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Remarks on structure of help file. This screen is never displayed by X(PLORE). Each screen of information begins with a centered title, and the last line is followed by CHR(200) ╚ . Following that is the list of screen numbers referred to by <A>, , ... on the screen. The screen number of this screen is 0, the first visible screen (the next screen) is number 1. The last number in the list of numbers before the terminator 9999 is the screen number of the current screen. ╚ 9999 X(PLORE) - The Mathematics Processor On-Line Help <A> How to use On-Line Help Capabilities of X(PLORE) <C> Introduction to X(PLORE) <D> Help for individual commands <E> Function keys and other special keys <F> Saving and loading your formulas and data files <G> Printing <H> Disk and Memory management ╚ 7 6 8 2 5 78 79 81 1 9999 Commands 1 <A> ! <F> AND <L> COLOR <Q> DIF <H> " <K> ASC <H> COLS <L> DIFFG <K> ' ASIN <H> COLVECTOR <C> DIV <A> * ASINH <H> COND <H,I> DISPLAY <A,K> + ATAN <D> CONJ <L> DOTGRAPH <A> - ATAN2 COS <L> DTG <H> .* ATANH COSH <H> EIG <H> ./ <H,I> AVERAGE COT <H> EIGENVAL <H> .^ <L> AXIS <L> CRCOLOR <T> ELSE <A> / <L> AXISCOLOR <L> CROSSX <T> END <W> // <L> BARG <L> CROSSY <L> ERASE <F> < <T> BEEP CSC <W> EVAL <F> <= <C> BIN <M> CURVE3D <J> EVALPOLY <F> <> <A> BINOM <L> CYAN <E> EXACTOFF <F> == <L> BKCOLOR <L> DARKGRAY <E> EXACTON <F> > <L> BLACK <K> DATE <T> EXIT <F> >= <L> BLUE DEGREE EXP <A> ABS <L> BROWN <E> DENOMINATOR <Y> EXPANDON ACOS <H> CHARPOLY <H> DET <Y> EXPANDOFF ACOSH <K> CHR <H> DIAG <X> more commands ╚ 21 22 23 24 25 26 27 28 35 36 37 38 57 2 64 67 68 69 2 71 78 79 8 3 83 2 9999 Commands 2 <H> EYE <H> GETMATRIXTOL <C> INOT <L> LIGHTMAGENTA <F> FALSE <O> GETSOLVETOL <T> INPUT <L> LIGHTRED <L> FIELD <L> GR <T> INPUT@ <L> LINE <L> FILL <M> GR3 <T> INPUTS <L> LINESTYLE <R> FIN <L> GRAPH <T> INPUTS@ <Y> LIST <E> FIX <M> GRAPH3D <C> INT <L> LISTG <E> FLOAT <T> GRAPHICS <L> INTEG LN <L> FONTSIZE <L> GREEN <C> INTEGER LNGAMMA <T> FOR <L> GRID <Q> INTOL LOAD <W> FORGET <Z> GSAVEOFF <C> IOR LOADGRAPH <C> FRAC <Z> GSAVEON <F> ISALIST LOADTEXT FREQUENCIES <C> HEX <F> ISBIGNUM LOG10 FSAVE <L> HISTOGRAM <C> IXOR <H> LU FSAVEGRAPHS <C> IAND <L> KEEPLINE <L> MAGENTA <O> FSOLVE <T> IF <I,K> LENGTH <L> MARK <T> FUNCTION <D> IM <L> LEVELC <H> MATRIX <L> GETBKCOLOR <O> IMPLICIT <L> LIGHTBLUE <M> MATRIXG <L> GETCOLOR <R> IN <L> LIGHTCYAN <H> MATRIXTOL <R> GETINDEPTH <R> INDEPTH <L> LIGHTGRAY <H,I> MAX <R> GETINTOL <R> INERROR <L> LIGHTGREE <X> more commands ╚ 21 22 23 24 25 26 27 28 35 36 37 38 57 3 64 67 68 69 3 71 78 79 8 4 34 63 3 9999 Commands 3 <L> MAXCOLOR <L> POLARG <H> ROWS TAN <H> MEDIT <L> POLARGRID ROWVECTOR TANH <H,I> MIN <K> POS <H> RREF <T> TEXT <C> MOD <W> PRECISION SAVEGRAPH <W> TEXTCOLOR <V> NEWPAGE <V> PRINT SAVE <L> THICKLINE <L> NOAXIS <V> PRINTGRAPH SEC <L> THICKTEXT <L> NOGRID <T> PROCEDURE <L> SETCRO <L> THINLINE <F> NOT <G,H,I> PRODUCT SINH <L> THINTEXT <E> NUMERATOR <H> QR SIN <H> TRACE <H> ONES <L> QU <L> SK <K> TIME <F> OR <L> QUICKG <L> SKETCH <L> TRAJ <L> PA RADIAN <O> SOLVE <H> TRANS <M> PA3 <A> RANDOM <J> SOLVEPOLY <F> TRUE <A> PAR <H> RANK <O> SOLVETOL <W> UNDEFINED <L> PARAMG <D> RE <H> SORT <L> UNMARK <M> PARAMG3D <L> RECT <A> SQRT <T> UNTIL <T> PAUSE <L> RED <H,I> STANDEV <K> UPCASE PI <H,I> REGR <K> STR <K> VAL <H> PINV <T> REPEAT <G,H,I> SUM <L> VARYLINE <L> PLOT <T> RETURN <H> SVD <L> VWRITE@ <L> PO <A> ROOT TABLE <X> more commands ╚ 21 22 23 24 25 26 27 28 35 36 37 38 57 4 64 67 68 69 4 71 78 79 8 5 18 4 9999 Commands 4 <T> WAIT <V> WINDOWTOP <W> ZERO <Z> Σ (alt-s) <X> WHILE <M> WORDWRAPON <F> ^ <A,F> () <V> WHITE <M> WORDWRAPOFF π (alt-p) {} <V> WINDOW <Q> WRITE <C> î (alt-i) <A,G,H> [] <V> WINDOWBOTTOM <V> WRITE@ ê (alt-e) <H,S> | <V> WINDOWLEFT <X> XOR <E> ² (alt-r) <J> % <V> WINDOWRIGHT <Y> YELLOW <E> √ (alt-q) & Function Keys and other special keys F1 Help (If graphics screen visible, abbreviated help) <K> F2 Print menu (If graphics screen visible, print graphic) <L> F3 Begin marking text Shift-F3 Mark entire entry <M> F4 Insert text cut with F3 Control-F4 Delete text buffers <M> F5 Scroll output up Shift-F5 Scroll up full screen <N> F6 Scroll output down Shift-F6 Scroll down full screen <N> F7 Save current work (If graphics screen visible, save screen) <O> F8 Load work from file <O> Shift-F8 View directory Control-F8 Change directory <O> F9 Display last graph Shift-F9 Toggle last graph to disk Control-F9 Redo configuration file <T> F10 Toggle between Input Window and Subroutine Window <Q> Insert Toggle Insert/Overwrite mode <R> Control-Enter Insert a blank line ╚ 34 22 24 26 21 10 30 37 29 82 7 79 18 11 78 63 71 11 35 19 25 38 28 23 17 27 5 9999 Capabilities of X(PLORE) Evaluate formulas <A> Using trigonometric, exponential and logarithmic functions Using real and complex numbers <C> hexadecimal and binary numbers <D> exact fractions and huge integers <E> strings <F> Algebraic Simplification <Z> Graph curves <G>, polar <H> and parametric curves Graph surfaces <J>, parametric surfaces <K>, space curves <L> and matrix data <M> Solve equations and systems of equations <N> Define implicit functions <O> Differentiate and integrate <Q> functions Perform vector and matrix calculations <R> including eigenvalues and eigenvectors <S> and statistical operations <T> Collect data into lists and perform statistical operations <V> Save and load data from external files <W> Print calculations, results and graphs <X> Write your own Pascal-like functions and procedures <Y> using X(PLORE)'s powerful mathematical and graphing operators. ╚ 21 22 24 23 25 37 40 41 42 58 59 60 61 64 65 68 69 28 31 32 34 35 78 79 71 83 6 9999 How to use On-Line Help Items on each help screen are accompanied by letters surrounded by angle brackets, like <*>. Pushing the corresponding letter on the keyboard will get you more help on this topic. To leave Help and return to X(PLORE), push <ESC>. To back up to the previous help screen, push <PgUp>. Throughout these screens, X(PLORE) commands are printed entirely in UPPERCASE letters. These help screens are stored in the file X(PLORE).HLP, which should be in the same directory as X(PLORE) unless you have set up a different location in your configuration file <A>. ╚ 19 7 9999 Introduction to X(PLORE) <A> Stopping X(PLORE) and saving your work so you can continue later Entering mathematical formulas into X(PLORE) <C> Understanding your results and controlling decimal precision PRECISION and UNDEFINED <D> Using variables for storing results FORGET <E> Defining your own functions EVAL <F> Creating two-dimensional graphs <G> Creating three-dimensional graphs <H> Writing your own procedures and functions Adding comments to your work with // <J> Using TEXTCOLOR to control the colors of your screen <K> Using cut and paste to copy and edit your formulas <L> Creating a configuration file for your own system <M> List of reserved words (4 screens) ╚ 9 10 12 13 14 38 57 15 16 17 18 19 2 8 9999 Stopping X(PLORE) and saving your work To stop X(PLORE), push Control-Q (for quit). You will be asked if you want to save your work. If you respond Y for yes, you will be asked to enter a file name for saving your work. Enter any memorable name, followed by a period, followed by an extension. The name can be up to 8 characters long, and the extension up to 3 characters. One common practice is to always use the extension XPL to identify X(PLORE) files. For example, you could call your work LEVERARM.XPL . To reload your work when you next use X(PLORE), start X(PLORE) with the command: XPL LEVERARM.XPL That is, follow the program name X(PLORE) with the name of the file containing your work. When you restart X(PLORE) you must execute all the lines in your Input Window. None of your functions or variables will be defined until you do this. Advanced information: Path information can be part of the file name. X(PLORE) will run files from remote directories, or CC can be in a remote directory and operate on a file in the current directory. See also <A> Using external files. ╚ 78 9 9999 Entering mathematical formulas into X(PLORE) Mathematical formulas are entered into X(PLORE) much as you would