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SymbMath 2.0: A Symbolic Calculator with Learning (Version 2.0) Copyright (C) 1990-1992 by Dr. Weiguang HUANG 5/6 Cara Road, Geelong, Vic. 3216, Australia Phone: (052)443282 This is Part One of manual. Please read Part Two in the file SymbMath.DO2. This manual is uncompleted at the moment (April 15, 1992) as SymbMath is growing. Welcome to report any error or suggestion in this manual to the author. Contents 1. Introduction 2. Capabilities 3. A List of Files in SymbMath Package 4. Pull-down Menu and Pop-up Menu 4.1 File Menu 4.2 Edit Menu 4.3 Run Menu 4.4 Output Menu 4.5 Color Menu 4.6 Help Menu 4.7 Print Menu 4.8 Edit Help Menu 5. Programming in SymbMath 6. Data Types 6.1 Numbers 6.2 Constants and Built-in Constants 6.3 Variables and Built-in Variables 6.4 Functions 6.4.1 Built-in Standard Functions 6.4.2 Calculus and Transformation Functions 6.4.3 User-defined Functions 6.5 Equations 6.6 Lists, Arrays, Vectors and Matrices 6.7 List Index 7. Operators 8. Expressions 9. Statements 9.1 Comment Statement 9.2 Evaluation Statement 9.3 Assignment Statement 9.4 Switch Statement 9.5 Command Statement 9.6 Compound Statement 9.7 End Statement 13. Library 15. Examples 15.1 Declaration and Simplification 15.2 Calculation 15.3 User-defined Function and Evaluation 15.4 Limits 15.5 Differentiation 15.6 Integration 15.7 Algebraic Equations 15.8 Differential and Integral Equations 15.9 Sums and Products 15.10 Series 15.11 Lists, Arrays, Vectors and Matrices 15.12 Complex Analysis 15.13 Tables of Function Values 15.14 Transformation and Pieces of Expressions 15.15 Chemical Calculation 15.16 Chemical Reactions 15.17 Learning from Users 15.17.1 Learning complicated indefinite integrals from a simple indefinite integral 15.17.2 Learning definite integral from indefinite integral 15.17.3 Learning integral from derivative 15.17.4 Learning integral from algebra 15.17.5 Learning complicated derivatives from a simple derivative 15.17.6 Learning complicated algebra from simple algebra 16. Interface with Other Software 16.1 With PlotData for Graph 17. Inside SymbMath 18. System Limits 19. Future 20. Keywords 20.1 Keywords in Alphabetical Order 20.2 Keywords in Functional Order 20.3 Keywords Dictionary 21. References 1. Introduction SymbMath (an abbreviation for Symbolic Mathematics) is not only a symbolic calculator but also an expert system that can solve symbolical math problems. SymbMath will also perform exact numeric computation. It can manipulate complicated formulas and return answers in terms of symbols, formulas and exact numbers. SymbMath is different from other software. (1) It is able to learn from users. If users only input one formula (e.g. one derivative) without writing any code, it will automatically learn many problems (e.g. derivatives, indefinite and definite integrals) related to this formula. (2) Users can easily edit the library (e.g. integral tables) in "formula format". (3) It requires only 640 KBytes RAM, so it can run on small IBM PCs under MS-DOS. 2. Capabilities Its capabilities include facilities to provide analytical and numerical answers for: o Differentiation: regular or higher order, partial or total, mixed and even implicit differentiation, one-sided derivatives. o Integration: indefinite or definite integration, multiple integration, infinity as a bound, parametric integration, iterated integration, line or surface integrals, discontinuous or implicit integration. o Solution of equations: roots of a polynomial, systems of algebraic or differential equations. o Manipulation of expressions: simplification, factoring or expansion, substitution, evaluation, user-defined functions, built-in standard functions, over 80 internal functions. o Calculation: exact and floating-point numerical computation of integer, rational, real and complex numbers in the range from minus to plus infinity, even with different units. o Limits: real or complex limits, one-sided limits, indeterminate forms. o Complex: calculation, functions, derivatives, integration. o Sum and product: finite or infinite, partial. o Chemical calculation: molecular and atomic weights, concentrations (all by entering the symbols of the chemical elements). o Chemical reactions: inorganic and organic. o Other: series, lists, arrays, vectors, matrices, etc. Also included are: o Pull-down and pop-up menus, sizable and colourable windows. o Twin screen text editor with syntax check. o Programming in integrated edit-interpret-debug environment. o Reading and generating BASIC and FORTRAN codes. o Displaying two-dimensional math notation. o On-line help, and on-line manual. o Interface with other software (e.g. with PlotData for graph). Its shareware version (SM20A.ZIP) is available free from the Calculator directory in SIMTEL20 archives on many anonymous FTP sites (wsmr-simtel20.army.mil, rana.cc.deakin.oz.au, garbo.uwasa.fi, etc) or by e-mail from listserv@vm1.nodak.edu, listserv@ndsuvm1.bitnet. SymbMath is growing though it has limited features at the moment. 3. A List of Files in SymbMath Package ------------------------------------------------------------------------- SymbMath.exe executable SymbMath system. SymbMath.ini initial file. SymbMath.sys configuration file. SymbMath.hlp help file, how to use help. SymbMath.hl2 help file, the list of the edit commands. SymbMath.hl3 help file, the list of keywords. SymbMath.hl4 help file, the keywords dictionary. SymbMath.txt introduction of SymbMath. SymbMath.doc part one of document file or manual file. SymbMath.do2 part two of document file or manual file. readme read me file, the latest updates and suggestion about SymbMath. *.li library files. *.sm user program files. problem.txt problems that other software cannot solved, but SymbMath can do. comment*.txt comments on SymbMath. statis.txt the download statistics at FTP site of garbo.uwasa.fi. exerci*.* exercise from the textbook. prolog.err error messages. shareware.txt shareware concept. software.txt software available from the author Dr. W.Huang. prolog.log log file of SymbMath. ------------------------------------------------------------------------ SymbMath has three versions: Shareware, Student and Advanced versions. 4. Pull-down Menu and Pop-up Menu The screen is divided into four areas: menu line, edit window, output window, and status line. The menu line has 6 pull-down menus: ------------------------------------------------------ | File Edit Run Output Color Help | ------------------------------------------------------ and 2 pop-up menus: print and edit help. Select the menu by the arrow keys or the first capital letter. 4.1 File Menu The File menu has 6 choices in the pull-down menu: ----------- |Load | |New File | |Save | |Directory| |OS shell | |Quit | ----------- The "Load" command loads a file from a disk into the editor. The "New file" command creates a new file. The "Save" command saves the program in the editor into a disk. The "Directory" command changes the current directory. The "OS shell" command executes a DOS command and automatically returns to the SymbMath system if users provide a special DOS command on the command window, otherwise it goes to the DOS shell, control must be returned to the resident SymbMath system with the EXIT command on the DOS shell. The "Quit" command exit from SymbMath. 4.2 Edit Menu SymbMath includes twin screen text editor for creating and modifying program files. Usage of the editor is similar to WordStar in nondocument mode, or to Borland's SideKick, or editor in C compiler. The problem files are ordinary ASCII files, and can be edited with any other text editor. Within the Edit window, press <F1> to pop-up the Edit Help menu (see 4.7 Edit Help Menu), or <Shift><F1> to call the manual. As the twin screen editor, use <F8> to call an auxiliary editor in another window, and <F7> to copy block from the auxiliary editor into the main editor. You can move the cursor around in the editor with the arrow keys, <Page Up>, <Page Down>, and some other keys listed below. The editor has two typing modes, Insert and Overwrite. In Insert mode, anything you type is inserted at the cursor position. In Overwrite mode, pressing a key replaces the current character at the cursor position with the new typed character. Initially, the editor operates in Insert mode, to switch modes, press the <Insert> key or <Ctrl>V. (<Ctrl>V means to hold down the <Ctrl> key on your keyboard while you press V. The V need not be in upper case.) Unlike a more standard word processing program, the editor does not have word-wrap. You must press <Enter> at the end of each line. The editor has an autoindent feature that allows subsequent lines to automatically indent. If you leave several spaces at the start of a line, use <Ctrl>OI to indent subsequent lines the same amount. The command sequence is a toggle, so you need only press <Ctrl>OI again to stop indenting. Sometimes you may want to see more than the area covered by the current window. You can "zoom" the window, expanding it to the size of the entire screen, by pressing <F5>. To return the window to its usual size, press <F5> again. For other window commands, such as moving and resizing a window, see the Window menu, below. Text in the editor always be saved after exit by press <Esc>. Table 4.2 Summary of the editing commands. --------------------------------------------------------------- Cursor Movement Commands (move cursor to): Character left <Ctrl>S or Left arrow <- Character right <Ctrl>D or Right arrow -> Word left <Ctrl>A or <Ctrl> <- Word right <Ctrl>F or <Ctrl> -> Line up <Ctrl>E or Up arrow Line down <Ctrl>X or Down arrow Scroll up <Ctrl>W Scroll down <Ctrl>Z Page up <Ctrl>R or <Pg Up> Page down <Ctrl>C or <Pg Dn> Beginning of line <Ctrl>QS or <Home> End of line <Ctrl>QD or <End> Top of screen <Ctrl>QE or <Ctrl><Home> Bottom of screen <Ctrl>QX or <Ctrl><End> Top of file <Ctrl>QR or <Ctrl><Pg Up> Bottom of file <Ctrl>QC or <Ctrl><Pg Dn> Beginning of block <Ctrl>QB End of block <Ctrl>QK Previous point <Ctrl>QP Marker 0 <Ctrl>Q0 Marker 1 <Ctrl>Q1 Marker 2 <Ctrl>Q2 Marker 3 <Ctrl>Q3 Set marker 0 <Ctrl>K0 Set marker 1 <Ctrl>K1 Set marker 2 <Ctrl>K2 Set marker 3 <Ctrl>K3 Insert & Delete Commands: Insert mode on/off <Ctrl>V or <Ins> Insert line <Ctrl>N Delete line <Ctrl>Y Delete to end of line <Ctrl>QY Delete left character <Ctrl>H or <Backspace> Delete character under cursor <Ctrl>G or <Del> Delete right word <Ctrl>T Block commands: Mark block begin <Ctrl>KB Mark block end <Ctrl>KK Mark word <Ctrl>KT Hide block <Ctrl>KH Copy block <Ctrl>KC or <F5> Repeat the last copy <Shift><F5> Move block <Ctrl>KV or <F6> Delete block <Ctrl>KY Read block <Ctrl>KR or <F7> Write block <Ctrl>KW Print block <Ctrl>KP Miscellaneous Commands: Quit edit <Esc>, <F10>, <Ctrl>KD, or <Ctrl>KQ Call the auxiliary editor <F8> Tab <Ctrl>I or <Tab> Tab mode toggle <Ctrl>OT Auto indent on/off <Ctrl>OI Restore line <Ctrl>QL Find <Ctrl>QF or <Ctrl><F3> Repeat last find <Ctrl>L or <Shift><F3> Find & replace <Ctrl>QA or <Ctrl><F4> Repeat last find & replace <Ctrl>L or <Shift><F4> Control key prefix <Ctrl>P Abort operation <Ctrl>U Restore error message <Ctrl>QW Search Options: L Local search G Global search N Replace without asking Y Replace with asking ------------------------------------------------------------------ 4.3 Run Menu The Run menu interprets the user program in the main editor, runs it into memory, and outputs the results in the output window. 4.4 Output Menu The Output menu has 3 choices: __________ |To disk | |Off disk| |Window | ---------- The "To disk" command forces the output to a disk file. The "Off disk" command returns the output control from the disk file to the output window. The "Window" command changes the output window size. SymbMath can write output to screen, a disk and a printer. Users select the "To disk" command in the Output window to force output to a disk file, and select the "Off disk" command to return the output into the Output window (screen). These output in the disk file can be read back into SymbMath by selecting the command "Load" in the File menu, but only the output in BASIC format can be run. Therefore, use the BASIC output format by setting the switch Output=BASIC when you write the output into a disk file. A reserved word "last" is the last output. 4.5 Color Menu The Color menus has 4 choices in the pull-down menu: --------------- |Menu line | |Edit window | |Output window| |Status line | --------------- When users choose one of them, the color pattern will come up. Users select the favor color by the arrow keys and <Enter>. 4.6 Help Menu The Help menu has 4 choices in the pull-down menu: ------------- |How to help| |Editor | |Keywords | |Dictionary | ------------- Each has its own help window and help file. The on-line help files are text (ASCII) files. In the help window, to search for a special word, you first press <Ctrl><F3>, then type the word you want to search, finally press <Ctrl><F3> again. e.g. if you want to search for the word "help", you first press <Ctrl><F3>, then type "help", finally press <Ctrl><F3> again. The cursor will go to the word "help". You press <Shift><F3> to repeat the last find. The commands within the help window are the same as that within the edit window, except that text is unable to be changed. Table 4.5 Commands Within the Help window ------------------------------------------------------------------- Commands Action arrows keys move the cursor. <F5> zoom the window, <F5> again to zoom out. <Esc> exit from the window. <F10> exit from the window, the same as <Esc>. <PgUp> page up. <PgDn> page down. <F2> go to nth line, end by <F2> instead of <Enter>. <Ctrl><F3> find the special word, end by <Ctrl><F3> instead of <Enter>. <Shift><F3> repeat last find. <Shift><F10> re-size window by arrow keys. ---------------------------------------------------------------------- There are other two help windows in the Edit window, press <F1> to pop-up the Edit Help Menu, or <Shift><F1> to call the manual window. 4.7 Print Menu Users pop-up the Print Menu by <Alt>P, and set "Printer On" in the Print window to toggles echoing to a printer, or/and set "Log File On" to toggles echoing to the log file "Prolog.Log". Another way to print, by using MS-DOS screen print command <Print Screen>, everything on screen is printed out. 4.8 Edit Help Menu Users pop-up the Edit Help menu by pressing <F1> within the editor. This menu has 7 choices: ------------------ |Show help file | |Cursor movement | |Insert & Delete | |Block functions | |Wordstar-like | |Miscellaneous | |Global functions| ------------------ Except for the first choice of the show help file, others have their own sub-menu. Users can select the edit command from the sub-menu. 5. Programming in SymbMath SymbMath is an interpreter, and accepts a program from the SymbMath Editor or a disk. It runs the program written by any editor in the text (ASCII) file format. After the first program was run into memory, users can call any part of the first program from a second program, as it seems the part of the first program has already been in the second program. Users can run many programs into SymbMath at a time. However, all names of the variables are public and name conflicts must be avoided. Users can clear all of them from memory by the command "clear_all". Note that the switch library "Switch.Li" should be run after clearing all from memory by "clear_all", otherwise the default switches may be changed. SymbMath language is procedure language, which is executed from top to bottom in a program, like BASIC, FORTRAN, or PACSAL. It also is an expression-oriented language. SymbMath program consists of a number of statements. The most useful statement contains expressions, the expression includes data, and the most important data is functions. The structure of SymbMath language is: data -> expression -> statement -> program Note that upper and lower case letters are different in SymbMath language, e.g. abc is different from ABC. 6. Data Types Data in SymbMath language is the numbers, constants, variables, functions, equations, lists, arrays, vectors, matrixes, and list index. All data can be operated. 6.1 Numbers The types of numbers are integer, rational, floating-point, power, real and complex numbers in the range from -infinity to infinity. In fact, the range of the input real numbers is -inf, -(10^307)^(10^307) to -(10^-307)^(10^-307), 0, (10^-307)^(10^-307) to (10^307)^(10^307), inf. The range of the output real numbers is the same as input when the switch Numerical=Off, but when the switch Numerical=On, it is -inf, -1.E307 to -1.E-307, 0, 1.E-307 to 1.E307, inf. It means that the number larger than 1.e307 is converted automatically to inf, the absolute values of the number less than 1.e-307 is converted to 0, and the number less than -1e307 is converted to -inf. For examples: ------------------------------------------- Numbers Type 23 integer 2/3 rational 0.23 floating-point 2.3E2 floating-point 2.3e-2 floating-point 2.3*10^20 power 10^1.E200 power 2+3*i complex inf real inf+i complex 3.14+inf*i complex inf+inf*i complex --------------------------------------------- The real numbers include the integer, rational, floating- point, and power numbers. They may be different when the switch Numerical=Off, and are the same when the switch Numerical=On (i.e. all converted to the floating-point numbers). That "a" and "b" are the same means a-b = 0, while that they are different means a-b<>0. For the floating-point numbers, the upper and lower case letters E and e are the same, e.g. 1e2 is the same as 1E2. 6.2. Constants and Built-in Constants Constants are the unchangable values. There are some built-in constants. The name of these built-in constants should be avoided in the user-defined constants. ------------------------------------------------------------------ Built-in Constants Meanings pi=3.1415926536 the circular constant. e=2.7182818285 the base of the natural logarithms. 1/2 i=(-1) inf infinity. -inf negative infinity. constant the integral constant discont discontinuity, e.g. 1/0. Evaluate the one- sided value by x=c+zero or c-zero if the value of expr is discont. c-zero approach c from negative (-inf) direction, limit as zero ->0. c+zero approach c from positive (+inf) direction, limit as zero ->0. undefined the undefined value, e.g. indetermiate forms: 0/0, inf/inf, 0*inf, 0^0, etc. -------------------------------------------------------------------- Notice that the constants discont and undefined are different. If the value of an expression at x=c is discont, the expression only has the one-sided value at x=c and this one-sided value is evaluated by x=c+zero or x=c-zero. If the value of an expression at x=c is undefined, the expression may be evaluated by the function lim(). e.g. evaluate 1/sgn(x) and sin(x)/x at x=0. Input: define(f(x)=1/sgn(x)) f(0) f(0+zero) Output: defined discont 1 Input: f2=sin(x)/x subs(x=0, f2) lim(x=0, f2) Output: f2 = sin(x)/x undefined 1 6.3. Variables and Built-in Variables The sequence of characters is used as the name of variables. The usual used independent variable is "x". A value can be assigned to the variable, by (1) the assignment "=", (2) the user-defined function f(), (3) the function subs(). e.g. (1) x=2, (2) f(2) if f(x)=x^2 has been defined, (3) subs(x=2 to x^2). The built-in variable is "last". The variable "last" is always be assigned the value of the last output. 6.4. Functions 6.4.1 Built-in Standard Functions Different versions of SymbMath have different built-in standard functions. The Advanced Version C has all of them. See the following table in detail for other versions. All below standard functions except n! or fac(n) can be differentiated and integrated symbolically. Table 6.1 Built-in Standard Functions ----------------------------------------------------------------------- Functions Meanings n! factorial of n. fac(n) the same as n!. sqrt(x) square root, the same as x^0.5. exp(x) the same as e^x. sgn(x) 1 when Re(x) > 0, or Re(x) = 0 and Im(x) > 0; 0 when x=0; -1 otherwise. abs(x) absolute value of x. ............... above functions in Shareware Version A ............... ln(x) natural logarithmic function of x, based on e. sin(x) sine function of x. cos(x) tan(x) csc(x) sec(x) cot(x) asin(x) arc sine function of x, the inverse of sin(x). acos(x) atan(x) acot(x) asec(x) acsc(x) ............... above functions in Student Version B ................. sinh(x) hyerbolic sine function of x. cosh(x) tanh(x) csch(x) sech(x) coth(x) asinh(x) arc hyerbolic sine function of x, the inverse of sinh(x). acosh(x) atanh(x) acoth(x) asech(x) acsch(x) erf(x) error function, or probability integral function of x. -------------------------------------------------------------------------- 6.4.2 Calculus and Transformation Functions Calculus functions are for calculus calculation while trans- formations functions are to move or take a piece of expression. Table 6.2 Calculus Functions ---------------------------------------------------------------------- Functions Meanings subs(x=c, y) substitutes x by c in y. subs(x=c to y) the same as subs(x=c,y). lim(x=c, y) gives the limit of y when x approaches c. Note that the correct answers only for the indeterminate forms: 0/0, inf/inf, 0*inf, 0^0, inf^0. d(y, x=c) gives the derivative of y with respect to an undefined variable x at x=c. d(y, x=c, n) gives the nth order derivative of y with respect to an undefined variable x at x=c. d((y)/d(x)) differentiate y with respect to x. d((y)/d(x)), n) gives the nth order derivative of y with respect to an undefined variable x. d(y) implicit differentiation, used in differential equations, e.g. x*d(x)+y*d(y) == 0. inte(y,x,a,b) find the definite integral of y with respect to an undefined variable x taken from x=a to x=b. inte(y, x from a to b) the same as inte(y,x,a,b). inte((y)*d(x)) find the indefinite integral of y with respect to an undefined variable x. inte(y) implicit integration, used to integrate the differential equations sum(y, x from a to b step c) sum of y. sum(y, x from a to b) sum of y step=1. sum([y]) transform list to sum prod(y, x from a to b step c) product of y. prod(y, x from a to b) product of y step=1. --------------------------------------------------------------------- Table 6.3 Transformation Functions --------------------------------------------------------------------- expand(F) expand F, e.g. expand((a+b)^2) gives a^2 + 2*a*b + b^2. factor(F) factorise from F, e.g. factor(a^2 + 2*a*b + b^2) gives (a+b)^2. solve(F==0, x) solve polynomial equation, systems of linear equations, linear differential equations. list(y, x from a to b step c) lists of y list(y) transform sum to list table(y, x from a to b step c) data table coef(expr, x^n) gives the coefficient of x^n in expr. assume(x>0) assume x>0. assume(x<0) assume x<0. assume(x == real) assume x is real. left(eq) left hand side of an equation. right(eq) right hand side of an equation. re(x) real part of complex numbers. im(x) imaginative part of complex numbers. nume(a/b) numerator of division, i.e. a. deno(a/b) denominator of a/b, i.e. b. ----------------------------------------------------------------------- For example: --------------------------------------------------------- differentiation diff() d((x^2)/d(x)) integration inte() inte((x^2)*d(x)) limit lim() lim(x=0, sin(x)/x) solving solve() solve(x^2+1==0, x) expanding expand() expand((a+b)^2) factoring factor() factor(a*c+b*c) ----------------------------------------------------------- where "x" is an undefined variable. Above functions can be operated and chained, like the standard functions. If the undefined variable d(x) is omitted in the functions d(y) and inte(y), they are implicit derivatives and integrals. If f(x) is undefined, d(f(x)/d(x)) is differentiation of f(x). These are useful in the differential and integral equations. (see later chapters). For examples: inte(inte(F * d(x)) * d(y)) is double integral with respect to both variables x and y. d(d(y/d(x)) x=c) is the second order derivative of y with respect to x at x=c. d(d(y/d(x))/ d(t)) is the mixed derivative of y with respect to x and t. d((y)/d(x)) is different from d(y)/d(x): (1) d((y)/d(x)) is an operation of differentiation, while d(y)/d(x) is not; (2) y and d(x) is inside the same () in d((y)/d(x)), while d(y) and d(x) can be separated in d(y)/d(x). d(y)/d(x) is the same as y', d(x) can be operated. The keywords "from" "to" "step" "," are the same as separators in multi-arguement functions. e.g. inte(f(x), x, 0, 1) are the same as inte(f(x), x from 0 to 1). The Shareware Version has not solve(), while the Student Version and the Advanced Version have all of these functions. 6.4.3 Users-defined Functions Users can define the new functions, which include the standard functions, calculus functions, and algebraic operators. Users define a new function f(x) by define(), define(f(x)=x^2) and then call f(x) as the standard functions. The function name can be any name, except for the keywords. The maximum number of arguments in the users-defined functions on this version is 2, e.g. f(x) and f(x,y). Alternately, users assign x^2 to a variable f by f=x^2, and then call the variable f by subs(x=c to f). f=x^2 seems a function. To clears a variable or function from memory by clear(x) clear x from memory. clear(f(x)) clear f(x) from memory. clear_all clear all from memory. Note that the switch library "Switch.Li" should be run after clearing all from memory by "clear_all", otherwise the default switches may be changed. 6.5 Equations An equation is an equality of two sides linked by an equation sign "==", e.g. x^2+p == 0, where the symbol "==" stands for an equation. Note the symbols "==" and "=" are different. Systems of equations are a list of equations, e.g. [a1*x+a2*y==a3, b1*x+b2*y==b3]. 6.6 Lists, Array, Vectors and Matrices A list is [a, b, c]. An array is a list of lists. e.g. [a, b, [c1, c2]]. A vector and matrix are similar a list and array, but in a matrix, the number of elements in each row should be the same, e.g. [[a11, a12], [a21, a22]]. 6.7 List Index and Built-in List Index The list index is the index for n-th element in a list. e.g. b[2] indicates the second element in the list b. The built-in list index is last[number]. The name of last output list is "last", e.g. last[1] is the first element in the last output list. 7. Operators Table 7.1 Operators ----------------------------------------------------------------- Operation Operators Examples Order comma , a=2, b=3 1 assignment = p=2+b 2 equation == x^2+x+1==0 3 equal == a==2 3 larger than > a>2 3 larger and equal >= a>=2 3 less than < a<2 3 less and equal <= a<=2 3 unequal <> a<>2 3 plus + a+b 4 minus - a-b 4 mutilation * a*b 5 division / a/b 5 power ^ a^b 6 power ** a**b 6 factorial ! n! 6 positive + +a 7 negative - -a 7 function f() sin(x) 7 list index f[] f[1] 7 parentheses () (a+b)*c 7 list [] [a,b] 7 -------------------------------------------------------------------- All functions have the same 7th order. Operations with higher order precede, otherwise operations are performed left to right. These are the usual algebraic conventions. a^b^c is the same as (a^b)^c. 8. Expressions An expression (i.e. expr) is combination of data and operators. e.g. a+b-f(c). 9. Statements The statements include: comment statement evaluation statement assignment statement switch statement command statement compound statement end statement SymbMath program must be terminated by the end statement. 9.1 Comment Statement # is the comment statement sign. You can add comments into a line, or even produce a single line which is entirely a comment, by preceding the comment sign with #. For example: # This is my program 3 + 4 # My first calculation Comments make your calculations more understandable, whether you are making a printed record of what you are doing or if you just want to jot some notes to yourself while you are working. 9.2 Evaluation Statement The evaluation statement has the format: expression SymbMath evaluates any expression which in a line and gives the value of the expression. e.g. Input: 3 + 4 Output: 7 # the value of 3+4 Input: d(x^6/d(x)) Output: 6 x^5 # the value of d(x^6/d(x)) The last output can be saved to a variable for the later use by the built-in variable "last", e.g. f=last. 9.3 Assignment Statement The assignment statement specifies that a new value of expression_2 be assigned to expression_1, and saved into memory. The form of an assignment is expression_1 = expression_2 where "=" is the assignment operator. Note that not only a single variable but also an expression can be assigned. e.g. x=2, sin(x)/cos(x)=tan(x). e.g. evaluate of d() and assign the result (6*x^5) to f. f=d(x^6/d(x)) 9.4 Switch Statement The switch statement sets or changes the switch status. The switch status is unchanged in memory until the new switch status is assigned. ---------------------------------------------------------------------- Switch Action Numerical=On convert numbers to floating-point numbers. Numerical=Off not convert numbers to floating-point numbers, this is default. NumIntegrate=Off disable numerically integration, this is default. NumIntegrate=On numerically integration. Output=TwoDim display output in two dimension, this is default. Output=BASIC display output in BASIC format. Output=FORTRAN display output in FORTRAN format. Output=Off not display output. ExpExpand=On exponent expansion. e.g. c^(a+b) to c^a*c^b. ExpExpand=Off disable exponent expansion, this is default. LnExpand=On log expand, e.g. ln(a*b) to ln(a)+ln(b) LnExpand=Off disable log expansion, this is default. ---------------------------------------------------------------------- Remember that first letter of the switch is upper case, and first letter of words are upper case, too. e.g. TwoDim means two dimension. 9.5 Command Statement The command statement neither evaluates any expression nor saves any expression. It executes a command. ---------------------------------------------------------------------- Commands Action last the last output. last[1] the first element of the last output list. show_all show all the variables in memory. clear(y) clear the variable y from memory. clear_all clear all the variables from memory. system(xx) execute the MS-DOS command xx. end the end of the program and library files. --------------------------------------------------------------------- 9.6 Compound Statement The all above statements are simple statements. The compound statement specifies that its component statements are to be executed in the same sequence as they are written. They are separated by the separators (comma ","). e.g. a+b, 2+3 9.7 End Statement Program must be terminated by the end statement. The end statement is "end" in a single last line. end 13. Library A library is a text file with a table of formula in mathematical notation, so users can edit them by any text (ASCII) editor. In order to expand the special ability of SymbMath, some libraries should be run. The library is loaded into the Editor by the command "Load" in the "File" menu, and then run into memory. Many libraries can be run into memory at a time, however, all names of the variables are public and name conflicts must be avoided. Users can clear all of them from memory by "clear_all". Note that the switch library "Switch.Li" should be run after clearing all from memory by "clear_all", otherwise the default switches may be changed. Alternately, a part of the library file rather than the whole library file can be copied into the Editor by pressing <F7> (external copy) and then run into memory in order to save memory. There are many libraries. The following are some of them. Table 13.1 Library ------------------------------------------------------------------------ File Name Library Function Switch.Li switches on the default state. Units.Li an units conversion. Function.Li the user-defined functions. Trig.Li reduction of trig functions. Hyerboli.Li reduction of hyerbolic functions. Integral.Li integrals. Derivati.Li derivatives of special functions. Diff.Li to find total derivatives, and implicit derivatives of an equation on one step. Chemical.Li the atomic weight of chemical elements. Inorgani.Li inorganic chemical reactions. Organic.Li organic chemical reactions. ----------------------------------------------------------------------- (continued in SymbMath.DO2)