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Version 7.1 ┌──┐┌──┐ ┌─────┐┌─────┐┌─┐ ┌─┐ ┌─────┐ ┌────┐ ┌────┐ │ └┘ │ │ ┌─┐ │└─┐ ┌─┘│ │ │ │ │ ┌─┐ │ │ ┌──┘ │ ┌──┘ │ ┌┐┌┐ │ │ └─┘ │ │ │ │ └─┘ │ │ └─┘ │ │ └──┐ │ └──┐ │ │└┘│ │ │ ┌─┐ │ │ │ │ ┌─┐ │ │ ┌─┐ │ └──┐ │ └──┐ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ┌─┐ │ │ │ │ ┌──┘ │ ┌──┘ │ ┌─┐ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └────┘ └────┘ └─┘ (C) 1993: Bernd Schultheiss, D-69168 Wiesloch, Hufschmiedstr. 3 C O N T E N T S : 1. Introduction ................................................. 1 2. Hardware-Requirements ........................................ 2 3. Copyright .................................................... 2 4. Installation ................................................. 3 5. Operation .................................................... 4 5.1 The Main Menu ............................................ 4 5.2 The Keyboard Layout ...................................... 5 5.3 The Texteditor ........................................... 7 5.4 Data Back Up, Snapshot and Hardcopy ...................... 8 5.5 The Calculators .......................................... 9 5.6 The Coordinate Systems .................................. 10 6. The Menu Info ............................................... 11 7. The Menu Algebra ............................................ 11 8. The Menu Geometry ........................................... 17 9. The Menu Analysis ........................................... 22 10.The Menu Stochastics ........................................ 30 11.The Menu Linear Algebra ..................................... 34 12.Appendix A : Syntax ......................................... 37 13.Appendix B : Supplements .................................... 38 14.Appendix C : File Formats ................................... 39 15.Appendix D : Printer Driver ................................. 41 16.Appendix E : Hints and Tricks ............................... 42 - 1 - ┌─────────────────────────────────────────────────────────────────┐ │ 1 . I N T R O D U C T I O N │ └─────────────────────────────────────────────────────────────────┘ The program MATH.ASS. contains a wide collection of routines and is designed to take away the horror of many mathematical problems. It is NOT a tutorial program for mathematics BUT a MATHematics ASSistant for teachers, students and any person facing mathematical problems. But of course it does not prevent students checking their homework with MATH.ASS. from improving their knowledge. All algorithms employed have been collected over the years and used to make up this program. If a certain type of problem is missing or in case you know of any other interesting algorithms I would be pleased to get a note. The program is continuously being revised and extended, and each registered user may be offered the latest version at a small update fee as soon as subtantial changes have been realized. «11 ┌────────────────────────────────────────┐ ┌──────────┤ W A R R A N T Y D I S C L A I M E R ├─────────────┐ │ └────────────────────────────────────────┘ │ │ │ │ --- PLEASE READ THIS INFORMATION CAREFULLY --- │ │ │ │ THE AUTHOR MAKES NO WARRANTY OF ANY KIND, EXPRESSED OR IMPLIED,│ │ INCLUDING WITHOUT LIMITATION ANY WARRANTIES OF MERCHANTABILITY │ │ AND/OR FITNESS FOR A PARTICULAR PURPOSE. │ │ │ │ THE AUTHOR DOES NOT ASSUME ANY LIABILITY FOR THE USE OF THIS │ │ SOFTWARE BEYOND THE REGISTRATION FEE OF THIS SOFTWARE. │ │ │ │ IN NO EVENT WILL THE AUTHOR BE LIABLE FOR ANY ADDITIONAL │ │ DAMAGES, INCLUDING ANY LOST PROFITS, LOST SAVINGS, OR OTHER │ │ INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING FROM THE USE OF, │ │ OR INABILITY TO USE, THIS SOFTWARE AND ITS ACCOMPANYING │ │ DOCUMENTATION, EVEN IF THE AUTHOR HAS BEEN ADVISED OF THE │ │ POSSIBILITY OF SUCH DAMAGES. │ »11 │ │ └─────────────────────────────────────────────────────────────────┘ By the way, It should be stressed particularly that the program is not qualified to dry wet poodles. - 2 - ┌─────────────────────────────────────────────────────────────────┐ │ 2 . H A R D W A R E R E Q U I R E M E N T S │ └─────────────────────────────────────────────────────────────────┘ The program runs on all IBM compatible computers with - 640 KB memory - a CGA, HGC, EGA or VGA graphic card - If existing, mathematical co-processors such as 8087, 80287 or 80387 will be supported. ┌─────────────────────────────────────────────────────────────────┐ │ 3 . C O P Y R I G H T │ └─────────────────────────────────────────────────────────────────┘ The program MATH.ASS. is Shareware, i.e.: - It may/should be tested, copied and distributed !!! - If you want to use the program furtheron, you may become an authorized user by remitting the registration fee. - Each change of the program or of the inherent files is a violation of copyright. ┌─────────────────────── REGISTRATION FEE ────────────────────────┐ │ │ │ amounts : 20 $ US for private users │ │ 40 $ US for schools, companies and other institutions│ │ │ │ payable to: Bernd Schultheiss, Hufschmiedstrasse 3 │ │ D-69168 Wiesloch, GERMANY │ │ VISA and MASTERCARD are accepted │ └─────────────────────────────────────────────────────────────────┘ «12 The amount of the registration fee depends on the payer. A teacher buying the program for his private use, pays the smaller fee, even if he wants to use the program in class. If the program is bought for the school by the governing body then the higher fee has to be paid. Upon payment of the registration fee you will receive your personal series number. Please write serial number, your name and address in the respective input fields of the program section Registration. This will generate the file MATHASS.REG and hence convert the shareware version into a full version lacking the discrete hint at the bottom of your screen and the shareware screen. ┌─────────────────────────────────────────────────────────────────┐ │ Before passing the program on to a third party you must delete │ │ the file MATHASS.REG containing your personal serial number. │ └─────────────────────────────────────────────────────────────────┘ »12 - 3 - ┌─────────────────────────────────────────────────────────────────┐ │ 4 . I N S T A L L A T I O N │ └─────────────────────────────────────────────────────────────────┘ To install MATH.ASS. on your hard disc, you only have to create a sub-directory called MATHASS and copy all files of the diskette into this sub-directory. For the DOS beginner the necessary steps are as follows: C: to change to the hard disc CD \ to change to the main directory MD MATHASS CD MATHASS COPY A:*:* as soon as the diskette is in drive A Of course you may give the MATHASS sub-directory a different name, and you do not necessarily have to create it in the main directory of C:. If you want to start MATHASS out of a menu system or with a batch file you must call it out of the MATHASS-sub-directory, otherwise printer drivers cannot be found . If this is not possible for one or another reason you may put in as parameter the files' path, for example: MATHASS \MATH\MA70\ The program automatically recognizes the type of graphic card and monitor (mono or color) being used and adjusts the graphic mode. If this does not work on your computer or if you want to run the pro- gram on a VGA card in a different mode you may start the program with the following parameters: MATHASS MONO with monochrome monitors MATHASS EGA to enforce EGA resolution MATHASS HGC to enforce HGC resolution MATHASS CGA to enforce CGA resolution «14 To define the printer mode use file MATHASS.PRN. You may install two printers at a time by writing the names of both printer drivers into this file. To do so you must choose the menu section INFO and start Installation. The window entitled " Available Files :" presents you all the files having the suffix "DRV" i.e. all printer drivers. In the input mask below you may assign the hardcopy routines initiated with keys F9 and F10 to their printer drivers. These files contain the control sequences for the graphic hardcopy. If neither your printer nor a compatible printer is listed you may easily create your own printer driver using a text editor. For detailed instructions see appendix D: Printer drivers »14 - 4 - ┌─────────────────────────────────────────────────────────────────┐ │ 5 . O P E R A T I O N │ └─────────────────────────────────────────────────────────────────┘ 5.1. The Main Menu ─────────────────────────────────────────────────────────────────── The program is started at the DOS prompt with MATHASS [parameters]. Then the main menu appears as follows: ╔══════════════════════════════════════════════════════════════════╗ ║ ┌──┐┌──┐┌─────┐┌─────┐┌─┐ ┌─┐ ┌─────┐┌────┐┌────┐ ║ ║ │ └┘ ││ ┌─┐ │└─┐ ┌─┘│ │ │ │ │ ┌─┐ ││ ┌──┘│ ┌──┘ ║ ║ │ ┌┐┌┐ ││ └─┘ │ │ │ │ └─┘ │ │ └─┘ ││ └──┐│ └──┐ ║ ║ │ │└┘│ ││ ┌─┐ │ │ │ │ ┌─┐ │ │ ┌─┐ │└──┐ │└──┐ │ ║ ║ │ │ │ ││ │ │ │ │ │ │ │ │ │ ┌─┐ │ │ │ │┌──┘ │┌──┘ │ ┌─┐ ║ ║ └─┘ └─┘└─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘└────┘└────┘ └─┘ ║ ╠═════╦════════╦══════════╦═════════╦════════════╦════════════╦════╣ ║ INFO║ ALGEBRA║ GEOMETRY ║ ANALYSIS║ STOCHASTICS║ LIN.ALGEBRA║ END║ ╠═════╩════════╬══════════╩═════════╩════════════╩════════════╩════╝ ║ General Info ║▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ║ Copyright ║▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ║ Operation ║▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ║ Installation ║▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ║ Registration ║▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ║ Annotation ║▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ╚══════════════╝▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒ ╔═══════════╦══════════════════════════════════════════════════════╗ ║ Vers. 7.1 ║ C'93: B.Schultheiss,D-69168 Wiesloch,Hufschmiedstr.3 ║ ╚═══════════╩══════════════════════════════════════════════════════╝ In the main menu you may select from seven menu sections (topics): INFO, ALGEBRA, GEOMETRY, ANALYSIS, STOCHASTICS, LINEAR ALGEBRA and END moving the cursor with the left/right keys. Below each section a number of subdivisions is displayed. To make a selection you move the cursor Up/Down and start it by pressing the Enter key. You may also start the menu sections directly with function keys F1 to F7 and their subdivisions with their initial letter. The data required for the solution of the chosen problem must be entered into one or more pages of so-called input masks. You may put in the terms according to the given order of the input fields or you may move the cursor independently from one input field to another. The actually active field is displayed in reverse mode. If you get stuck at some point press function key F1 for a display of the keyboard layout or press function key F2 for detailed infor- mation on the immediate problem. - 5 - 5.2 The Keyboard Layout ─────────────────────────────────────────────────────────────────── F1 = displays this help page │ ESC = M A I N M E N U F2 = context sensitive help │ Enter = next field F3 = saves data in a file │ PAGE down = move on F4 = loads data from a file │ PAGE up = move back F5 = Texteditor │ Ctrl Y = clear input field F6 = undelete │ Ctrl L = clear all fields F7 = save graphic as IMG │ Insert = TAB-/EDIT-mode F8 = save graphic as PCX │────────────────────────────────── F9 = Hardcopy 1 (see Install.) │ TAB-mode (without cursor) : F10= Hardcopy 2 (see Install.) │ cursor keys = switch field ─────────────────────────────── │ Home = first field Shift F1 store input data in │ End = last field : Registers R1 to R10 │────────────────────────────────── Shift F10 │ EDIT-mode (with cursor) : ─────────────────────────────── │ Up/Down = switch field Alt F1 restore input data │ left/right = move cursor : form registers R1 to │ Home = begin. of field Alt F10 R10 │ End = end of field ─────────────────────────────────────────────────────────────────── «01 F1: If you press F1 (Help) while working in a menu subdivision the keyboard layout will appear on your screen. The word "H E L P" appears in the upper right corner of your screen instead of "F1/F2=Help" and of "ESC = Quit HELP" at the bottom. From this you know that you are in a help text. In addition, if working on a color monitor in EGA- or VGA-mode the background color of your screen changes as soon as you get into a help text. F2: If you press F2 ( the second help key ) while working in a menu subdivision the respective chapter of the manual will appear on the screen. In longer help texts you may switch from one page to another with Page Down/Page Up keys and with Down/Up keys. A small bar at the right margin of the text will show you the position in the text. F3: As soon as you have saved your input data in a file by pressing F3 you may load it again with key F4. F4 will display a list of all files with the suffix DAT in the current directory as well as an input mask for the file name you want to select. F4: see F3 F5: With the help of a small-scale text editor you may write e.g. captions or remarks in a function graph. For a more detailed description see page after next. F6: Working in the TAB-mode means that the contents of each input is automatically deleted prior to each new input. Pressing F6 restores the former contents. F7: If you want to transfer graphic data to a suitable word proces- sing program you may save it in the format *.IMG by pressing F7 or in the format *.PCX by pressing F8. F8: see F7 F9: Hardcopy 1 and 2 are routines which produce a printout of the screen display. You may install two printers by listing the corresponding printer drivers in the file MATHASS.PRN. F10:see F9 - 6 - The ESC key: To quit the current subdivision and return to the main menu press the ESC (=escape) key. Apart from this you must press the the ESC key to leave the Help text or the Editor. The Page keys By pressing the PgDn / PgUp keys you can switch from one page to another within a menu subdivision. E.g. there are six pages for the curve discussion : The input of the function term, the display of the derivatives, the display of zeros, extrema and points of inflection, the input of the range of the graph, the output of the graph and the output of the table of values. The input pages may contain several input fields. The necessary input may be inserted in any order you like. In former MATHASS versions you could only delete and not edit the input if it had to be corrected. As of version 7 you can switch between the TAB and EDIT modes with the INS key. The TAB mode: This is the usual input mode which allows you to move from one input field to another with the cursor keys. The cursor is in- visible and characters are appended in the active input field (displayed in reverse mode). The EDIT mode: In the active input field the blinking cursor indicates where the next character will be inserted or where it will be deleted when pressing the DEL key. You can move the cursor within the input field by pressing the cursor keys left or right and Home or End. To move to the adjacent input field when working in the EDIT mode you must press TAB /Shift TAB or Ctrl right/Ctrl left. In doing so the TAB mode is automatically reactivated. The Delete keys: In addition to the usual delete keys DEL and Backspace you can delete the contents of the active input field by pressing Ctrl Y and the contents of the complete input page by pressing Ctrl L. The registers R1 to R10: Terms you have entered in one section of the program may be used in another section if you have stored them beforehand in one of the registers R1 to R10. By pressing Shift F1 .... F10 you save the contents of the currently active input field. You can restore them again at any time by pressing Alt F1 ... F10. The last registers are used in some menu subdivisions to save the produced results ( e.g. polynomials, regression...) for the use in other menu subdivisions. »01 - 7 - 5.3 The Text Editor (F5) ─────────────────────────────────────────────────────────────────── To add remarks or captions to the results of a menu subdivision prior to their printing you can activate a text editor out of the text mode as well as out of the graphic mode with key F5. You will know from the caption in the upper right corner that you are in the editor. In addition, if working with a color monitor in the EGA- or VGA-mode the background color of your screen changes as soon as you are in the editor. «02 The Keyboard Layout of the text editor: Home = moves the cursor to the beginning of a line End = moves the cursor to the end of a line PgUp = moves the cursor to the top of the screen PgDn = moves the cursor to the bottom of the screen Enter <─┘ = moves the cursor to the beginning of the next line F1/F2 = shows you the keyboard layout of the editor F3 to F5 = not defined F6 = restores the background F7 = saves graphic date as IMG F8 = saves graphic data as PCX F9 = hardcopy of printer 1 (see Installation) F10 = hardcopy of printer 2 (see Installation) Shift F1 to F10 = SAVE MA1.SCR, MA2.SCR, ..., MA10.SCR Alt F1 to F10 = LOAD MA1.SCR, MA2.SCR, ..., MA10.SCR Ctrl F1 to F10 = MERGE MA1.SCR, MA2.SCR, ..., MA10.SCR The insert mode is not available since you write directly into the display memory. You can only overwrite the input. If you call the editor out of the graphic mode the contents of the screen will be saved. By pressing F6 you may re-load those parts of curves which have been overwriten. Apart from this in the graphic mode the cursor moves at half line spacing thus improving the legibility of indices and exponents. With SAVE, LOAD and MERGE the screen contents which have been annotated in the editor may be saved, re-loaded or merged with other screen contents. The file names used are MA1.SCR, MA2.SCR etc. MERGE only works in the graphic mode and of course gives only proper results on condition that the merged coordinate systems have the same scales. »02 You may switch to the editor via menu section INFO/Annotation in order to edit a graphic page which has been saved beforehand by pressing Shift F1, ..., Shift F10. ─────────────────────────────────────────────────────────────────── WARNING : Each file created in MATHASS overwrites any other file of the same name. - 8 - 5.4 Backup, Snapshot and Hardcopy ─────────────────────────────────────────────────────────────────── You have several options for saving your results: a) Data Backup ────────────── Press F3 to save the contents of your input data in a file. This saves the contents of an input screen at a compressed form. To re- store the work session at a later time press key F4 to reload the data. Additionally in the menu subdivisions Statistics and Regression of the menu section STOCHASTICS, you may save all values or pairs of values in an ASCII file so that they can be used in other programs, too. The files in Statistics provide one line for each value, those in Regression provide one line for each pair of values separated by a comma. The file names consist of maximum eight letters and fixed suffixes i.e. '.st' or '.rg' resp. b) Snapshot ─────────── In each menu subdivision you can save the contents of the current screen by pressing F7 or F8. Both ways text is saved in ASCII files named MA1.TXT, MA2.TXT, etc.. With F7 graphic screens are saved in the IMG format at MA1.IMG, MA2.IMG etc. and with F8 in the PCX format at MA1.PCX, MA2.PCX etc.. ┌─────────────────────────────────────────────────────────────────┐ │ The snapshot routines do not support the VGA graphic mode ! │ │ If you are working with VGA cards and you want to use the snap- │ │ shot function you must run the program in the EGA mode i.e. │ │ you must start it with MATHASS EGA. │ └─────────────────────────────────────────────────────────────────┘ If the word processor or publishing program you want to import your snapshots shows them white on black start MATHASS by using the parameter REVERSE. The snapshot routines are not available in the main menu nor in the help mode. c) Hardcopy ─────────── In former MATHASS versions F9 and F10 were used to produce hard- copies on EPSON FX-80 and NEC P6 resp. or on compatible printers. As of version 7 a number of printer drivers are available and the file MATHASS.PRN tells you which hardcopy routine, either F9 or F10, should be used. In the annex D: you find details of some printer drivers and their respective structures. With the help of this list you should find it easy to create your own printer drivers and/or different output formats. - 9 - 5.5 The Calculators ─────────────────────────────────────────────────────────────────── «2A The chapters Operations with Fractions, Large Numbers, Complex Num- bers and Place Value Systems have been realized as calculators working in the "Inverse Polish Notation" thus differing from the "Algebraic Notation" in terms of the input methods. To perform a calculation you must firstly enter the two operands separately and then indicate the operation to be performed. ┌─────┐ ┌─────┐ ┌─┐ ┌─┐ Example: 4 │ENTER│ 12 │ENTER│ 5 │+│ │*│ corresponds to 4∙(12 + 5) = └─────┘ └─────┘ └─┘ └─┘ The intermediate results obtained by the calculation of terms are stored in four batch registers which are mostly named x, y, z and t. Functions like SIN or COS affect register x. Operations such as + or * combine x and y , the result is being stored in register x and the other registers move up. The following example demonstrates the function of the four batch registers x, y, z and t: Calculator for Large Numbers. ( 9 + 8 ) · ( 7 + 2 ) problem ───────────────────── 8 + 3 · 4 input contents of x, y, z and t ─────── ───────────────────────────────────────── 9 x = 9 ENTER x = 9 y = 9 8 x = 8 y = 9 + x = 17 ENTER x = 17 y = 17 7 x = 7 y = 17 ENTER x = 7 y = 7 z = 17 2 x = 2 y = 7 z = 17 + x = 9 y = 17 * x = 153 ENTER x = 153 y = 153 8 x = 8 y = 153 ENTER x = 8 y = 8 z = 153 3 x = 3 y = 8 z = 153 ENTER x = 3 y = 3 z = 8 t = 153 4 x = 4 y = 3 z = 8 t = 153 * x = 12 y = 8 z = 153 + x = 20 y = 153 / x = 7.65 Firstly the operation 9 + 8 is performed and put into the batch register, then 7 + 2. Both sums are multiplied and the product is put into the batch register (ENTER) etc. »2A - 10 - 5.6 The Coordinate Systems ─────────────────────────────────────────────────────────────────── «03 In all program sections plotting function graphs you must indicate at the beginning the range of values, the scaling of the axes, the resolution to be used for the graphic and the angle mode and apart from this, whether or not the chart shall fill the complete screen. a) To improve the depiction of the interval the program automati- cally adjusts the indicated range of values. For examples of the range adjustment see below. b) For both axes the scaling may be normal i.e. with constant scale divison or logarithmic. With logarithmic scaling, decimal powers are used as labels and the lower limit if negative is set at 0.1. c) To a great extent the resolution accounts for the accuracy of the graph. If you choose to work in the low resolution (0) only every eighths horizontal pixel is used for the calculation of the function values. Thus the function graph is plotted rela- tively fast but maybe too angular. In this case you can later still increase the resolution. In the highest resolution (3) the function value is calculated for each individual pixel. d) You may choose between measure of angle in radians, in degrees or in grades. Correspondingly the arguments of the trigonometric functions are calculated as radians (RAD) with a perigon angle of pa=2π, as degrees (DEG) with pa=360° or as grade (GON) with pa=400 gon. pa stands for plenus angulus. e) Zoom determines whether only the right half of the screen (0) or the whole screen (1 to 5) is used for the display of the graph. In the full screen mode you can indicate the quadrant into which you want to write the text. Zoom = 5 implies a display without text in full screen mode. f) Examples for the adjustment of the range of values: - with normal scaling ┌────┬────┬────┬────┬────┬────┬────┬────┐ -1.5 ≤ x ≤ 2.1 1 0 1 2 ┌──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┐ -1.5 ≤ x ≤ 9.4 -2 0 2 4 6 8 10 ┌────┬────┬────┬────┬────┬────┬────┬────┬───┐ -0.8 ≤ x ≤ 0.1 -.8 -.6 -.4 -.2 0 ┌────────┬────────┬────────┬────────┬───────┐ -7 ≤ x ≤ 12 -10 0 10 - with logarithmic scaling ┌────────────────────┬────────────────────┐ -4 ≤ x ≤ 4 .1 1 10 ┌─────────────┬─────────────┬─────────────┐ 0.01 ≤ x ≤ 12 .01 1 10 100 »03 - 11 - ┌─────────────────────────────────────────────────────────────────┐ │ 6 . T H E M E N U I N F O │ └─────────────────────────────────────────────────────────────────┘ The menu subdivisions General Info, Copyright and Operation provide the essentials already described in chapters 1 to 5 of this manual. With menu subdivision 'Installation' you can adjust and run your printer, with 'Annotation' you can edit and print saved graphic data. For a more detailed description of both functions please refer to pages 7 and 8. 'Registration' displays the input mask for your personal serial number, your name and address which you must fill in to remove the shareware reminder at the bottom of the screen. Upon receipt of the registration fee ( see 3. Copyright ) you will get your personal serial number . ┌─────────────────────────────────────────────────────────────────┐ │ 7 . T H E M E N U A L G E B R A │ └─────────────────────────────────────────────────────────────────┘ 7.1 Prime Numbers ─────────────────────────────────────────────────────────────────── «21 This part of the program calculates all prime numbers and all twin primes between two given numbers. The result may be directed to the screen, to the printer or to the file PRIM.TXT . Prime numbers are all the natural numbers with exactly two divisors Hence number 1 is not a prime number and number 2 is the only even prime number. Euklid already proved that there exists an infinite number of prime numbers. And it is proven too that the intervals in the sequence of primes are not limited whatsoever. Twin primes are two primes with a difference of two e.g. 10007 and 10009 or 1000018709 and 1000018711. If the difference between the upper and the lower limit exceeds 50000 the range is automatically limited to 50000. If the output is routed to the screen the program displays only as many primes as will fit onto the screen. »21 7.2 Prime Factorization ─────────────────────────────────────────────────────────────────── The program factorizes natural numbers into their prime powers. «22 The prime factorization or canonical representation of a number is unique except for the order of the factors. Examples : 123456789 = 3^2 ∙ 3607 ∙ 3803 1234567890001 = 304643 ∙ 4052507 12345678900001 = prime number (takes a little longer) »22 - 12 - 7.3 G.C.F. and L.C.M. ─────────────────────────────────────────────────────────────────── The program determines for two numbers a and b the greatest common factor ( G.C.F. ), the lowest common multiple ( L.C.M. ) and their respective set of divisors. «23 Example : a = 1001 b = 3575 greatest common factor G.C.D = 143 lowest common multiple L.C.M = 25025 T(a) = { 1 7 11 13 77 91 143 1001 } T(b) = { 1 5 11 13 25 55 65 143 275 325 715 3575 } The G.C.F. is the biggest element in the intersection of the set of divisors of a and b. In fractional arithmetics the G.C.F. of the numerator and of the denominator is the biggest number by which the fraction may be cancelled. The L.C.M. is the smallest element in the intersection of the set of multiples of a and b. In fractional arithmetics the L.C.M. of two denominators is called the common denominator. If you have already determined the G.C.F.(a,b), the L.C.M.(a,b) is determined by the formula L.C.M(a,b) = a∙b / G.C.D(a,b) »23 7.4 Decimal Fractions -> Vulgar Fractions ─────────────────────────────────────────────────────────────────── «24 Each decimal fraction may be represented as a vulgar fraction. In the case of terminating decimal fractions you simply move the decimal point to the right and take the corresponding power of ten as its denominator. As for recurring decimal fractions see formulas below: _ _ _ 0.1 = 1/9 , 0.2 = 2/9 , ... , 0.9 = 9/9 = 1 _ _ 0.01 = 1/90 , 0.02 = 2/90 , ... __ __ ___ 0.01 = 1/99 , 0.02 = 2/99 , ... 0.000001 = 1/999000 The program transforms recurring as well as terminating decimal fractions into vulgar fractions after you have put in the non- recurring part of the decimal and the recurring decimal seperately. Example 1 : non-recurring part : 1.20 recurring decimal : 045 ___ 1.20045 = 120/100 + 1/2220 = 533/444 Example 2 : non-recurring part : 1.20 recurring decimal : 9 1.209 = 120/100 + 1/100 = 121/100 »24 - 13 - 7.5 Vulgar Fractions -> Decimal Fractions ─────────────────────────────────────────────────────────────────── «25 Each vulgar fraction may be represented as a decimal fraction. If a series of digits is repeated ad infinitum in the decimal fraction it is called a recurring decimal fraction. The repeating series of digits is called the recurring decimal and it is marked by a line above. The program transforms vulgar fractions into terminating or recur- ring decimal fractions and determines the recurring decimal as well as its length after you have put in the numerator and the denomina- tor of the fraction. Example 1 : numerator : 533 denominator: 444 ___ 533/444 = 1.20045 The recurring decimal starts with the 3rd digit following the decimal point and is 3 digits long. Example 2 : numerator : 124 denominator: 125 124/125 = 0.992 a terminating decimal fraction If the depiction of a decimal fraction exceeds one line three dots mark its abortion. »25 7.6 Binomials of n-th degree ─────────────────────────────────────────────────────────────────── «26 One of the most popular formulars of school-level mathematics is certainly the binomial formular (a + b)² = a² + 2ab + b² The program calculates the more general theorem (a∙x + b∙y)^n mit 2 ≤ n ≤ 44 Example : (3x - 4y)^7 = +2187∙x^7 -20412∙x^6∙y +81648∙x^5∙y^2 -181440∙x^4∙y^3 +241920∙x^3∙y^4 -193536∙x^2∙y^5 +86016∙x∙y^6 -16384∙y^7 If a=1 and b=1 you get the numbers of the Pascal triangle i.e. the binomial coefficients where each of them is the sum of the two numbers overhead. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 »26 - 14 - 7.7 Equations of the 4th degree ─────────────────────────────────────────────────────────────────── The program determines the real solutions of an equation of the 4th or of a lower degree. You must put in the coefficients a to e, with a being the coefficient of x^4. «27 For equations of a higher degree than 4 no algebraic solutions except approximate computation (zeros in the menu subdivision Curve Discussion). Sometimes the polynomial division offers a solution. For example if q is the result of trial the polynomial division by (x-q) provides an equation of lower degree whose solution contains the rest of the results. Examples : a) x^4 + 2x^3 - 8x^2 -18x - 9 = 0 Input a=1, b=2, c=-8, d=-18 and e=-9 Solution L = ( -3, -1, 3 ) b) x^5 - x^4 - 16x + 16 =0 Solution resulting from trial x = 1. With the program polynomial division the left side of equation is divided by (x-1). The zeros of the polynomial x^4 - 16 provide the rest of the results. »27 7.8 Diophantine Equations ─────────────────────────────────────────────────────────────────── «28 Named after Diophantus of Alexandria ( ca. 250 A.D. ), who in his book Arithmetica seeks to solve linear and square equations and especially to find their integral solutions. The program computes the integral solutions of the equation a∙x - b = m∙y with m > 0 This for example permits the determination of the integral points in a straight line. Example: The straight line with the equation y = 7/3∙x - 5/3 <=> 7∙x - 5 = 3∙y comprises the integral points L = { (x/y) │ x=2+3t, y=3+7t and t integral } = { (2/3),(5/10),(-1/-1),(8/17), ... } »28 7.9 Pythagorean Tripels ─────────────────────────────────────────────────────────────────── «29 Pythagorean tripels are integral solutions (x,y,z) of the equation x² + y² = z² , standing for the sides of right-angled triangles. The program computes all Pythagorean tripels not bigger than a determined number. With x, y, z < 60 you get: 3 4 5 5 12 13 8 15 17 7 24 25 20 21 29 9 40 41 12 35 37 28 45 53 »29 - 15 - 7.10 Calculator for Big Numbers ─────────────────────────────────────────────────────────────────── «04 The program works as a scientific calculator of Reverse Polish Notation (RPN), i.e. at first you put in the operands and then the operation (see manual 5.5). The program calculates with 60 digits behind the decimal point and displays the first 48 numbers. If there are more than 48 digits before the decimal point the program changes the ordinary display to a floating point display with tenths power exponents of maximum four digits. When calculating with Big Numbers please note that only 48 positions are displayed while the internal calculatory process may include 2000 and more digits. Apart from the basic arithmetic operations and from exponentiation the RPN Calculator provides the functions SIN, COS, TAN, ATN, LN, EXP, FAC (x!) as well as the constant Pi () and in addition, as is normal with RPN calculators, batch operations such as scrolling (arrow keys) or exchanging (x - y). With MODE you may change the angle for trigonometric functions. Examples : ┌─────┐ ┌─┐ 1 234 567 890 │ENTER│ 9 876 543 210 │*│ └─────┘ └─┘ amounts to 12 193 263 111 263 526 900 ┌─────┐ ┌─┐ 2 │ENTER│ 64 │^│ amounts to 184 467 440 737 709 551 616 └─────┘ └─┘ ┌─┐ 40 │F│ amounts to └─┘ 815 915 283 247 897 734 345 611 269 596 115 894 272 000 000 000 »04 7.11 Calculator for Fractions ─────────────────────────────────────────────────────────────────── «05 For the computation of fractional arithmetics the program provides a calculator of Reverse Polish Notation, i.e. you must at first enter the operands and then the calculatory operation ( see manual 5.5 ). Apart from the four basic arithmetic operations and exponentiation with integer exponents the RPN calculator provides batch operations such as scrolling (arrow keys) and exchanging (x<->y). Numerators and denominators of a fraction must be entered separat- ly. To switch from one field to another press <- or ->. ┌──┐ ┌─────┐ ┌──┐ ┌─┐ Example: 9 │->│ 16 │ENTER│ 24 │->│ 5 │+│ means 9/16 + 24/5 └──┘ └─────┘ └──┘ └─┘ and amounts to 93/80 »05 - 16 - 7.12 Calculator for Place Value Systems ─────────────────────────────────────────────────────────────────── «06 For Place Value Systems the program provides a calculator of Reverse Polish Notation, i.e. at first you enter the operands and then the calculatory operation (see manual 5.5) Apart from the four basic arithmetic operations and exponentiation with natural exponents the RPN calculator provides batch operations such as scrolling (arrow keys) and exchanging (x<->y). It calculates only in the area of natural numbers including zero, i.e. negative differences are put to zero and remainders are trun- cated. By pressing Ctrl B you may change the basis. The biggest basis is 16 (hexadecimal) with the digits 0,...,9, A, B, C, D, E, F. Example with basis of 2 : 10000000 decimal 128 Enter 101011 decimal 43 + amounts to 10101011 decimal 171 = 128+0+32+0+8+0+2+1 Ctrl B 16 change to hexadecimal system Enter amounts to AB decimal 171 = 10∙16 + 11 »06 7.13 Calculator for Complex Numbers ─────────────────────────────────────────────────────────────────── «07 For Complex Numbers the program provides a calculator of Reverse Polish Notation, i.e. at first you enter the operands and then the calculatory operation (see manual 5.5). Apart from the basic arithmetic operations and from exponentiation the RPN Calculator provides the functions SIN, COS, TAN, ATN, LN, EXP, KONJ., √, as well as the constant Pi (π) and in addition, as is normal with RPN calculators, batch operations such as scrolling (arrow keys) or exchanging (x<->y). With MODE you may change the angle for trigonometric functions. The real part and the imaginary part of the numbers are entered in separate fields. To switch from one field to another press <- or ->. Example: ( 3 + 4i )( 3 - 4i ) = 25 ┌──┐ ┌─────┐ ┌────┐ ┌───┐ 3 │->│ 4 │ENTER│ │KONJ│ │ * │ makes 25 + 0i └──┘ └─────┘ └────┘ └───┘ »07 ─────────────────────────────────────────────────────────────────── When using the RPN calculators for your data input there are less functions available than in the rest of the program, e.g. your input may not be edited or stored. - 17 - ┌─────────────────────────────────────────────────────────────────┐ │ 8 . T H E M E N U G E O M E T R Y │ └─────────────────────────────────────────────────────────────────┘ 8.1. Right-angled Triangles ─────────────────────────────────────────────────────────────────── «31 A right-angled triangle mostly is well-determined by two magnitudes and the right angle. In the program you are asked to choose two of the following elements and then to enter their values: - catheti a and b - hypotenuse c - hypotenuse sections p and q - altitude h - angles alpha and beta - area A If you enter the values of two out of these nine magnitudes and confirm your input by pressing Page Down the program will compute the others. You may plot the triangle with menu subdivision Triang- les by three magnitudes. »31 8.2 Triangles defined by three magnitudes ─────────────────────────────────────────────────────────────────── «32 Enter three exterior magnitudes (sides or angles) and confirm your input with Page Down, then the program will compute the sides, the angles, the altitudes, the medians and the bisectors of the angles, the circumference and the area as well as the centres and radiuses of the inscribed and the circumscribed circle of the triangle. In addition the program draws the triangle with its inscribed and circumscribed circles. If you enter two sides and the angle opposite of the shorter side, you will get the first solution. To get the second solution press any key. Example : a = 3, b = 4, Alpha = 36.87° 1. solution : sides 3 4 5 angles 36.87° 53.13° 90° : 2. solution sides 3 4 1.4 angles 36.87° 126.9° 16.26° »32 8.3 Triangles defined by three Vertices ─────────────────────────────────────────────────────────────────── «33 If you enter the coordinates of the three vertices of a triangle the program computes all exterior and interior magnitudes, i.e. the sides, the angles, the altitudes, the radians and the bisectors of the angles, the circumference and the area as well as the centres and the radiuses of the circumscribed and the inscribed circle of the triangle. In addition the program plots the triangle with its circumscribed and inscribed circle. »33 - 18 - 8.4 Polygons ─────────────────────────────────────────────────────────────────── «34 At first you must enter the coordinates of the vertices of the polygon. The program provides an input mask for a maximum of 14 vertices A to N. The last vertex must differ from the origin. As soon as you have confirmed the input by pressing Page Down the program seeks the last vertex different from the origin of the ordinates so as to compute the number of vertices. Then the area, the circumference and the coordinates of the centroid are displayed and the polygon is plotted. »34 8.5 Mappings ─────────────────────────────────────────────────────────────────── «35 With MATHASS it is possible to produce up to five mappings of a polygon. The input and output of the relevant data requires four masks. ┌──────────────────────────────────┐ │ Enter the polygon │ │ ┌──────────────────────────────────┐ │ │ Select type of mapping │ │ │ ┌──────────────────────────────────┐ │ │ │ Define mapping │ │ │ │ ┌──────────────────────────────────┐ │ │ │ │ Plot the mappings of the polygon │ └────│ │ │ │ │ │ │ │ └────│ │ │ │ │ │ └────│ │ └──────────────────────────────────┘ Mask 1: At first you must enter the coordinates of the polygon into an mask with a maximum of 14 vertices A to N. As soon as you have confirmed the input by pressing Page Down the program seeks the last vertex different from the origin of the ordinates so as to compute the number of vertices. Mask 2: You may connect up to five mappings. For each mapping you may choose from six types of mappings (see below): - translation by dx in direction x and by dy in direction y - axial symmetry (PQ) - point symmetry Z - rotation around point Z with angle α - homothetic stretching out of Z by factor k - shear transformation by line PQ with angle α It is also possible to choose several mappings of the same kind. This means that you can produce multiple symmetries of a polygon of different lines. - 19 - Mask 3: You define the mappings by entering the parameters required for the selected type of mapping, i.e. - the vector (dx/dy) for the translation - two points P and Q of the axe of symmetry - center Z for the point symmetry - point Z and angle α for the rotation - point Z and factor k for the homothetic stretching - two points P and Q of the axe, angle α for the shear transf. Mask 4: For each of the connected mappings the image and the coordinates of the vertices are plotted. If you enter zero in the field line styles the image will not be plotted. This facilitates e.g. the immediate display of the image of a composition of rotation and stretching or translation. The original polygon is always plotted in line style 1, i.e. in a full black line. »35 8.6 Coordinate Systems ─────────────────────────────────────────────────────────────────── «36 With MATHASS you may transform three-dimens. cartesian coordinates into three-dimensional polar coordinates or cylindrical coordinates and vice versa. In a cartesian coordinate system (x/y/z) a point is located by its distance from each of three mutually perpendicular intersecting lines with the same unit of length. In a polar coordinate system (r/phi/Theta) a point is located by its radius vector, the angle of rotation phi on the equatorial plane and the angle of elevation Theta from the equatorial plane. In a cylindrical coordinate system (rho/phi/z) a point is located by its distance rho from the cylinder axis, the angle of rotation phi around the axis and the altitude z above the origin. Setting z at 0 or Theta at 0 the program computes the two-dimen- sional cartesian coordinates or two-dimensional polar coordinates respectively. Ex.: ┌─── cartesian ────┐ ┌────── polar ─────┐ ┌─── cylindrical ──┐ │ │ │ │ │ │ │ x = 1 │ │ r = 1.7320508 │ │ rho = 1.4142136 │ │ y = 1 │ │ phi = 45 │ │ phi = 45 │ │ z = 1 │ │Theta= 54.735610 │ │ z = 1 │ │ │ │ │ │ │ └──────────────────┘ └──────────────────┘ └──────────────────┘ »36 - 20 - 8.7 Plane determined by three points ─────────────────────────────────────────────────────────────────── «37 Three non-collinear points determine exactly one plane. Given the coordinates of three points the program computes the equation of this plane in the point-slope form and in the form of coordinates as well as its distance from the origin with the directional vectors and the normal vectors being increased to integers. In addition to this the position of the plane in the space is plotted including its lines intersecting a cube symmetric to the axes as well as its trace points. Example: Plane determined by points A(2/0/1), B(3/3/6), C(4/-1/2) ┌ ┐ ┌ ┐ ┌ ┐ -> │2│ │1│ │ 2│ x = │0│ + r∙│3│ + s∙│-1│ v.v. 8x + 9y - 7z = 9 │1│ │5│ │ 1│ └ ┘ └ ┘ └ ┘ »37 8.8 Sphere determined by four points ─────────────────────────────────────────────────────────────────── «38 Four non-coplanar points determine exactly one sphere. Given the coordinates of four points the program computes the equation of the sphere and a cube symmetric to the axes is plotted including the sphere's great circle parallel to the image's plane. Example: A sphere determined by A(11/1/3), B(7/1/7), C(3/-5/7) and D(3/-8/-2) has a centre M(5/-2/1) and a radius r=7 »38 8.9. Intersection of two lines ─────────────────────────────────────────────────────────────────── «39 Given two lines the program will compute their point of inter- section, the angle of intersection and their distances from the origin. The lines must be defined as a parametric representation. If the lines don't have any point in common you will see either of the two following messages: 'Parallel lines' or 'Skew lines'. Example : ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ │ 1│ │ 1│ │ 3│ │ 2│ g: x = │ 0│ + r∙│-1│ and h: x = │-2│ + s∙│ 3│ │ 2│ │ 1│ │ 4│ │ 0│ └ ┘ └ ┘ └ ┘ └ ┘ Point of intersection : S(3/-2/4) ( r=2, s=0 ) Angle of intersection : 99.2° Distances from origin : d(g,O) = 1.41 , d(h,O) = 5.39 »39 - 21 - 8.10 Intersection of two planes ─────────────────────────────────────────────────────────────────── «3A Given two planes the program will compute the line of intersection, the line's distance from the origin and the angle of intersection between the two planes. The program plots the intersection of the planes in a cube symme- tric to the axes and the line of intersection of the two planes. Example : Planes E1: x + y + z = 12 and E2: x - y = 5 intersect at ┌ ┐ ┌ ┐ -> │ 0│ │ 1│ g: x = │-5│ + r∙│ 1│ │17│ │-2│ └ ┘ └ ┘ Distance from origin : d = 7.778175 Angle of intersection : α = 90° »3A 8.11 Intersection of two spheres ─────────────────────────────────────────────────────────────────── «3B Given the coordinates of the center and the radii of two spheres the program computes the centre and the radius of the circle of intersection as well as the coordinate equation of the plane of intersection. The three-dimensional depiction of all elements is only provisionary. So far I have not found a reasonable solution for the depiction of two circles and their circle of intersection. If you set a sphere's radius to zero and the centre to a point on the other sphere the program computes the equation of the tangen- tial plane. Example : Spheres k1: M(1/3/9), r=7 and k2: M(2/-1/5), r=4 intersect in a circle around M(2/-1/5) with radius r=4, situated in plane x - 4y - 4z = -14 . »3B 8.12 Intersection of plane and sphere ─────────────────────────────────────────────────────────────────── «3C Given a plane's coordinate equation and a sphere's centre and radius the program computes the center and the radius of the circle of intersection. Example: Given plane E and sphere k with E: 2x + y -2z = 11 K: (x-2)^2 + (x+1)^2 + (x-5)^2 = 49 and centre M(2/-1/5) and radius r=7 Result: The plane and the sphere intersect in a circle around M(6/1/1) with radius r=3.6056 »3C - 22 - ┌─────────────────────────────────────────────────────────────────┐ │ 9 . T H E M E N U A N A L Y S I S │ └─────────────────────────────────────────────────────────────────┘ 9.1 Polynomials ─────────────────────────────────────────────────────────────────── «41 The program calculates the product and the quotient of two poly- nomials. Example: 1. Polynomial : x^4 + 4x^3 + 6x^2 + 4x + 1 2. Polynomial : x^2 + 2x + 1 Product : x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1 Quotient : x^2 + 2x + 1 Remainder: 0 The two polynomials as well as their product, their quotient and the remainder polynomial are automatically written to the registers R6 to R10 ; they can, for example, be read as function terms in the function plotter by means of Alt F6 to Alt F10. »41 9.2 Function plotter 1 ─────────────────────────────────────────────────────────────────── «42 Up to five functions can be plotted in one coordinate system at one time. Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). Also allowed are compositions or derivatives of already defined functions. For this purpose, f1(x), f2(x), etc. are simply substi- tuted by y1, y2, etc. in the function. The derivatives of f1(x), f2(x), etc. are entered as y1', y2', ... respectively y1", y2", ... Compositions of functions originating from compositions or deri- vations, are not allowed. Examples : Given f1(x)=sin(x) and f2(x)=3*sqr(x), then f3(x)=2*y1^2-y2 means f3(x)=2*sin(x)^2-3*sqr(x) f4(x)=f2(y1) " f4(x)=3*sqr(sin(x)) f5(x)=y2' " f5(x)=3/(2*sqr(x)) For each of the defined functions one line style can be selected; in case of EGA and VGA graphics the user may chose one of the colors red, green, blue or black. »42 Range, scaling, accuracy, angle mode, full or half screen will be determined on the next page. - 23 - 9.3 Function plotter 2 ─────────────────────────────────────────────────────────────────── «43 This little routine was added by me so as to enable the plotting of several functions in separate diagrams. Six functions can be plotted graphically in one print-out. Range, axial scale, plotting accuracy, the angle mode for trigono- metrical functions and the number of derivations can be determined independently for every function. ┌────── scaling ──────┐┌── accuracy ──┐┌─ angle ─┐┌─ derivations─┐ │ 0 : x and y linear ││ 0 : raw ││ 0 : RAD ││ 0 : only f │ │ 1 : x linear, y log. ││ 1 : medean ││ 1 : DEG ││ 1 : f and f' │ │ 2 : x log., y linear ││ 2 : fine ││ 2 : GON ││ 2 : f, f',f" │ │ 3 : x and y log. ││ 3 : very f. ││ ││ are plotted│ └──────────────────────┘└──────────────┘└─────────┘└──────────────┘ Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). The text editor (F5) allows the marking of the diagrams, e.g. with numbers of page and problem. »43 9.4 Function plotter 3 ─────────────────────────────────────────────────────────────────── A segmentwise defined function, determined by partial function f1 to f2, is plotted. «44 Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). Domain of definition, interval mode and color are entered for every partial function. Interval 0 : open on both sides 1 : open on left side and closed on right side 2 : closed on left side and open on right side 3 : closed on both sides A decision on plotting the border elements or not is also possible. They are described as discs or cubes, depending on whether they belong to the domain of definition or not. »44 - 24 - 9.5 Families of curves ─────────────────────────────────────────────────────────────────── «45 The program plots diagrams of any function containing a parameter k. The values for k can be listed or determined by initial value, final value and step. The distinction of individual curves is possible by means of color, line style, or both. Range, scaling, accuracy, angle mode and full or half screen will be determined on the next page. The curves can be lettered with the parameters by means of text editor (F5) before being printed out. Examples : overview of power functions f(x,k) = x^k k out of { 0, 1/2, -1/2, 1, -1, 2, -2 } range 0 ≤ x ≤ 4 , 0 ≤ y ≤ 4 Sine curves with different phase displacements. f(x,k) = sin(x+k) k from -2 tos 2 step 1 range -7 ≤ x ≤ 7 , -2 ≤ y ≤ 2 »45 9.6 Curve discussion ─────────────────────────────────────────────────────────────────── «46 The program can discuss any function. This means: The derivations are determined; the function is investigated in regard of zeros, extrema and points of inflection for a range which had been determined beforehand; the diagrams of f, f', and f" are plotted; a table of values is issued. ┌──────────────────────────────────┐ │ Input of function term │ │ ┌──────────────────────────────────┐ │ │ Output of derivations │ │ │ ┌──────────────────────────────────┐ │ │ │ Output of curve discussion │ └────│ │ ┌──────────────────────────────────┐ │ │ │ Input of coordinate system │ └────│ │ ┌──────────────────────────────────┐ │ │ │ Output of diagram │ └────│ │ ┌──────────────────────────────────┐ │ │ │ Output of table of values │ └────│ │ │ │ │ │ └────│ │ │ │ └──────────────────────────────────┘ - 25 - Page 1: Input of function term ──────────────────────────────────── The function term is entered together with range and accuracy of examination and angle mode. Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). The range of examination is the interval, in which the function is examined in regard of zeros, extrema and points of inflection. It must not be too large, as this consequently increases the step investigating the function in regard of reversal of sign. If a low accuracy is selected (0:raw), the examination will proceed faster than in case of high accuracy. For functions with very quick reversal of sign, however, zeros might not be noticed. Page 2: Output of derivations ───────────────────────────────── The derivations f' and f'' of f are determined by means of symbolic calculus according to the usual derivative rules. After this a number of basic simplifications are made. Although it is easy to verify the first part of the algorithm, mistakes might easily creep into the second part. If you feel that a derivation is not correct, it is recommended to enter the calculated term as well as the term you found out yourself, into function plotter 1 , and to have the two terms plotted with different colors. Should the curves not coincide, I would appreciate an adequate notice. Page 3: Output of curve discussion ────────────────────────────────────── Zeros, local maxima, local minima and the points of inflection of the function in the investigated range are issued. Gaps in the domain of definition are not recognized by the program, simply because they often don't lie within the number domain or are skipped, due to the binary arithmetics. For this reason, extrema or points of inflection can be indicated there by mistake. What was told about the gaps in the domain of a definition is also valid for the continuity and for the existance of a derivative of f, f' and f". Inevitably, the user himself / herself will have to make some effort. Page 4: Input of coordinate system ──────────────────────────────────────── See section 5.6 - 26 - Page 5: Output of diagram ───────────────────────────────── The diagram of function f (red continuous line), of 1st derivation f' (green dashed line) and of 2nd derivation (blue dotted line) are issued. The points of inflection of function f are marked. If accuracy is selected too high ( fine, very f.), the line styles in flat curve ranges cannot be differed any more. Page 6: Output of table of values ────────────────────────────────── Range and step of the table of values for f, f' and f" can be determined here. The examination range of the curve discussion is given. Places, where one of the functions is not defined, are marked by ---. Example : f(x)=4/x-4/x^2 examination range from -10 to 10 examination accuracy 1 derivations : f'(x) = -4/x^2+8/x^3 f"(x) = 8/x^3-24/x^4 ┌───────────────────┬───────────────┬────────────────────┐ │ zeros: f' │ extrema: │ inflexions: f' │ ├───────────────────┼───────────────┼────────────────────┤ │ N(1/0) 4 │ H(2/1) │ W(3/0.8889) -0.148 │ │ │ │ │ │ │ │ │ │ │ │ │ └───────────────────┴───────────────┴────────────────────┘ range : -10 ≤ x ≤ 10 and -2 ≤ y ≤ 2 table of values from -10 to 10 step 2.5 ┌───────────┬──────────────┬──────────────┬──────────────┐ │ x │ f(x) │ f'(x) │ f"(x) │ ├───────────┼──────────────┼──────────────┼──────────────┤ │ -10 │ -0.44 │ -0.048 │ -0.0104 │ │ -7.5 │ -0.604444 │ -0.090074 │ -0.026548 │ │ -5 │ -0.96 │ -0.224 │ -0.1024 │ │ -2.5 │ -2.24 │ -1.152 │ -1.1264 │ │ 0 │ --- │ --- │ --- │ │ 2.5 │ 0.96 │ -0.128 │ -0.1024 │ │ 5 │ 0.64 │ -0.096 │ 0.0256 │ │ 7.5 │ 0.462222 │ -0.052148 │ 0.0113778 │ │ 10 │ 0.36 │ -0.032 │ 0.0056 │ └───────────┴──────────────┴──────────────┴──────────────┘ »46 - 27 - 9.7 Newton-Iteration ─────────────────────────────────────────────────────────────────── «47 Newton-Iteration is an approximation method for the calculation of a zero of f(x). If an initial value x0 is entered, which is close enough to the desired zero, then the next approximation calculated is the intersection of tangent to graph of f in point P(x0/f(x0)). This leads to the recursion formula f(x(n)) x(n+1) = x(n) - ───────── f'(x(n)) Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). The procedure converges, if f(x0) ∙ f"(x0) > 0 Example : f(x) = x-cos(x) x(0) = 1 x(1) = .75036387 x(2) = .73911289 x(3) = .73908513 x(4) = .73908513 »47 9.8 Integral calculus ─────────────────────────────────────────────────────────────────── «48 The oriented and the absolute content of the area between two function curves in a desired interval, i.e. the two integrals, are calculated b b ⌠ ⌠ A1 = │(f1(x)-f2(x))dx and A2 = │ │f1(x)-f2(x)│dx ⌡ ⌡ a a Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). Also are determined: the twisting moments for rotation around x-, respectively y-axis, the bodies of revolution covered, and the centroid of the area Example : f1(x)=4-x^2 , f2(x)=(x-1)^2 interval from 0 to 1.5 oriented and absolute content A1 = A2 = 4.5 twisting moments Mx=8.1563 My=3.0938 bodies of revolution Vx=51.247 Vy=19.439 centroid S(0.6875/1.8125) »48 - 28 - 9.9 Parameter curves ─────────────────────────────────────────────────────────────────── «49 Curves, which are not determined by an explicit function term, but by two functions for the horizontal and vertical direction, can be plotted with this program. On the first page, the two function terms and the range to be covered by the parameter, are entered. Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). On the second page, range, scaling, accuracy, angle mode and full or half screen are determined. For the minimum accuracy, 100 points of the curve are calculated. Every increase of accuracy will double the amount of points. Examples : 1. The circle x(k)=sin(k), y(k)=cos(k), k from -Pi to Pi range -2 ≤ x ≤ 2 , -2 ≤ y ≤ 2 2. The spiral x(k)=k*sin(k), y(k)=k*cos(k), k from 0 to 20 range -20 ≤ x ≤ 20 , -20 ≤ y ≤ 20 3. The Lissajous figures obtained when two a-c voltages with different frequencies are applied to one oscilloscope: x(k)=sin(3*k), y(k)=cos(5*k), k from -Pi to Pi range -2 ≤ x ≤ 2 , -2 ≤ y ≤ 2 »49 9.10 Series Expansion ─────────────────────────────────────────────────────────────────── «4A Plotter for functions given as a series Σ f(x, k). The expansions in a series can be compared with different parameter ranges. For easier distinction, they can be displaced in y-direction. Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). For k and positive integer terms of k you may use the function fac(k) to calculate the factorial. Example : f(x,k) = x^(2*k-1)/fac(2*k-1)*(-1)^(k+1) k from 1 to 16 To calculate the first 16 terms of the taylor expansion of the the sine curve. »4A - 29 - 9.11 Area functions ─────────────────────────────────────────────────────────────────── «4B An area function f(x, y) is plotted, i.e. the three-dimensional diagram of a function with two variables. In many area functions, terms are used several times; therefore it is possible to define a term u(x, y) separately and to use it as a u in the function term. Besides the signs + - * / ^ ( ) the function terms may also contain the functions sqr (√x), abs (│x│), int, sgn, exp (e^x), ln, lg, ld, sin, cos, tan, cot, asin, acos, atn ( arcus sinus, arcus cos, arcus tangens ), sh, ch, th ( hyperbolic sinus, cosinus and tangens ), sec and csc (secans and cosecans). The area is described as an adjustable amount of plane sections. The amount of points per line is determined by the selection of the accuracy. Moreover, the angle mode for the trigonometrical functions and the display as full or half screen (Zoom) is determined. Hidden lines are not plotted, and those parts of the area which are seen from below are described in another color ( only EGA and VGA with color monitor). Examples : a) f(x,y) = sin(u)/u u(x,y) = sqr(x*x+y*y) -10 ≤ x ≤ 10 , -10 ≤ y ≤ 10 , 0 ≤ z ≤ 1 b) f(x,y) = cos(u)*(cos(u)+1) u(x,y) = sqr(x*x+y*y) -10 ≤ x ≤ 10 , -10 ≤ y ≤ 10 , 1 ≤ z ≤ 5 c) f(x,y) = u*cos(y/u) u(x,y) = 1/(1+x*x/10) 0 ≤ x ≤ 6 , 0 ≤ y ≤ 10 , 0.5 ≤ z ≤ 2 d) f(x,y) = abs(cos(x))+abs(cos(y)) -5 ≤ x ≤ 5 , -5 ≤ y ≤ 5 , 1 ≤ z ≤ 3 e) f(x,y) = (abs(x)+abs(y))/u u(x,y) = sqr(x*x+y*y) -10 ≤ x ≤ 10 , -10 ≤ y ≤ 10 , 1 ≤ z ≤ 2 »4B - 30 - ┌─────────────────────────────────────────────────────────────────┐ │ 1 0 . T H E M E N U S T O C H A S T I C S │ └─────────────────────────────────────────────────────────────────┘ 10.1 Statistics ─────────────────────────────────────────────────────────────────── «51 Mean ( arithmetic mean ), median, variance and standard deviation are determined for a prime notation. In addition, the distribution is presented as a histogram. The prime notation can be read from a file; it can also be entered in one to four input pages with 64 values each in the program. The inputs can also be saved in a file. Here, all values of the prime notation are written to the file in ASCII-format, whereas function F4 'Save data in a file' will save only one input page and apply an internal format. Mean x = 1/n ∙ Σ x(i) Medean is the value in the centre of the assorted prime notation; if the amount of values is even, then it is the mean value of the two central values. _ _ Variance σ² = 1/(n-1)∙Σ(x(i) - x) v.v. 1/n∙Σ(x(i) - x) Standard deviation σ = √σ² »51 10.2 Regression ─────────────────────────────────────────────────────────────────── With this routine it is possible to execute a fitting of curves for a series of measurements with a maximum of 192 pairs of values. The pairs of values can be read in from a file; it can also be entered in one to four input pages in the program. The inputs can be saved in a file; one pair of values, separated by a comma, is written to each line of the file. You may choose between the following types of regression and may displace or stretch all points in x- or y-direction, if necessary. 0 : Proportional regression y = b∙x 1 : Linear regression y = a + b∙x 2..9: Polynom regression n-ter order y = a(0) + ... + a(n)∙x^n 10 : Geometrical regression y = a∙x^b 11 : Exponential regression y = a∙b^x 12 : Logarithmic regression y = a + b∙ln(x) - 31 - «52 Range, scaling, accuracy, angle mode and full or half screen are determined for the diagram after selection of the type of regres- sion. The range is determined in a way that the given points are plotted. However, it can be modified at will, to enable a preview to different ranges. The function term of the approximation curve, the coefficient of determination, the correlation coefficient and the standard devia- tion are issued together with the diagram. On the next page, these values are issued once more, together with a table of values of the function. The function term of the approximation curve is filed automatically in register R9. Except for the polynom regression, additionally the function term of the inverse function is written to register R10. This allows the curve discussion of the results, or the calculation of the areas below the curves. However, you should bear in mind, that the coefficients are integrated into the calculation with eight-figure exactness only, so that further-on calculation can soundly propagate rounding errors. The following formula are used for linear regression. In case of geometrical regression, x and y have to be substituted by ln x and ln y. In case of exponential regression only y, and in case of logarithmic regression only x has to be substituted. b = (n∙Σxy - Σx∙Σy)/(n∙Σx² - Σx∙Σx) , a = (Σy - b∙Σx)/n Coeff. of determiantion r² = h1/h2 Coeff. of correlation r = √r² Variance σ² = (h2 - h1)/(n-2) Standard deviation σ = √σ² with h1 = b∙(Σxy - 1/n ∙Σx∙Σy) and h2 = Σy² - 1/n ∙Σy∙Σy Example : x │ 0 3 -1 1 ──┼───────────────────────────────────── y │ 4 10 6 7 ┌────────────────────────────────┬────────────────────────────────┐ │ Regression type : 1 │ Regression type : 3 │ ├────────────────────────────────┼────────────────────────────────┤ │ Linear regression : │ Polynom regression : │ │ │ │ │ y = 1.228571∙x + 5.828571 │ y = 4 + 1.25∙x + 2.5∙x^2 │ │ │ - 0.75∙x^3 │ │ │ │ │ Coeff.of determin.: 0.70438 │ Coeff.of determin.: 1 │ │ Coeff.of correl. .: 0.83927 │ Coeff.of.correl. : 1 │ │ Standard deviation: 1.66476 │ Standard deviation: 0 │ └────────────────────────────────┴────────────────────────────────┘ The geometrical regression and the logarithmic regression cannot be executed with these data, as the third pair has a negative x-value. »52 - 32 - 10.3 Combinatorial analysis ─────────────────────────────────────────────────────────────────── «53 The amounts of possibilities to select k out of n elements are calculated, setting value on sequence or not ( i.e. arrangement or combination), and permitting repetitions or not. Example : n = 49, k = 6 Arrangements without repetitions = 10 068 347 520 Arrangements with repetitions = 13 841 287 201 Combinations without repetitions = 13 983 816 Combinations with repetitions = 25 827 165 Permutations of k : k! = 720 »53 10.4 Binomial Distribution ─────────────────────────────────────────────────────────────────── For a b(k;n;p) distributed random variable X with fixed n and p you can calculate - a histogram of the probabilities P( X = k ) - a table of their values from k-min to k-max - the probability P( k-min < X < k-max) «54 Theory: n balls are drawn with replacement out of a container with a portion p of red balls. The random variable X stands for the amount of red balls drawn. The probability of k of the balls drawn being red, is characterized by P(X=k) = b(k;n;p). The values for n and p are entered, where p as probability has to lie between 0 and 1. After this, a simple histogram gives a first survey over the values of P(X=k). The numeric values are issued in a table of values. Example : n = 50 p = .3 P( X = 8 ) = .010989 P( X = 9 ) = .021978 8 ▄▄ P( X = 10 ) = .038619 10 ▄▄▄▄▄▄ P( X = 11 ) = .060185 12 ▄▄▄▄▄▄▄▄▄▄▄▄▄▄ P( X = 12 ) = .083830 14 ▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄ P( X = 13 ) = .105017 16 ▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄▄ P( X = 14 ) = .118948 18 ▄▄▄▄▄▄▄▄▄▄▄▄▄ P( X = 15 ) = .122347 20 ▄▄▄▄▄▄ P( X = 16 ) = .114700 22 ▄▄ P( X = 17 ) = .098314 24 ▄ P( X = 18 ) = .077247 P( X = 19 ) = .055757 P( X = 20 ) = .037039 P( X = 21 ) = .022677 P( X = 22 ) = .012811 P( X = 23 ) = .006684 P( X = 24 ) = .003223 P( 8 ≤ X ≤ 24 ) = .990366 »54 - 33 - 10.5 Hypergeometric distribution ─────────────────────────────────────────────────────────────────── For a h(k;n;m;r) distributed random variable X with fixed n, m and r you can calculate a histogram and a table of values for the probabilities P( X = k ). «55 This routine is especially useful, because hardly any tables for hypergeometric distribution exist due to the four input variables, and the calculation of probabilities requires a great deal of expenditure. Theory: A container is filled with m balls, r of which are red. If n balls are drawn without replacement, then the random variable X tells, how many red balls were drawn. The probability of k of the balls drawn being red, is characterized by P(X=k) = h(k,n,m,r). The amount of balls drawn n, the total amount m and the amount of red balls r are entered. As the drawing proceeds without dis- carding, verify that n<m, and also r<m. »55 10.6 Normal distribution ─────────────────────────────────────────────────────────────────── For a N(µ,σ²) distributed random variable X with given expected value µ and variance σ², you can calculate 1 -u(x)²/2 x - µ - density function f(x) = ─────── ∙ e with u(x) = ───── σ∙√(2π) σ x ⌠ - and distribution function Φ(x) = │ f(t) dt ⌡ -∞ «56 The diagram of the density function f(x) is often called Gaussian curve, or bellshaped curve. The distribution function Φ(x) is designated as Gaussian error function, because, according to Gauss, this distribution is assumed for the random errors in astronomical observations. Expected value µ and variance σ² are entered. For µ=0 and σ=1, you will receive the standardized normal distribution. Subsequently, range, scaling, accuracy, angle mode and full screen or half screen are determined for the graphical construction. The diagrams of f(x) and Φ(x) are plotted, and their function terms are filed in the registers R9 and R10. In program section 'integral calculus' it is now possible to calcu- late the areas below the bellshaped curves and to solve problems like P(x1< X <x2). Finally, a table of values for f(x) and Φ(x) is issued with optional steps. »56 - 34 - ┌─────────────────────────────────────────────────────────────────┐ │ 11 . T H E M E N U L I N E A R A L G E B R A │ └─────────────────────────────────────────────────────────────────┘ 11.1 Systems of linear Equations ─────────────────────────────────────────────────────────────────── «61 The program determines the solution vector of a system of linear equations ( SLE ) with n equations and n variables ( n ≤ 10 ). First enter then number of equations and then the coefficients of the system. The system must be transformed to : ┌ ┐ │ a(1,1)∙x(1) + ... + a(1,n)∙x(n) = b(1) │ │ : : : │ │ a(n,1)∙x(1) + ... + a(n,n)∙x(n) = b(n) │ Example: If you are looking for a parabola through P(1/3), Q(2/1) and R(4/9) you have to solve the following system of equ. 1∙x(1) + 1∙x(2) + 1∙x(3) = 3 The solution vector is 4∙x(1) + 2∙x(2) + 1∙x(3) = 1 16∙x(1) + 4∙x(2) + 1∙x(3) = 9 ( 2, -8, 9 ) Thus the parabola is discribed by y = 2x^2 - 8x + 9. »61 11.2 Linear Combination ─────────────────────────────────────────────────────────────────── «62 The program determines the linear combination of a vector out of three given vectors. An error message will be displayed if these vectors are linearly dependent. With this routine you may proof the linear independence of three vectors what means to proof if they are situated in a plane. Example 1 : ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ │ 1 │ │ 2 │ │ 0 │ │ 2 │ a∙│ 2 │ + b∙│ 1 │ + c∙│ 1 │ = │ 3 │ │ 0 │ │ 1 │ │ 0 │ │ 7 │ └ ┘ └ ┘ └ ┘ └ ┘ Result : a = -12 , b = 7 , c = 20 Example 2 : ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ │ 1 │ │ 2 │ │ 1 │ │ 2 │ a∙│ 2 │ + b∙│ 1 │ + c∙│ 5 │ = │ 3 │ │ 0 │ │ 1 │ │-1 │ │ 7 │ └ ┘ └ ┘ └ ┘ └ ┘ Result : The vectors are linearly dependent »62 - 35 - 11.3 Scalar Product ─────────────────────────────────────────────────────────────────── «63 Given two vectors the scalar product, the length of the two vectors and the included angle will be calculated. ┌ ┐ ┌ ┐ Example : -> │ 1 │ -> │ 5 │ a = │ 3 │ b = │ 0 │ │ 1 │ │ 3 │ └ ┘ └ ┘ Scalar product of the vec. = 8 Length of the first vector = 3.32 Length of the second vector = 5.83 included angle ß = 65.56° »63 11.4 Vector product ─────────────────────────────────────────────────────────────────── «64 Given two vectors the vector product and its magnitude will be cal- culated. The vector product is a vector orthogonal to the paralle- logram which is put up by the given vectors. Its magnitude is equal to the area of the parallelogram. ┌ ┐ ┌ ┐ Example : -> │ 1 │ -> │ 7 │ a = │ 2 │ b = │ 1 │ │ 3 │ │ 4 │ └ ┘ └ ┘ ┌ ┐ -> -> │ 5 │ -> -> a x b = │ 17 │ │ a x b │ = 21.98 │-13 │ └ ┘ »64 11.5 Triple Product ─────────────────────────────────────────────────────────────────── «65 Given three vectors the program will calculate the triple product. Its magnitude is equal to the volume of the parallelepipedon which is put up by the three vectors. Linear dependent vectors has zero as triple product. ┌ ┐ ┌ ┐ ┌ ┐ Example : -> │ 2 │ -> │ 2 │ -> │ 3 │ a = │ 3 │ b = │ -1 │ c = │ 9 │ │ 5 │ │ 7 │ │ 2 │ └ ┘ └ ┘ └ ┘ -> -> -> ( a x b ) ∙ c = 26 »65 - 36 - 11.6 Matrix Inversion ─────────────────────────────────────────────────────────────────── «66 Given a square matrix the program will calculate the inverse matrix as well as the order, the rank and the determinant of the matrix. Example : Order of the matrices = 3 ┌ ┐ ┌ ┐ │ 0 1 1 │ │ 0.25 -0.25 0.5 │ │ 0 1 3 │ is inverse to │ 1.5 -0.5 0 │ │ 2 0 1 │ │ -0.5 0.5 0 │ └ ┘ └ ┘ »66 11.7 Matrix Multiplication ─────────────────────────────────────────────────────────────────── «67 Given two square matrices of same order the product matrix will be calculated. Non-square matrices must be completed with zeros. Example : Order of the matrices = 4 1. Matrix : 2. Matrix : ┌ ┐ ┌ ┐ │ 1 2 3 : 0 │ │ 1 2 3 4 │ │ 4 5 6 : 0 │ │ 5 6 7 8 │ │ . . . . . . . . . │ │ 9 10 11 12 │ │ 0 0 0 : 0 │ │ . . . . . . . . │ │ 0 0 0 : 0 │ │ 0 0 0 0 │ └ ┘ └ ┘ Product matrix : ┌ ┐ │ 38 44 50 56 │ │ 83 98 113 128 │ │ . . . . . . . . │ │ 0 0 0 0 │ │ 0 0 0 0 │ └ ┘ »67 - 37 - ┌─────────────────────────────────────────────────────────────────┐ │ 12 . A P P E N D I X A : S Y N T A X │ └─────────────────────────────────────────────────────────────────┘ 12.1 Notation of numbers ─────────────────────────────────────────────────────────────────── Fixpoint notation with decimal point: 1000.5 for 'thousand point 5' Floating point notation : 1.2e3 for 1.2∙10^3 π ≈ 3.1415926535898 may be entered as pi . 12.2 Arithmetic signs ─────────────────────────────────────────────────────────────────── + Addition * Multiplication ^ Raise to power - Subtraction / Division ( ) Brackets The multiplication sign * may never be omitted. The usual rules of hierarchy are valid. Fractions must be written in one line. The numerator and denominator are bound by parenthesis if necessary. 5x - 3 f(x) = ─────────── notified as f(x) = (5*x-3)/(2*x+4) 2x + 4 f(x) = 3x² - 5x + 1 notified as f(x) = 3*x^2-5*x+1 12.3 Functions ─────────────────────────────────────────────────────────────────── sqr(x) for the square root function √x abs(x) for the absolute value │x│ int(x) for the Gaussian parenthesis [x] sgn(x) for the signe function ( -1 for x<0, 0 for x=0, 1 for x>0 ) exp(x) for the natural exponential function e^x ln(x), lg(x), ld(x) logarithms to base e, 10 and 2 sin(x), cos(x), tan(x), cot(x) always with paranthesis !!! asin(x), acos(x), atn(x) the arcus functions of sin, cos and tan sh(x), ch(x), th(x) the hyperbolical sine, cosine and tangens sec(x) = 1/cos(x), csc(x) = 1/sin(x) secans and cosecans fac(n) = 1∙2∙ ... ∙n the factorial of a natural number n norm(x) the distribution function of the standard normal distrib. - 38 - ┌─────────────────────────────────────────────────────────────────┐ │ 13 . A P P E N D I X B : S U P P L E M E N T S │ └─────────────────────────────────────────────────────────────────┘ To save space and clearness some features have not been discribed until now. 13.1 The Parameter P ─────────────────────────────────────────────────────────────────── In addition to the parameters MATHASS \[path]\ directiory of MATHASS-files MATHASS \[Graphicmode]\ VGA, EGA, HERCULES oder CGA MATHASS MONO for monochrome monitors MATHASS INVERS for snapshot routines you may tell MATHASS which part of the program should be called first. To do this start the MATHASS with MATHASS P [number of menu] [number of subdivision] Example: MATHASS P42 leads directly to the function plotter 1 , MATHASS P2A to the RPN calculators ( A = hexadez. 10 ). 13.2 The Control Cursor Keys ─────────────────────────────────────────────────────────────────── In masks with many fields the following key combinations may be useful: Ctrl Home to jump to the first field of the mask even in the EDIT mode. Ctrl End to jump to the last field of the mask even in the EDIT mode. Ctrl Crs_Up to jump to the first input line Ctrl Crs_Dn to jump to the last input line Using an old keyboard driver these combinations may not work. Ctrl PgDn same as PgDn. Additionally the calculation time is displayed in lower right corner. 13.3 Automatic Error Correction ─────────────────────────────────────────────────────────────────── To make input of function terms saver missing multiplication signs are completed if possible. Example: 4x^2-3x is automatically completed to 4*x^2-3*x. - 39 - ┌─────────────────────────────────────────────────────────────────┐ │ 14 . A P P E N D I X C : F I L E F O R M A T S │ └─────────────────────────────────────────────────────────────────┘ The following files must be on your MATHASS-Disk: START .BAT gives first instructions and starts program MATHASS.EXE the program MATHASS MATHASS.DOC the manual to be printed with COPY MATHASS.DOC PRN MATHASS.HLP used by help function MATHASS.PRN contains the selected printer drivers MA .PRM contains the prime factors EPS_FX .DRV printer driver for EPSON FX-80 and compatibles EPS_LQ .DRV printer driver for EPSON LQ " " EPS_MX .DRV printer driver for EPSON MX-80 " " HP_DESK.DRV printer driver for HP Deskjet " " NEC_P6 .DRV printer driver for NEC P6 " " The program itself writes files with the following extensions: *.DAT opened by pressing F3 to save your inputs. They use an internal format and can only be read by MATHASS using F4. *.SCR opened by pressing Shift F1, ..., Shift F10 to save your graphics in an internal format. They can later be re-loaded with Alt F1, ..., Alt F10 or the program point annotation to add text. *.IMG opened by pressing F7 to save your graphics as IMG file. The files are successively called MA1.IMG, MA2.IMG etc. *.PCX opened by pressing F8 to save your graphics as PCX file. The files are successively called MA1.PCX, MA2.PCX etc. *.TXT opened by pressing F7 or F8 to save text screens as ASCII. The files are successively called MA1.TXT, MA2.TXT etc. *.ST they hold the prime notation for doing statistics in ASCII. Each line holds one number. *.RG they hold the pairs of values for doing regression in ASCII. Each line holds one pair of values seperated by semicolon. *.TMP are temporary files which will be deleted after regular termination of the program. - 40 - ┌─────────────────────────────────────────────────────────────────┐ │ 15 . A P P E N D I X D : P R I N T E R D R I V E R S │ └─────────────────────────────────────────────────────────────────┘ The first line of each printer driver defines the type of printer. 0 stands for Laser printer, 9 for nine pins dot matrix printers and 24 for twenty-four pins dot matrix printers. Five lines for each screen mode ( VGA, EGA, HGC and CGA ) follow containing the vertical sacling factor and the printer control sequences for initializing, for line spacing and for graphics mode. Non-printable characters are notified by their corresponding ASCII code in brackets. ( i.e. [27] for ESC ). 13.1 HP_DESK.DRV ─────────────────────────────────────────────────────────────────── 0 printer type ( 0 for Laser ) VGA: 1 vertical scaling factor : 1 ( ≥ 1 ) [27]E[27]&l0C intialize printer : ESC E line spacing 0/48 Zoll : ESC & l 0 C [27]*t100R[27]*r0A graphics mode 100 dpi : ESC * t 100 R beginning at left border : ESC * r 0 A [27]*b graphics line header : ESC * b <Bytes> W Bytes and W are set by the program [27]*rB[27]&l6D end of graphics : ESC line spacing 6 lpi : ESC & l 6 D EGA: HGC CGA 1.3 1.5 2.3 [27]E[27]&l0C [27]E[27]&l0C [27]E[27]&l0C [27]*t100R[27]*r0A [27]*t100R[27]*r0A [27]*t100R[27]*r0A [27]*b [27]*b [27]*b [27]*rB[27]&l6D [27]*rB[27]&l6D [27]*rB[27]&l6D 13.2 NEC_P6.DRV ─────────────────────────────────────────────────────────────────── 24 printer type ( 24 pins dot matrix printer ) VGA 2 vertical scaling factor : 2 (integer) [27]@ initialize printer : ESC @ [27]3[23] graphics line spacing : ESC 3 <23> [27]*[38] graphics mode : ESC * <38> [27]2 reset line spacing : ESC 2 EGA: HGC CGA 3 3 4 [27]@ [27]@ [27]@ [27]3[22] [27]3[24] [27]3[27] [27]*[38] [27]*[38] [27]*[38] [27]2 [27]2 [27]2 - 41 - 13.3 EPS_MX.DRV ─────────────────────────────────────────────────────────────────── 9 printer type ( 9 pins dot matrix printer ) VGA: 1 vertical scaling factor : 1 (integer) [27]@ initialize printer : ESC @ [27]3[24] graphics line spacing : ESC 3 <24> [27]Y graphics mode : ESC Y ( bzw. L ) [27]2 reset line spacing : ESC 2 EGA: HGC CGA 1 1 2 [27]@ [27]@ [27]@ [27]3[24] [27]3[24] [27]3[24] [27]Y [27]Y [27]Y [27]2 [27]2 [27]2 13.4 EPS_FX.DRV ─────────────────────────────────────────────────────────────────── 9 printer type ( 9 pins dot matrix printer ) VGA: 1 vertical scaling factor : 1 (integer) [27]@ initialize printer : ESC @ [27]3[21] graphics line spacing : ESC 3 <21> [27]*[4] graphics mode : ESC * <4> [27]2 reset line spacing : ESC 2 EGA: HGC CGA 1 1 1 [27]@ [27]@ [27]@ [27]3[24] [27]3[24] [27]3[24] [27]*[6] [27]*[1] [27]*[4] [27]2 [27]2 [27]2 13.5 EPS_LQ.DRV ─────────────────────────────────────────────────────────────────── 24 printer type ( 24 pins dot matrix printer ) VGA: 2 vertical scaling factor : 2 (integer) [27]@ initialize printer : ESC @ [27]3[22] graphics line spacing : ESC 3 <22> [27]*[38] graphics mode : ESC * <38> [27]2 reset line spacing : ESC 2 EGA: HGC CGA 3 3 4 [27]@ [27]@ [27]@ [27]3[21] [27]3[22] [27]3[25] [27]*[38] [27]*[38] [27]*[38] [27]2 [27]2 [27]2 - 42 - ┌─────────────────────────────────────────────────────────────────┐ │ 16 . A P P E N D I X E : H I N T S A N D T R I C K S │ └─────────────────────────────────────────────────────────────────┘ The snapshot routines do not support VGA mode. If you have a VGA card installed and want to use the snapshot routines you must start the program with MATHEASS EGA or use a memory residend snapshot program like VGA2GIF. If your printer works with none of the hardcopy routines you can save snapshots and use a graphics program like Graphics Workshop (GWS) from Alchemy Mindworks to do the printing. VGA2GIF and GWS are shareware and are available from every good shareware vendor. Some keyboard driver may use the exponentiation key ^ to enter the french characters â, ê, î, ô and û. In that case you have to press space after ^ to enter an exponentiation, especially when you are in the calculators. The function plotter accept not only rational but arbitrary func- tions. Because of that there is no milky way to recognize gaps. To suppress the plotting over poles try a higher value for accuracy. Line styles 2, 3 and 4 are not discernible if you choose a to great value for accuracy because the distance between calculated points is sometimes less then the length of the dashes. Prime numbers are shown in the so named program point only as many as fit to the screen. For longer lists use output to file PRIM.TXT. They will then be displayed for browse. In the program point mappings it may occur that the last coordinate pairs do not fit to the screen. In that case set the line styles of the preceding mappings to zero. Much effort was done to avoid input errors. Should nevertheless the program hang at any time you can quit with Ctrl Break. In that case type thereafter MODE CO80 to reset the colors.