ordinarily write them. They are evaluated when you press <ENTER>. The usual order of operations is followed, with exponentiation following multiplication and division following addition and subtraction. Parentheses () can be used to alter this order. The main difference from ordinary mathematical notation is that everything must be written on one line. Here are some examples: Mathematics X(PLORE) 2 2(3.2-4.3) 2*(3.2-4.3)^2 4.2 - 2.3 --------- (4.2 - 2.3)/(2.1 + 3.2) 2.1 + 3.2 Sin(4.2) SIN(4.2) See also editing entries <A>, entering numbers , algebraic functions <C>, transcendental functions <D>, hexadecimal and binary numbers <E>, matrices <F>, complex numbers <G>, huge integers and exact fractions <H> and output . ╚ 11 20 21 22 23 28 24 25 12 10 9999 Editing entries When typing an entry to X(PLORE), you can correct your entry by erasing with the Backspace and Delete keys, moving the cursor with the arrow keys, and toggling between insert and overwrite mode by pressing the Insert key. You can leave your formula with the vertical arrow keys, but if you do the formula will not be evaluated. To evaluate a formula, you must press <ENTER> while the cursor is somewhere in the formula. Formulas that have been evaluated are displayed as normal text; unevaluated formulas, including the one you are currently typing, are displayed in boldface or a different color. Formulas longer than one line can be entered into X(PLORE), but you must not press <ENTER> while typing an entry. Just keep typing and let the entry flow from one line to the next. If you press <ENTER>, you will cause your (partial) entry to be terminated and evaluated. After using a formula once, you can use it again. Move the cursor to it, modify it if desired, and push <ENTER>. CC is controlled temporally like a calculator rather than spatially like a spreadsheet. Your results depend on the order in which you execute your formulas, not on the order in which they are listed on your screen. You can scroll more rapidly up and down your list of entries with the <PgUp> and <PgDn> keys. You can add blank lines between formulas by pressing Control-Enter. See Cut and Paste <A> for information on advanced editing. ╚ 18 11 9999 Understanding your results and controlling decimal precision When you enter a formula into X(PLORE) and press <ENTER> <A>, the result appears after the variable ANS in the Output Window. If you assigned your formula to a variable , it will appear after the variable in the Output Window. If the formula you entered had an undefined result, for example 1/0, then the value UNDEFINED will be returned. If UNDEFINED is combined with any other value, the result is UNDEFINED. You can even enter UNDEFINED from the keyboard. You can control the number of significant digits displayed by X(PLORE) with the command PRECISION(n), where n is a number between 1 and 9 (15 for CC87). PRECISION(0) causes results to be displayed in scientific notation: 1,234,567,890 = 1.23456789E9, -0.0000001 = -1E-7. Numbers displayed with î are complex values <C>. These may be created by apparently real formulas like LN(-2) or ASIN(3) as well as by overtly complex formulas. ╚ 10 13 24 12 9999 Using variables for storing results You can assign the result of any formula to a variable with a command like: X = SIN(4.2) . Variable names can be up to 12 characters long. They must begin with a letter and use only letters and digits. X(PLORE) ignores upper and lower case distinctions in variable names, so AB, ab, Ab and aB are the same variable. All of your variables are listed in the Output Window. If there are more variables than can be shown at one time, you can scroll the Output Window with the function keys F5 and F6. Once a variable is defined, it can be used in another formula. Having defined X as above, you could define Y = ASIN(X). You can also redefine a variable by storing a new value in it. If you do this, you will lose the original value entirely. Variables are useful for long calculations. You can save yourself a lot of recalculations by always storing the results of your calculations in your own variables instead of leaving them in the default variable ANS. You can destroy the variable X with the command FORGET(X). If X stored a large list, you could regain a considerable amount of memory by FORGETting X. Also, since a variable and a function <A> cannot have the same name, you must forget a variable before using its name for a function, and vice versa. Greek letters can be used in variable names: α=alt-a, ß=alt-b, δ=alt-d, φ=alt-f, Γ=alt-g, ε=alt-h, Ω = alt-o, and Θ=alt-t. ╚ 14 13 9999 Defining your own functions X(PLORE) allows you to define your own functions for evaluating or graphing <A>. Anytime you find yourself using the same expression over and over, you should turn it into a function. Here are some sample function definitions: f(x) = SIN(x + PI/3) g(x,y) = x^2 - y^2 Once defined, functions can be used in expressions or in other function definitions: y = f(3.2) z = g(4.1, -5.3) Variables can be used in function definitions, and that leads to an important difference between functions and variables. When a variable is defined, all terms of the definition are evaluated. When a function is defined, its terms are not evaluated; they are evaluated when the function is evaluated. That means that if a component of the function definition is changed, that change will be reflected the next time the function is evaluated. You can force all the components of a function to be evaluated when the function is defined by preceding the function with the command EVAL: f(x) = EVAL(x^2+a) ╚ 38 14 9999 Writing your own procedures and subroutine functions X(PLORE) allows you to extend its power with multi-line procedures and functions written in a Pascal-like programming language. All standard control structures are available. These functions can have value parameters and function-name parameters. You can write a routine that operates on an arbitrary function, like an equation-solving routine. Subroutines can call one another irrespective of the order in which they are defined, and recursion is fully supported. Subroutines can be used to implement algorithms from numerical analysis, to write interactive data analysis routines, or even to develop dynamic presentations and interactive mathematics lessons. Programming commands are available to pause a routine or make it wait a specified number of seconds; to build complex graphics screens and permit the user to select one region with the crosshairs; to write messages to the user and to acquire input from text or graphics screens. Text and graphics color can be controlled completely, and complex graphs can be recalled from disk for quick display. For more information on subroutines, see <A>. ╚ 71 15 9999 Adding comments to your work with // You can add comments to your list of X(PLORE) entries to remind yourself why you did what you did or to inform another user how to use your routines. Comments can be used in both the Input Window and the Subroutine Window. To add a comment to the end of an entry, preface the comment with the symbol //, for example: s = 16*t^2 // Law of falling bodies A line can be started with //, in which case the entire entry is a comment. ╚ 16 9999 Using TEXTCOLOR to control the colors of your screen If you are using a color screen, you can specify the color of executed entries and all but the last output, the color of new entries and the last output, and the background color with the command: TEXTCOLOR(foreground,highlight,background) Foreground, highlight and background must be integers between 0 and MAXCOLOR, or color words which are just synonyms for these integers. The color words corresponding to 0..15 are: BLACK BLUE GREEN CYAN RED MAGENTA BROWN LIGHTGRAY DARKGRAY LIGHTBLUE LIGHTGREEN LIGHTCYAN LIGHTRED LIGHTMAGENTA YELLOW WHITE The author's favorite is TEXTCOLOR(BLUE, RED, LIGHTGRAY) These colors can also be used for the graphics screens <A>. ╚ 51 17 9999 Using cut and paste to copy and edit your formulas You can copy part or all of one formula to another part of your Input Window or Subroutine Window as follows: move the cursor to one end of the text to be copied, press F3, move the cursor to the other end (the selected text will be highlighted) and press <ENTER>. At any later time, you can press F4, and the selected text will be copied at the cursor. If you press <DEL> instead of <ENTER>, the selected text will disappear from the screen. However, pressing F4 will still make it reappear at the position of the cursor. You can insert the selected text at several points; just move the cursor to each place you want the text and press F4. Pressing Shift-F3 will select the entire entry at the cursor, then pressing <ENTER> or <DEL> will either select or cut and select this entry. X(PLORE) has 10 buffers for saving text. If you make a second selection, your first selection is retained. You can cause it to be copied at the cursor by pressing Alt-1. Each time you make a new selection, your previous selections are promoted through the text buffers and can be accessed with the keys Alt-1 through Alt-9. If you need to recover the memory used by the text buffers, you can empty all of them by pressing Control-F4. Normally X(PLORE) will break long lines at a "natural" place. To force lines to to use the full width of the screen, enter WORDWRAPOFF; to restore word wrap, enter WORDWRAPON. ╚ 18 9999 Creating a configuration file for your own system X(PLORE) starts with a number of default assumptions. It starts in radian mode <A>, in insert mode , with varying linestyles and colors <C>, and with certain screen colors <D>. You can change these assumptions, you can change the location of the help file <E> and the last graph file <F>, and you can specify your printer type <G> with a configuration file. To create a configuration file, press Control-F9 and answer the questions on the screen. X(PLORE) will create a file X(PLORE).CFG in the same directory as X(PLORE). It will read this file the next time it starts and use the information found there to determine its default assumptions. If you already have a configuration file, you can change it by pressing Control-F9 and answering the questions on the screen. ╚ 22 11 49 17 7 63 79 19 9999 Entering Numerical values into X(PLORE) Numerical values are entered into X(PLORE) just as you would ordinarily write them. You can precede numbers with optional + or - signs. The following are all valid numerical entries for X(PLORE): 3.47 0.347 .347 -347 Blanks are not permitted in numbers; the expression - 34 is not valid. Numbers can be entered in scientific notation. The exponent is preceded by E or e. The following numbers are all valid and all equal: 3.47 0.347E1 34.7e-1 347E-2 0.003470e3 See also complex numbers <A>, huge integers and exact fractions , hexadecimal and binary numbers <C>, and display precision <D>. ╚ 24 25 23 12 20 9999 Arithmetic and algebraic functions Numbers <A> can be combined with + (addition), - (subtraction), * (multiplication), / (division) and ^ (exponentiation). For example: 3+4 = 7 3-4 = -1 3*4 = 12 3/4 = 0.75 3^4 = 81 Where it is not ambiguous, multiplication can be indicated by adjacency: 3(4-1) = 3*(4-1) = 9. Implicit multiplication is potentially dangerous, and it is generally preferable to use * to indicate multiplication. The following functions are available (z is complex, x real, n,m integers): SQRT(z) = square root of z -- abbreviation: √=Alt-q z² = square of z -- ² entered as Alt-r ABS(z) = absolute value of z; n! = n-factorial RANDOM(n) = random integer in range 1..n RANDOM = random real in (0,1) INT(z) = integer part of x FRAC(z) = fractional part of x ROOT(x,n) = nth real root of x LNGAMMA(z) = natural log of Γ(z); LNGAMMA(n+1) = LN(n!) BINOM(n,m) = binomial coefficient = n!/(m!*(n-m)!) ╚ 20 21 9999 Trigonometric and exponential functions The value π = 3.14159... is entered into X(PLORE) as PI or alt-p. Available trigonometric functions are: SIN(x), COS(x), TAN(x), COT(x), SEC(x) and CSC(x). Inverse trigonometric functions are: ASIN(x), ACOS(x) and ATAN(x). Trig functions are evaluated in radian mode unless you have changed to degree mode by entering the command DEGREE. You can switch back to radian mode by entering the command RADIAN. The current mode is indicated in the upper right corner of the screen. The function ATAN2(x,y) returns the polar angle of the point (x,y). The base of the natural logarithms, ê = 2.718... is entered as alt-e. Available exponential functions are EXP(x) = ê^x, LN(x) = natural logarithm of x, and LOG10(x) = real log (base 10) of x. Hyperbolic functions are also defined: SINH(x), COSH(x), TANH(x), ASINH(x), ACOSH(x) and ATANH(x). In all of these functions (except LOG10), x can be any real or complex value <A>. ╚ 24 22 9999 Integers: hexadecimal and binary numbers X(PLORE) can distinguish integers from non-integers, and it will operate on integers with n DIV m (integer part of n/m) and n MOD m (remainder of n divided by m). Integers can be combined bitwise with the commands n IAND m, n IOR m and n IXOR m. You can invert the bits in an integer with the command INOT n . Integers can be entered in hexadecimal or binary notation. Hexadecimal numbers are preceded by $, binary by #. The following numbers are all equal: 27 = $1a = #11011 Integers, hexadecimal and binary numbers are all 32-bit quantities, but integers are signed and the others are not. One kind of number can be changed to another with the command INTEGER(n) (change to integer), HEX(n) (change to hexadecimal) and BIN(n) (change to binary). There is a function PAR(n) that returns 1 if n is even and -1 if n is odd. This is useful for defining sums and products <A>. See also huge integers and exact fractions ; ╚ 25 23 9999 Complex numbers X(PLORE) can use complex numbers as easily as real numbers. Any of X(PLORE)'s functions which are mathematically defined for complex values, for example SIN(x) and EXP(x), will accept complex arguments. Complex numbers are entered into X(PLORE) like this: 3.2 - 4.7î, where î is entered with alt-i. Special complex functions include: RE(z) = real part of z IM(z) = imaginary part of z CONJ(z) = conjugate of z If z is complex, then INT(z) returns INT(RE(z)) + INT(IM(z))î, and FRAC(z) is defined by INT(z) + FRAC(z) = z. Also ABS(z) is the real modulus of z. The polar argument of z can be computed as ATAN2(RE(z),IM(z)) <A>. Functions which do not use complex values include ATAN2, LOG10 and ROOT. Complex numbers can be defined with huge integer or exact fraction coefficients . ╚ 22 25 24 9999 Using huge integers and exact fractions X(PLORE) has special routines to accommodate huge integers and exact fractions with huge numerators and denominators. To enter a huge integer, precede it with &: &12345678987654321 will be stored exactly. To create an exact fraction, use two exact parts: &2/&3 is stored as 2/3, not 0.666666667. Huge integers and exact fractions can be added, subtracted, multiplied, divided, and operated on by integer exponents. The functions DIV, MOD, ! and BINOM work on huge integers; ABS, INT and FRAC work on exact fractions. If any other function is applied to an exact fraction, the fraction will be changed to a decimal before the function is applied. Matrices <A> can be defined with exact fraction entries, in which case addition, multiplication and division of matrices will be exact. If x is an exact fraction, then NUMERATOR(x) and DENOMINATOR(x) are the numerator and denominator of x, and FLOAT(x) is the floating point approximation to x. Conversely, if x is a floating point number, FIX(x) is the exact fraction which is the floating point representation of x. This is useful for turning integers into huge integers. Note: because of the inaccuracies in the floating point representation of numbers, FIX(1/3) is different from &1/&3 = 1/fix(3). To make X(PLORE) treat all integers as huge and all fractions as exact, enter the command EXACTON. To leave exact mode, enter EXACTOFF. You can try &3^100 and &100! to see huge integers, and &1/&3 + &2/&7 to see exact fractions. ╚ 28 25 9999 Boolean expressions Real numbers and strings <A> can be compared with the operators a == b ( a equals b ) a <> b ( a does not equal b ) a b ( a is greater than b ) a <= b ( a is less than or equal to b ) also denoted by ≤ = alt-l a >= b ( a is greater than or equal to b) also denoted by ≥ = alt-m The results of these comparisons is always one of the values TRUE, FALSE or UNDEFINED. The result is UNDEFINED if one side is undefined or symbolic. These comparisons are used mostly in conjunction with the decision-making commands in subroutines . Complex numbers can be compared with == and <>. Note that the equals operator is 2 equal signs: == . This is so the expression x == y won't be confused with the variable assignment x = y. The boolean values TRUE and FALSE can be combined with the usual logical operators AND, OR, XOR and NOT, for example: a d Boolean values can be combined with numbers, since TRUE=1 and FALSE=0. Variables can be tested for being a list or a huge number or exact fraction with the boolean functions ISALIST(x) or ISBIGNUM(x). ╚ 37 71 26 9999 Sums and Products You can defined finite sums (and analogous products) with X(PLORE). For example, to sum the first 20 squares, enter: SUM(i^2,i = 1 to 20) To take the product of the first 10 even inverses (1/2)*(1/4)*..*(1/20), enter: PRODUCT(1/n,n = 2 to 20 step 2). Functions can be defined with sums and products. To define the nth Fourier approximation to the square wave, enter: f(x,n) = SUM(SIN(i*x)/i,i=1 to n step 2) The operator SUM can be abbreviated with Σ = alt-s. You can also take the sum (product) of a list x with SUM(x) <A> or sum the columns of a matrix m with SUM(m) . ╚ 35 32 27 9999 Matrices <A> Entering matrices into X(PLORE) {...} MATRIX MEDIT LOAD DISPLAY Defining special matrices ONES ZERO EYE RANDOM DIAG <C> Matrix operations + - * / \ ^ .* ./ .^ <D> Matrix functions CONJ TRANS " DET TRACE LU QR RREF RANK ROWS COLS CHARPOLY EIGENVAL EIG SVD PINV COND <E> Sorting and statistical operations SORT MIN MAX SUM PRODUCT AVERAGE STANDEV REGR <F> Controlling the accuracy of matrix operations MATRIXTOL GETMATRIXTOL ╚ 29 29 30 31 32 33 28 9999 Entering matrices and defining special matrices There are four ways to enter a matrix into X(PLORE): (i) enclose the elements in braces, separating the elements with commas and the rows with semicolons: M = {1,2,3;4,5,6} (ii) define each matrix element: M = MATRIX(3*i+j-3, i = 1 to 2, j = 1 to 3) (iii) make X(PLORE) display a spreadsheet for entering the elements of the matrix: MEDIT(M). Use the arrow and tab keys to move around the spreadsheet: (iv) read the elements of the matrix from a specially prepared file: LOAD(M) (see <A>) To display a previously constructed matrix, enter the command DISPLAY(M). Certain matrices are predefined: EYE(n) is an n x n identity matrix ZERO(n,m) is an n rows by m columns matrix of zeros ONES(n,m) is an n x m matrix of 1's RANDOM(n,m) is an n x m matrix of random elements in the range (0,1) DIAG(x) is a diagonal matrix whose elements are taken from the list (or row or column vector) x. ╚ 78 29 9999 Matrix operations Matrices can be combined in the usual way with + - * , provided the sizes are correct. You can raise a matrix to a positive or negative integer power: A^n. In addition, you can evaluate A/B = A*B^(-1) and A\B = A^(-1)*B. Element-by-element multiplication or division between two matrices of the same size can be computed with the commands A.*B and A./B . The elements of a matrix can be raised to a power or their roots can be taken with A.^x or Root(A,n) . Matrices and scalars can be multiplied and divided. If M is a matrix and x a scalar, then x*M and M*x multiplies each element of M times x . Also, M/x divides each element of M by x , and x/M = x*(M^(-1)). A quick way to define the inverse of M is 1/M . Submatrices are also defined. If M is a matrix, then M[i,j] is the i,j'th element of M, and M[i1:i2, j1:j2] is the submatrix of M with rows i1..i2 and columns j1..j2. M[i1:i2,j] is part of the j'th column of M , and M[i,j1:j2] is part of the i'th row of M . If an ordinary scalar function like SIN is applied to a matrix, it is applied to every element of the matrix. ╚ 30 9999 Matrix functions Suppose M is a matrix. The following functions are defined: COLS(M) number of columns of M ROWS(M) number of rows of M TRACE(M) trace of M DET(M) determinant of M RANK(M) rank of M RREF(M) row-reduced form of M TRANS(M) transpose of M CONJ(M) conjugate each element of M PINV(M) pseudo-inverse of M COND(M) condition number of M DIAG(v) if v is a single row or column matrix, returns diagonal matrix M" conjugate-transpose (Hermitian) of M CHARPOLY(M) characteristic polynomial of M, expressed as a list <A> EIGENVAL(M) list of eigenvalues of M EIG(M) list of two matrices A and D. A*D = M*A . D is diagonal and the diagonal elements are the eigenvalues of M; the columns of A are the eigenvectors of M. LU(M) list of two matrices L and U; L*U = M, L is a permuted lower triangular matrix and U is upper triangular. QR(M) list of two matrices Q and R; Q*R = M, the columns of Q are orthonormal and R is upper triangular. SVD(M) list of three matrices A,B,C; A and C have orthogonal columns, and B is an invertible diagonal matrix. A*B*C" = M The diagonal elements of B are the singular values of M ╚ 34 31 9999 Sorting and statistical operations on matrices If M is a matrix and i an integer, then SORT(M,i) returns a matrix of the same size as M with the same rows as M but sorted on the i'th column. The data in the i'th column can be real (not complex) numbers or strings <A>. To sort on multiple columns, first sort on the least significant column, then work your way up to sorting on the most significant column. The commands MIN(M), MAX(M), PRODUCT(M), SUM(M), AVERAGE(M) and STANDEV(M) all work the same way. They return a row vector the same size as one row of M, containing in each position the minimum, maximum, etc of the elements in the corresponding column of M . The operation of REGR is more complex. Suppose A is an n x s matrix. Let A1 be the matrix obtained by deleting the last column of A and adding a first column of 1's to A. Let c be the deleted column. Then x = REGR(A) is a column vector which is the least squares solution to the equation A1*x = c. For example, let A be a n x 2 matrix. If each row represents a point, and if y = a+bx is the regression line through the points, then: ┌ ┐ │a│ │b│ = REGR(A) └ ┘ ╚ 37 32 9999 Controlling the accuracy of matrix operations There is a small constant that controls the accuracy of matrix operations like inverting matrices, the LU decomposition and row reduction. Basically, whenever a matrix element smaller than this constant appears, it is set to 0. This procedure is necessary to overcome the inherent inaccuracies in floating point arithmetic. The current value of this constant is returned by GETMATRIXTOL. To give this constant a new value, for example 0.001, execute the command MATRIXTOL(0.001). Executing MATRIXTOL(0) restores the constant to its default value. Too small a value for this constant makes even singular matrices appear non-singular to X(PLORE); too large a value makes non-singular matrices with large condition numbers <A> appear singular. Fiddling with this constant can sometimes make a delicate matrix operation converge. ╚ 31 33 9999 Lists A list is a sequence of values of any sort: real or complex numbers, strings <A>, even other lists. Many different kinds of values can be in a single list. To define a list, either enclose its elements in parentheses: (1,2,3,4,5) or defined it with a formula: LIST(i,i = 1 to 5). The indexing variable in LIST need not increment by 1. You could define the list (0, 0.01, 0.02, ... , 1.00) with the command LIST(x, x=0 to 1 step 0.01). The expression () means the empty list. Defining a singleton list is tricky, because (x) just means x. One way is to use the formula LIST(x,i=1,1). If t is a list, then t[i] is the i'th element of t, and t[i:j] is the sublist of t consisting of elements t[i]..t[j]. The expression t[i,j] means the same thing as t[i][j]; it indicates the j'th element of the i'th element of t, so it is not defined unless t[i] is itself a list. Lists can be added, subtracted, multiplied and divided, term by term. The length of the result is the length of the shortest component. If a scalar is combined with a list by any of these arithmetic operations, it is combined with each element of the list. Thus (1,2) + (3,4) = (4,6) and 1 + (2,3) = (3,4). Scalar functions like SIN applied to a list are applied to every element of the list. For a description of list functions, see . ╚ 37 35 34 9999 List Functions If x and y are lists, then we have the functions: LENGTH(x) length of x x|y concatenation of x and y SUM(x) sum of elements of x PRODUCT(x) product of elements of x MIN(x) minimum of x MAX(x) maximum of x AVERAGE(x) average of x STANDEV(x) standard deviation of x ROWVECTOR(x) make x a row matrix COLVECTOR(x) make x a column matrix DIAG(x) diagonal matrix <A> from elements of x. FREQUENCIES(x,a,b,n) divides interval (a,b) into n subintervals and returns list of number of elements of x in each subinterval. See HISTORGRAM or BARG . REGR(x,y) returns list (a,b,r) of data for regression line through the points (x[i],y[i]). The regression line is y = a + bx and the correlation coefficient is r. If an ordinary scalar function is applied to a list, for example SIN(x), then the function is applied to every element in the list. Once a list has been created, you can view it with the command DISPLAY(x) ╚ 28 52 35 9999 Polynomials For X(PLORE), a polynomial is a list <A> of coefficients. The polynomial n a + bx + ... + cx is represented by the list (a,b,...,c). If p is a list, the value of the polynomial p at the point x can be computed with the function EVALPOLY(p,x). For example, EVALPOLY((1,2,3), -1) = 2. All the roots, real and complex with multiplicity, of the polynomial p can be computed with the function SOLVEPOLY(p) which returns a list of roots. For example, SOLVEPOLY((2,-3,1)) = (1,2). If some roots cannot be found, they are replaced by UNDEFINED in the list of roots. ╚ 34 36 9999 Strings In X(PLORE), a string is a sequence of characters. Strings are defined by placing the desired characters between apostrophes, like this: 'This is a string' Strings are used in INPUT and WRITE statements, they can be stored in variables, and they can be manipulated as follows. If s is a string, then s[i] is the i'th character of s, and s[i,j] or s[i:j] is the substring of s consisting of characters i through j . Strings s and t can be concatenated with the operator s|t or s+t. Strings can be compared with the operators ==, <>, < , <=, >, >= <A>, and strings can be put into list and matrices. A list of strings can be operated on by MAX and MIN , and a matrices can be sorted on a column of strings <C>. The following functions apply to strings: LENGTH(s) number of characters in s. ASC(s) list of ascii codes of characters of s. If s is a single character, then ASC(s) is an integer, not a list of integers. CHR(x) is x is an integer or list of integers, CHR(x) is the string of characters whose codes are in x. Only codes 32..255 are recognized. UPCASE(s) the same string as s , but with a..z replaced by A..Z. POS(s,t) returns the position of the substring s in the string t, or 0. STR(v) v is a value, returns a string representing the value VAL(s) s is a string representing a math expression, returns value DATE, TIME return the current date and time as strings ╚ 26 35 32 37 9999 Two-dimensional Graphing <A> Defining Windows: WINDOW ERASE WINDOWTOP WINDOWBOTTOM WINDOWLEFT WINDOWRIGHT Graphing y = f(x): GRAPH GR DOTGRAPH DTG QUICKG QU SKETCH SK <C> Graphing r = f(Θ) in polar coordinates: POLARG PO <D> Graphing parametric curves x = f(t), y = g(t): PARAMG PA <E> Graphing level curves f(x,y) = c: LEVELC <F> Vector fields and level curves: FIELD TRAJ <G> Drawing lines and rectangles, filling regions: LINE RECT FILL <H> Plotting and marking points: MARK UNMARK PLOT Adding labels to your graphs: WRITE@ VWRITE@ THICKTEXT THINTEXT FONTSIZE <J> Axes and Grids: AXIS NOAXIS AXISCOLOR GRID POLARGRID NOGRID <K> Making successive graphs appear different: KEEPLINE VARYLINE <L> Setting line styles: THICKLINE THINLINE LINESTYLE <M> Setting screen colors: COLOR BKCOLOR GETCOLOR GETBKCOLOR MAXCOLOR BLACK BLUE GREEN CYAN RED MAGENTA BROWN LIGHTGRAY DARKGRAY LIGHTBLUE LIGHTGREEN LIGHTCYAN LIGHTRED LIGHTMAGENTA YELLOW WHITE <N> Bar graphs of list data: BARG or HISTOGRAM <O> Graphing lists of points: LISTG MARK UNMARK PLOT Using crosshairs: CROSSX CROSSY SETCROSS,CRCOLOR <Q> Graphing integrals and derivatives: INTEG DIFFG <R> Using Zoom and UnZoom to change the graphics window ╚ 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 38 9999 Defining Windows Two-dimensional graphs appear in a window which you define with the command: WINDOW(a,b,c,d) where a,b,c,d are real numbers. The graphics window is then a ≤ x ≤ b, c ≤ y ≤ d . For example, if you want to graph the function y = sin(x) in the range 0 ≤ x ≤ 2π and show values -1 ≤ y ≤ 1, execute WINDOW(0,PI,-1,1) before the graphics command <A>. When you create a new window, it is empty except for optional axes . X(PLORE) actually expands the window you specify by 10% to assure that your entire graph will be visible. The ends of the axes, which are not quite at the edge of the window, correspond to your specified window limits. You can erase all the figures in a window without rescaling the window with the command: ERASE The startup window parameters are -1.4 ≤ x ≤ 1.4, -1 ≤ y ≤ 1. The aspect ratio 1.4:1 makes circles appear round, not oval, on most monitors. The command WINDOW(0,0,0,0) restores these parameters. There are functions which return the boundaries of the current window: WINDOWTOP WINDOWBOTTOM WINDOWLEFT WINDOWRIGHT These are useful mostly for programming operations <C>. ╚ 40 48 71 39 9999 Graphing y = f(x) Once you have defined a graphing window <A>, you can view the portion of the graph of y = f(x) which lies within the window by executing the command: GRAPH(f(x),x) f(x) can be either a function with bound variable x or an expression involving x. For example, you could graph the parabola with the command GRAPH(x^2,x). The abbreviation for GRAPH is GR. The operator GRAPH plots one point (x,f(x)) for each pixel on the x-axis, and connects successive points. To omit the connections and just plot the points, use the command: DOTGRAPH(f(x),x) or its abbreviation DTG(f(x),x). To plot only every third point (and connect the points), use QUICKG(f(x),x) or its abbreviation QU(f(x),x). To plot only every 20th point, which is useful for rapidly graphing complicated functions, use SKETCH(f(x),x) or its abbreviation SK(f(x),x). ╚ 39 14 40 9999 Graphing r = f(Θ) in polar coordinates Once you have defined a graphing window <A>, you can graph the portion of a polar curve r = f(Θ) that lies in the window with the command: POLARG(f(t),t) f(t) can be a function with bound variable t or an expression involving t. For example, to graph the cardoid r = 1+cos(Θ), execute the command POLARG(1+cos(t),t). The variable t covers the domain (0, 2π); you can change this default by specifying a range for t: POLARG(f(t),t = a to b). Polar curves are, by default, graphed with 200 line segments. To change that default to any number of line segments up to 1000, add the desired number of segments after the last parameter. Using a large number of segments can make a smoother curve; using a smaller number can draw a faster curve or even a polygon. Try WINDOW(-1.4, 1.4, -1, 1) POLARG(1, T = 0 TO 4*PI, 5) The abbreviation for POLARG is PO. ╚ 39 41 9999 Graphing parametric curves x = f(t), y = g(t) Once you have defined a graphing window <A>, you can graph the portion of a parametric curve x = f(t), y = g(t) for a ≤ t ≤ b that lies in the window with the command: PARAMG(f(t),g(t),t=a to b) f(t) and g(t) can be a functions with bound variable t or expressions involving t. For example, to graph the Lissajous curve x = sin(6t), y = sin(7t), 0 ≤ t ≤ 2π , execute the command PARAMG(SIN(6*t), SIN(7*t),t=0 to 2*PI) Parametric curves are, by default, graphed with 200 line segments. To change that default to any number of line segments up to 1000, add the desired number of segments after the last parameter. Using a large number of segments can make a smoother curve; using a smaller number can draw a faster curve or even a polygon. Try PARAMG(SIN(6*t), SIN(7*t),t=0 to 2*PI,1000) The abbreviation for PARAMG is PA. ╚ 39 42 9999 Graphing level curves f(x,y) = g(x,y) To graph the solution to the equation f(x,y) = 0, use the command: LEVELC(f(x,y)=g(x,y), x = a, y = b) LEVELC(x^2+y^2 = 1, x =.7, y = .7) The graph will start at the solution point nearest (a,b) and extend in two directions. Normally 50 steps are used in each direction. To change the number of steps, add the number you wish at the end of the command: LEVELC(f(x,y) = g(x,y), x = a, y = b, n). If you specify the number of steps, you can optionally specify the step size as a number between -5 and 5 (the default is 0--the scale is logarithmic). Examples: LEVELC(x^2+y^2 = 1, x =.7, y = .7, 100) LEVELC(x^2+y^2 = 1, x =.7, y = .7, 100, 2) LEVELC(x^2+y^2 = 1, x =.7, y = .7, 100, -2) ╚ 43 9999 Vector fields and trajectories To graph the vector field (f(x,y), g(x,y)), issue the command: FIELD(f(x,y), g(x,y), x, y) FIELD(-y/(x^2+y^2), x/(x^2+y^2), x, y) Normally a field of 14 x 14 vectors will be displayed. Larger vectors show as larger on the screen, but the proportions are not necessarily correct. To show more or fewer vectors, add an integer parameter after the first and second variable indicating how many vectors to show in each direction: FIELD(-y/(x^2+y^2), x/(x^2+y^2), x, 10, y, 10) To draw a trajectory following a vector field (f(x,y), g(x,y)), use: TRAJ(f(x,y), g(x,y), x=a, y=b) TRAJ(-y/(x^2+y^2), x/(x^2+y^2), x=0.5, y=0.5) The trajectory will start at (a,b). Usually 50 steps are used; you can optionally specify a different step number and step size as for LEVELC <A>. ╚ 43 44 9999 Drawing lines and rectangles, filling regions You can add lines and rectangles to your graphics screens or fill bounded regions on the screen. To draw a line between (a,b) and (c,d) (assuming that both these points lie in the current graphing window <A>), use the command: LINE(a,b,c,d) To draw a rectangle with opposite corners (a,b) and (c,d), execute: RECT(a,b,c,d) The color and style of the line or rectangle can be determined by COLOR and LINESTYLE <C>. To fill a region, first pick a point (a,b) inside the region. Then pick a color c1 to fill with and a fill pattern p, 0 ≤ p ≤ 11. Use p = 1 for solid fill. Suppose c2 is the color of the boundary of the region (the region must be bounded by curves of one color). To fill the region, execute the command: FILL(a,b,c1,c2,p) ╚ 39 51 50 45 9999 Plotting and marking points If (a,b) is a point in the current graphing window <A>, you can plot a point of the current color at (a,b) with the command: PLOT(a,b) To draw a cross of the current color at (a,b), use the command: MARK(a,b) To delete this cross, use: UNMARK(a,b) Individual points created with PLOT are very hard to see, but numbers of them can be used to shade regions or produce curves. See <C> for information on using these commands with lists. ╚ 39 51 53 46 9999 Adding labels to your graphs You can put alphanumeric labels anywhere on your two-dimensional graphic screens. They can be horizontal or vertical, and they can be printed in various sizes. To print a string s <A> horizontally on the screen with the lower-left corner at Window coordinates (a,b), use the command: WRITE@(a,b,s) To write vertically bottom to top, use VWRITE@(a,b,s) You can change the size of your labels by entering an optional size parameter after the string. The parameter should be an integer in the range 1..7; the default is 2. For example, to label a curve you might enter: WRITE@(0.5,0.5,'Parabola',3) To change the default font size on graphic labels, enter FONTSIZE(n), where n is the desired size. FONTSIZE(0) returns you to the default of 2. this command also affects the size of coordinate labels on graphs. You can use the crosshairs to locate labels. First, while viewing the graphics screen, move the crosshairs to the desired point. Then press control-W (Control-V for vertical labels), enter the font size, and type the desired label. To use thicker characters on the screen (only in size 3 or greater), enter the command THICKTEXT. To return to the default characters, execute THINTEXT. ╚ 37 54 47 9999 Showing and hiding axes, and changing their color When you create a new graphics screen with WINDOW or ERASE <A>, X(PLORE) normally adds axes the screen. You can suppress the display of axes by entering the command: NOAXIS Axes are not displayed on any subsequent screen until you restore them with: AXIS Labeled tics every dx, dy can be added to the axes with nx, ny subdivisions: AXIS(dx, nx, dy, ny) You can change the color of the axes for the next new screen with the command: AXISCOLOR(c) where c is one of the color words or numbers . A rectangular grid with separations dx and dy can be added to two-dimensional graphs with the command: GRID(dx, dy) A polar coordinate grid can be created with: POLARGRID(dr, dΘ) Grids are turned off with the command: NOGRID ╚ 39 51 48 9999 Making successive graphs appear different X(PLORE) allows you to draw as many graphs as you want on the same two-dimensional graphics screen. Normally these graphs are distinguished by being drawn with different line styles <A> (monochrome systems) or colors (color systems). To stop X(PLORE) from changing line styles or colors between graphs, issue the command KEEPLINE To start changing again, issue: VARYLINE If you set the color for a graph with the command COLOR , the next graph will always be drawn in that color. Subsequent graphs will have different colors or the same color depending on whether KEEPLINE or VARYLINE is active. Lines drawn with LINE <C> and LISTG <D> are always drawn with the current line style and do not change the style, even if VARYLINE is active. Only graphs drawn with GRAPH <E>, POLARG <F> and PARAMG <G> are affected. ╚ 50 51 45 53 40 41 42 49 9999 Setting line styles On both monochrome and color systems, X(PLORE) will draw lines and graphs with different line styles. (Since the line segments used to draw graphs are usually very short, different line styles may not be distinguished in graphs, but they will be distinguished on long lines.) Linestyles are indicated by integers in the range 0..3, and are set by the command: LINESTYLE(n) See <A> for information on varying line styles with graphs. Lines and graphs can be either thin (the default) or thick. You set this default with the commands: THICKLINE THINLINE Once set, the thickness stays the same until changed. ╚ 49 50 9999 Setting screen colors If you are using a color system, you can set the color of the next line or graph with the command: COLOR(c) where c is one of the color words: BLACK BLUE GREEN CYAN RED MAGENTA BROWN LIGHTGRAY DARKGRAY LIGHTBLUE LIGHTGREEN LIGHTCYAN LIGHTRED LIGHTMAGENTA YELLOW WHITE or an integer in the range 0..15. The color of successive graphs may vary unless you hold it constant <A>. You can set the background color for your graphs with the command BKCOLOR(c) You can determine the current color or background color with the functions: GETCOLOR GETBKCOLOR The highest color number available on your system is MAXCOLOR. To erase a curve or line, redraw it in the background color, using linestyle 0 . Color commands also apply to three-dimensional graphs. <C> ╚ 49 50 62 51 9999 Bar Graphs of List Data If x is a list <A> of data, you can plot a bar graph of the data with the command: BARG(x,a,b,n) or HISTOGRAM(x,a,b,n) X(PLORE) divides the interval (a,b) into n subintervals, computes the number of elements in x contained in each subinterval, and plots a bar graph of these numbers. Outliers are included in the lowest and highest categories respectively. BARG is also a function; it returns as its value a list of the frequencies or numbers of elements contained in each subinterval. BARG makes its own graphics window, ignoring the current window as defined by WINDOW <C>. ╚ 34 35 39 52 9999 Graphing lists of points If x and y are lists of real numbers, then the command: LISTG(x,y) draws line segments from the point (x[i],y[i]) to the point (x[i+1],y[i+1]) for i = 1..length(x)-1. If x and y have different lengths, the shorter length is used. If one end of a line is outside the current window, the line will not be drawn. You can plot these points instead of connecting them with the command: PLOT(x,y) and you can mark them with crosses with: MARK(x,y) and erase the marks with: UNMARK(x,y). See for applying these commands to pairs of real numbers. ╚ 54 45 53 9999 Using Crosshairs When you are viewing the two-dimensional graphics screen, you will see a circle somewhere on the screen. On a new screen it appears in the upper-left- hand corner. This is the crosshairs. The coordinates of the center of the crosshairs are printed at the lower-right hand corner of the screen. You can move the crosshairs with the cursor (arrow) keys. Try it, and see how the coordinates change. You can effect finer movements with the shifted-cursor keys, or the number keys 2,4,6,8. The crosshairs are useful for locating the intersections of curves or points where graphs cross the axes. The functions CROSSX and CROSSY return the last x- and y-coordinates of the crosshairs. Suppose you have graphed a function y = f(x), and you want to solve the equation f(x) = 0. Using the crosshairs, you can find the point where the graph of f(x) crosses the x-axis. The x-coordinate of this point is the solution to f(x) = 0. Return to the Input Window, and enter the command <A>: SOLVE(f(x) = 0, x = CROSSX) That is, after you move the crosshairs to the x-intercept of the graph, you can use CROSSX as an accurate approximation to the desired root. The command SETCROSS(x,y) moves the crosshairs to the window point (x,y). This is useful for pointing out something on the screen to the user of a procedure. CRCOLOR(colorword) sets the color of the crosshairs. ╚ 64 54 9999 Graphing integrals and derivatives You can illustrate the processes of integration and differentiation while computing integrals and derivatives. Suppose you wanted to compute the integral of sin(x) between x = 0 and x = π . You could use the command IN(SIN(x), x = 0 to PI) <A>, but you can watch the integral being calculated by using the command: INTEG(SIN(x), x = 0 to PI) X(PLORE) will draw the graph of SIN(x) on the screen and fill in the area being integrated (if it is inside your current graphing window ) as well as computing the integral. Actually, you will see X(PLORE) draw a sequence of vertical lines. X(PLORE) draws a line each time it evaluates the function being integrated. Watching the lines will show you how X(PLORE) computes its integrals. Similarly, you could computed the derivative of a function f(x) at x = a with the command DIF(f(x), x = a) <C>, but if you use the command: DIFFG(f(x),x = a) X(PLORE) will draw the graph of f(x) and the tangent line at x = a as well as computing the derivative. ╚ 69 39 68 55 9999 Using Zoom and UnZoom to change the graphics window Once you have displayed a two-dimensional graph on your screen, you can outline and magnify part of it, or increase the size of the graphics window to seem more of it. Move the crosshairs <A> to one corner of the region to be magnified. Press Control-Z. The crosshairs will change to Z. Move the Z (with the cursor keys) to the opposite corner of the region to be magnified. A box will outline the designated region. Press <ENTER>. The region you have outlined will be expanded to fill the screen. To cancel Zoom any time after starting, just press <ESC>. If you have already started Zooming, you can restart at the location of the Z by pressing Control-Z again. To undo the effect of zooming, press Control-U while the graph is visible to return to the previous graph. If you press Control-U without a graphic to UnZoom to, then the current screen will be doubled in size and the graphs redrawn. Only graphs drawn with GRAPH, DOTGRAPH, QUICKG, SKETCH, POLARG, PARAMG, DIFFG, INTEG, FIELD, LEVELC and TRAJ can be Zoomed and UnZoomed. Written labels , lines drawn with LINE <C>, and area fills <D> will not be repeated. ╚ 54 47 45 56 9999 Three-dimensional graphing <A> Graphing z = f(x,y) GRAPH3D GR3 Graphing parametric surfaces x = f(s,t), y = g(s,t), z = h(s,t) PARAMG3D PA3 <C> Graphing parametric curves x = f(t), y = g(t), z = h(t) CURVE3D <D> Graphing matrix data MATRIXG <E> Using color with three-dimensional graphs COLOR BKCOLOR AXISCOLOR GETCOLOR GETBKCOLOR After you have displayed a three-dimensional graph on the screen, you can rotate it in space to view it from different directions by pressing the keys x X y Y z Z. Multiple keypresses effect larger rotations. You can also change the way the figure is shaded by pressing the keys 1 2 3 4. ╚ 58 59 60 61 62 57 9999 Graphing z = f(x,y) If f(x,y) is a function or expression in two variables, its graph is a surface. You can sketch the portion of the graph over the rectangle a ≤ x ≤ b, c ≤ y ≤ d with the command: GRAPH3D(f(x,y), x = a to b, y = c to d) Each side of the rectangle is divided in to 15 segments to make the drawing, and the range shown on the vertical axis is sufficient to show all points of the graph. You can control the number of subdivisions on each side of the domain rectangle, and you can control the size of the vertical axis. To specify the number of subdivisions for one of the horizontal axes, place the number after the upper limit for that axis. To specify the vertical axis, place the lower and upper limits after the specifications for the second horizontal axis. IF YOU ADD SOME BUT NOT ALL OF THESE SPECIFICATIONS, YOU MUST SEPARATE THE PARTS OF THE GRAPHING COMMAND WITH SEMICOLONS. Here are some sample graphing commands: GRAPH3D(x^2 - y^2, x = a to b step 10, y = c to d step 20, -2 to 2) GRAPH3D(x^2 - y^2, x = a to b; y = c to d; -2 to 2) GRAPH3D(x^2 - y^2, x = a to b step 10; y = c to d step 20) The abbreviation of GRAPH3D is GR3. ╚ 58 9999 Graphing parametric surfaces x = f(s,t), y = g(s,t), z = h(s,t) Parametric surfaces are graphed much like ordinary surfaces <A>. You must specify the range for each parameter. You can optionally specify the number of steps to use within each parameter (the default is 15) and the visible range of the x-, y- and z-axes. A sample minimal command for graphing a sphere is: PARAMG3D(SIN(S)COS(T),SIN(S)SIN(T),COS(S), S = 0 to PI, T = 0 to 2*PI) If you use some but not all of the options, you must separate the parts of the command with semicolons. If you specify the range of one axis, you must specify all the axes. PARAMG3D(SIN(S)COS(T),SIN(S)SIN(T),COS(S), S = 0 to PI step 30, T = 0 to 2*PI step 30, -2,2,-2,2,-2,2) PARAMG3D(SIN(S)COS(T),SIN(S)SIN(T),COS(S), S = 0 to PI; T = 0 to 2*PI; -2,2,-2,2,-2,2) PARAMG3D(SIN(S)COS(T),SIN(S)SIN(T),COS(S), S = 0 to PI step 30; T = 0 to 2*PI step 30) When you enter commands this long, just keep typing and let them spill over onto a second or third screen line. Don't push <ENTER> when you come to the end of your first line; that will execute your partially completed command. The abbreviation for PARAMG3D is PA3. ╚ 58 59 9999 Graphing parametric curves x = f(t), y = g(t), z = h(t) Parametric curves are graphed just like parametric surfaces <A>, but only one parameter need be specified. To graph the helix, for example, enter: CURVE3D(COS(T),SIN(T),T, T = 0 to 4*PI) You an optionally specify the number of segments to use for the parameter (the default is 100) and the size of the x-,y- and z-axes. You must specify all axes or none, and if you don't specify all options you must separate the parts of the command with semicolons. Here are some examples: CURVE3D(COS(T),SIN(T),T, T = 0 to 4*PI step 20, -2,2, -2,2, 0,2) CURVE3D(COS(T),SIN(T),T, T = 0 to 4*PI; -2,2, -2,2, 0,2) CURVE3D(COS(T),SIN(T),T, T = 0 to 4*PI step 20) ╚ 59 60 9999 Graphing matrix data If M is a matrix, you can graph a surface whose heights are the entries of the matrix with the command: MATRIXG(M) The size of the vertical axis is determined by the entries in the matrix You can change this default and specify the size of the vertical axis yourself by adding the lower and upper limits after the name of the matrix: MATRIXG(M,-3,4) ╚ 28 61 9999 Using color with three-dimensional graphs You can select the color of a three-dimensional graph by executing COLOR(c) before drawing the graph. The parameter c can be any of the color words <A>. You can also select the color of the axes with AXISCOLOR(c) or the background color with BKCOLOR(c) Three dimensional surfaces are actually drawn with two colors representing the two sides of the surface; the colors used are the one you selected and either its brighter or dimmer version, for example YELLOW and BROWN. The functions GETCOLOR and GETBKCOLOR return the current color values. ╚ 51 62 9999 Last Graph When you leave a graphics screen by pressing <ENTER> or <ESC>, your graph is saved in memory. (Actually, data that permits reconstruction of the graph is saved.) You can see your last graph again by pressing F9, even if you have already told X(PLORE) to erase it by executing ERASE or WINDOW <A>. After redisplaying the graph, you can manipulate it by moving the crosshairs (two-dimensional) or rotating it (three-dimensional). If you need more memory, you can save the last graph data on disk instead of in memory by pressing shift-F9. Pressing shift-F9 again returns the last graph data to memory. If X(PLORE) senses that it is running short of memory, it will automatically shift the last graph data to disk. If you are saving your last graph on disk, the file of last graph data will normally be located in the same directory as X(PLORE) (even if you called X(PLORE) from a remote directory). You can use the configuration file to change the location of the last graph file. You can stop saving the last graph, for example if you are drawing an animation, with the command GSAVEOFF. Start saving again with GSAVEON ╚ 39 19 63 9999 Solving equations To solve the equation f(x) = g(x) , you must choose an estimate a for the solution and enter the command: SOLVE(f(x) = g(x), x = a) The solution is returned in the variable x . SOLVE is not a function; it is a procedure that changes the variable x. For example, to solve Cos(x) = x , enter SOLVE(COS(x) = x, x = 1); You can solve systems of equations too, provided there are the same number of variable as equations. For example: SOLVE(x^3 - y^2 = 3, x^2 + y^3 = 0, x = 2, y = -2) The solutions show up in the variables x and y . You can speed up equation solving, at the expense of accuracy, by substituting FSOLVE for SOLVE. See also: <A> Implicit functions IMPLICIT Controlling speed and accuracy of equation solving SOLVETOL GETSOLVETOL ╚ 65 66 64 9999 Implicit functions X(PLORE) allows you to define functions implicitly. For example, the equation Cos(x) = t*x defines a function of t; for each value of t you get a different solution. You can define this function in X(PLORE) with the command: f(t) = IMPLICIT(COS(x) = t*x, x = 0.5) The starting estimate x = 0.5 works well for t ≥ 0. IMPLICIT also works with systems of equations <A>, returning a list of values. You can make implicit functions evaluate faster by changing the parameter SolveTolerance . ╚ 64 66 65 9999 Controlling speed and accuracy of equation solving Equation solving is controlled by a constant called SolveTolerance. If you are solving an equation f(x) = g(x) , X(PLORE) stops when it finds x such that │f(x) - g(x)│ < SolveTolerance . (If no such x can be found, but if X(PLORE) can find x such that │f(x) - g(x)│ < 100*SolveTolerance, X(PLORE) will accept this solution.) You can change the value of SolveTolerance with the command: SOLVETOL(t) t must be a positive real number. The larger the value of t , the faster your solutions will converge but the less accurate they may be. The default value of SolveTolerance is 0.0000001. You can restore SolveTolerance to this value with the command SETSOLVETOL(0). The function: GETSOLVETOL returns the current value of this SolveTolerance. ╚ 66 9999 Creating tables of function values If f(x) is a function or expression using x, you can display a table of functional values: x f(x) 0.0 ... 0.1 ... 0.2 ... ........... 1.0 ... with the command: TABLE(f(x),x=0 to 1 step 0.1) Of course you can use any values you like for the lower and upper limits and step value of the parameter. The step must be specified. You can tabulate one or two functions. You can also enter, for example, TABLE(SIN(x), COS(x), x = 0 to PI/2 STEP PI/20) ╚ 67 9999 Differentiation If f(x) is a function or expression using x, then you can differentiate it at the point x = a with the command: DIF(f(x), x = a) Differentiation is symbolic, not approximate. You can see the symbolic derivative of f(x) (provided x is an undefined variable) with the command: DIF(f(x), x) Higher derivatives can be computed by putting the order of the derivative at the end of the command: DIF(f(x),x = a, 3) To compute the third symbolic derivative, you must somehow tell X(PLORE) that the 3 is not the value of x but the order of differentiation. You do this by placing it after a comma: DIF(f(x),x,3) To avoid ambiguities, you must use '=' in the DIF statement when assigning a value to the variable of differentiation and a comma before the order. See <A> for graphing derivatives with DIFFG. ╚ 55 68 9999 Integration X(PLORE) will compute the numerical approximation to the integral of f(x) for a ≤ x ≤ b: b ⌠ │ f(x) dx ⌡ a with the command: IN(f(x),x = a to b) For example: IN(EXP(x), x = 1 to 2) You can compute integrals faster, if less accurately, by substituting FIN for IN. See <A> for information on using INERROR INTOL INDEPTH GETINTOL and GETINDEPTH to control the speed and accuracy of integration. For graphing integrals with INTEG, see . ╚ 70 75 69 9999 Controlling the speed and accuracy of integration X(PLORE) uses an adaptive Gauss-Kronrod algorithm for computing integrals. You can get a very conservative estimate of the error in an integral with the command INERROR which returns an error estimate for the last integral. The algorithm is controlled by two constants, InTolerance and Depth. The current values of these algorithms are returned by the functions: GETINTOL GETINDEPTH You can reset these constants with the commands: INTOL(t) INDEPTH(n) t must be a positive real number, and n a positive integer. Using 0 for either value will reset the corresponding constant to its default value, which are 0.00001 and 7 respectively (0.0000001 and 9 for X(PLORE)87). The smaller the value of InTolerance and the larger the value of depth, the slower and more accurate will be the computation of integrals. ╚ 70 9999 Subroutines Subroutines are entered in the Subroutine Window. To see the Subroutine Window, press function key F10. Press F10 again to return to the Input Window. There are two kinds of subroutines, procedures and functions. Functions can return a value, procedures cannot. Both consist of a header line which may include parameters, followed by a series of ordinary X(PLORE) entries and control structures, followed by the word END on its own line. Subroutine names can be up to 12 characters long and must begin with a letter. They cannot duplicate the name of any of X(PLORE)'s built-in operators or any of your variables or functions. Subroutines CANNOT use variables and functions from the Input Window. To return a value from a subroutine function, place the command RETURN(v) on one line of your subroutine, where v is the value to be returned. See also <A> Subroutine headers and parameters PROCEDURE FUNCTION % Displaying messages and accepting user input WRITE INPUT INPUTS PAUSE WAIT BEEP <C> Decision and looping operators IF..ELSE FOR WHILE REPEAT..UNTIL EXIT <D> Programming and graphics GRAPHICS TEXT INPUT@ INPUTS@ <E> Examples ╚ 82 75 76 77 72 71 9999 Example 1 -- Taking a test PROCEDURE Test(n) // ask n addition questions, accept no wrong answers FOR i = 1 to n a = RANDOM(9) b = RANDOM(9) REPEAT INPUT(STR(a)|' + '|STR(b)|' =',x) UNTIL x = a+b END END How to use: in the Input Window, enter: Test(5) To see another example, push <A>. ╚ 73 72 9999 Example 2 -- Random Walk PROCEDURE RandomWalk // Depict a two-dimensional random walk WINDOW(-4.2,4.2,-3,3) GRAPHICS x=0 y=0 WHILE x > WINDOWLEFT AND x < WINDOWRIGHT AND y > WINDOWBOTTOM AND y < WINDOWTOP t = RANDOM*2*PI x1 = x+COS(t) y1 = y+SIN(t) LINE(x,y,x1,y1) x = x1 y = y1 END TEXT END To use this procedure, enter the command: RandomWalk. To see another example, push <A>. ╚ 74 73 9999 Example 3 -- Displaying Newton's Method FUNCTION Newton(%f,x) // Solving f(t) = 0 with Newton's method starting with t = x, illustrated GRAPHICS GRAPH(f(t), t) REPEAT x1 = x - f(x)/dif(f(t),t=x) LINE(x,f(x),x1,0) LINE(x1,0,x1,f(x1)) x = x1 INPUTS('x = '|STR(x)|'. