and cosine hyperbolicus tanh(),coth() : Tangens and cotangens hyperbolicus asinh(),acosh() : Arcussine and arcuscosine hyperbolicus atanh(),acoth() : Arcustangens and arcuscotangens hyperbolicus exp() : Exponentialfunction (e^x) sqr() : Squareroot log() : decadentic logarithm (to the basis 10) ln() : Logarithm naturalis (to the basis e) abs() : Absolut value (Betragsfunktion) k : 'Constant' of the curve bands pi : Circle number (ca. 3.14159265358979323846) e : Eulers number (ca. 2.71828182845904523536) epso : Electric fieldconstant (ca. 8.8542*10^(-12)) myo : Magnetic fieldconstant (PI*4.0*10^(-7)) c : Lightspeed (ca. 2.997925*10^8) h : PLANK-Constant (ca. 6.6256*10^(-34)) qe : Basic electric charge (ca. 1.6021*10^(-19)) me : Mass of an electron (calm) (ca. 9.1091*10^(-31)) u : Atomic Massunit (ca. 1.660277*10^(-27)) gk : Gravityconstant (ca. 6.67*10^(-11)) Numeric constants can be given in this way (e.g.) : 1 ! -2 ! 5.6521 ! -3.998 ! 2.543E+08 ! 42.4432E-32 Physical units will not be accepted! To use functions not implemented please build other functions e.g. from the appendix of the Basic-Manual. (e.g. logarithm to the basis b => log(x)/log(b) ) Further there is something special with negating signs. This sign has the highest priority. -x^2 is calculated intern as (-x)^2. To set the term correct should be used brackets : -(x^2). But you should not only use brackets by using this sign, also if you use several divisions like 3/x/k, where always should been set brackets (either (3/x)/k or 3/(x/k) )! All open brackets have to be closed! Draw functions <AMIGA>+C opens again a requester, where the functions to be drawn could be choosen. The first letters of the functions are written behind the individual gadgets. After the selection the plotter starts with 'OK' or cancels the selection with 'CANCEL'. Pressing Space stops the drawing process. Not the whole interval is needed to be drawed in order to zoom or to print the graph (only if you wish an hardcopy you should draw the graph completly on the screen!). Coordinates <AMIGA>+W draws again a cross on the screen at the actual mouseposition and its coordinates in the graphs world in the right Titlebar. This function can be ended after pressing the left mousebutton. Discussion <AMIGA>+K asks at first the function to be discussed. A second requester asks the interval to be searched, which 'k' constant of a band should be used and how high the search density should be. As higher the density as better the result and the longer takes the calculations. The given value is enough for the most functions. Then there will be opened a window where are two gadgets. 'GO ON' turns the 'page' if it was fully written. 'CANCEL' will end the discussion. At first the first and second differentiation of the choosen function will be generated and after that the zero digits, extrema (Max = maximum, Min = minimum), turning points (T = turning point, S = saddle point), pole digits and definition gaps as well as symmetry like (periodical) axis- and pointsymmetry through any (!) axis/point. To leave the discussion at the end by selecting the 'GO ON' gadget. If you want to get easily good differentiations you should do this : Get the first differentiation of the wanted function and optimize it by yourself. Then start a new function with that optimized differentiation and make again a discussion of the (new) function. The first differentiation (that is the second of the basic function) can also be optimized and so on. Using this method as often as needed gives you simple and quite good optimized differentiations that else have to be calculated by yourself. Calculate integrals <AMIGA>+I wants to know what type of integration should be used. There are different types to be choosen : Integral : Calculates the normal integral Area : Calculates the area between the graph and the x-axis (always positiv!) Area between two functions : Calculates the area between two functions (always positiv!) Rotation body : Calculates the volume of a body that will formed by turning the function around the x-axis three dimensionaly, so the function is the outline of the body Surface of rotation body : Instead calculating the volume, the surface of such a body is calculated Curve length : Calculates the length of the curve in an interval The Simpson-rule is used to calculate the integrals. It could take some seconds to get the result, depending on the density set. As higher the density as preciser the result and longer the time needed. The given value holds the error of the result down by some ten thousandth. The error of the Simpson-rule could also easily be estimated more precise by using this term (b-a)^5/(180*(2*m)^4)*(max[a,b] f[4](x)) with a = start of the interval, b = end, m = density of integration, max[a,b] f[4](x) = maximum value of the fourth differentiation in the interval [a;b]. So you can see that by calculating polynoms up to the third order nearly no errors may occur! Calculator <AMIGA>+T opens a window with a scientific calculatoremulation. This calculator should only make small conrollcalculations possible and should not replace a full calculator! Clicking on the 'OK' gadget will leave the function. The key shortcuts are these : Expression: Shortcut: Meaning: 1/x k Reciprocal value x^2 q Square y^x y y high x SQR w Squareroot EE e Exponetialdisplay and -input x! f Faculty SIN s Sinue COS c Cosine TAN t Tangens LOG l Decantic logarithm LNx n natural logarithm ASIN S Arcus Sine ACOS C Arcus Cosine ATAN T Arcus Tangens 10^x L 10th potency e^x N Exponentialvalue DRG d Change of the angleunit sDRG D Change of the unit and value % % Percentage xWy z y root of x PI p Circle number PI e a Eulers number x<>y <TAB> Changes value with memory +/- ` Changes the sign C <DEL> Clear display = <RETURN> Calculates the input Lin. equations <AMIGA>+G is a module to solve lineare equations with two and up to seven unknown. At first the number of unknown has to give in. Then a new window opens with the right number of coefficients. Behind the equals sign had to put the results of each term. Until all values are correct, the computer calculates after selecting the 'CALC' gadget the variables. It places the results under the coefficients. The program uses Gaußs-eliminationrule, so it is quite fast and needs only some seconds to calculate a result. Prime factor reduction <AMIGA>+M asks for an integer number to reduct it into their prime factors. The operation is similar to the discussion. Calculate prime numbers <AMIGA>+N opens a window and asks the start and end value. Then this function calculates prime numbers until the user stops or the end value is reached. The operation is also similar to the discussion. Printer : Print functions <AMIGA>+P tests if the printer preferences was set and if not it first calls the preferences. Please read the documentation at 'Printer preferences'. After set all values needed, the computer opens a statuswindow to show the progress in print and to stop the print by selecting the 'CANCEL' gadget. It could take some seconds to abort the print, because the computer calculates the actual stripe and print it before the computer cancels. The computer prints everything (and only that) what is on the screen (at least at a part of the screen). If functions were edited or the coordinate system was changed since the last call of 'Draw functions' the curves have to drawn again (it must not fill the screen completly, because the print is not a hardcopy!). The computer calculates all functions new for the printer to get an highresolutive Image. The DMA-Screen turn off is possible with 'HELP'. This increases the speed up to the factor two! See also the section about the general use of keys. Print coordinate system <AMIGA>+O prints the intervals of the coordinate system, its marks and further informations (like e.g. a logarithmic display) as Text. Discussion <AMIGA>+D is excatly the same menupoint like in the menu 'Plotter' only here all screenoutput is also send to the printer. Table of function values <AMIGA>+V asks for the number of values, the startvalue, the step that will be added and a value for the constant 'k' (if existent in the term). Then the printer gives out a table for the selected function in this order : X-value, function value, first three differentiations values and the root function value. So can easiely own graphs be drawn, if the quality of the print is to bad or no graphics printer is available. (You can also use a program to send the printer graphics into a file, so you can read it afterwards.) Prime factor reduction <AMIGA>+X Calculate prime numbers <AMIGA>+Y See the documentation under the menu 'Plotter' only the output is also send to the printer. Printing Preferences <AMIGA>+F opens a window in order to edit the printer preferences. The commen preferences have to be set before the program is started. With the keys '1'-'7' or selecting the individual gadget the printing density is choosen. It has the same effect as in the usual Preferences. The actual printing density is written under these gadgets in DPI (Dots per Inch). These values are given by the hardware of the choosen printer. Density 7 is always the highest density and so the highest resolution. Under these values the size of the inserted paper is given. There can also changed the size of the print to any smaller values. So scaled prints are possible (e.g. in the coordinate system with X and Y from -5 to 5 and 10cm x 10cm printing size every unit is nearly excact 1cm long! So can easily printing models for school and university be created). But as higher the resolution as longer will take the time to print. The last input is the distance from the left margin. So two or more graphs can be printed side by side. To exit these preferences select 'OK' or 'CANCEL', where 'CANCEL' will not accept the changes made. Screenhardcopy <AMIGA>+H is only needed if the print is urgent or only a rough print of the graph is needed. This print uses also the printing preferences. To make a hardcopy the screen have to be completly drawn, because only the screencontent is printet! The screenbar will not be printed. General use of the keys : In nearly all parts of the program have particular key a special effect. Here is a list of the keys that can be used instead of selecting the gadgets (that is only possible if no stringgadget is activated!) : <SPACE>,'w','o' OK or GO ON, CALC (positive gadget) 'c' CANCEL, etc. (negative gadget) 'n' Next (in Enter functions) 'v' Last (in Enter functions) 'x','y' Set logarithmic system on/off (in Coordinate system) <RETURN> Switches to the next stringgadget. If none was activated the first will be activated. With the help of <RETURN> can all gadgets be selected until none is activated any more. Then the booleangadgets can be selected by pressing the individual keys (e.g. 'OK', 'CANCEL'). <HELP> Turns of all DMA (Direct Memory Access). The program will run twice the speed on a usual AMIGA computer. Information to the copyright : © 1990-1993 by Andre Wiethoff. This program is Shareware and copyright- protected. This program may only given away unmodified together with the documentations. Those who often use this program, should recompense my work with some money. Depending on the interest on my program, it can be improved (e.g. vectorcalculations, etc.) and definitly freed from errors. Those who found errors in this program or have some ideas how to improve it should contact me please. My contactaddress : Andre Wiethoff Höhenweg 2 57392 Schmallenberg Germany P.S. : This documentation is a bit funny, eh? I think that there are a lot of new word creations and hard grammatical mistakes. I gave my best, but I was the worst pupil in my english class. So if anybody has a desire to correct this documentation and make it readable I would be pleased to get a corrected copy.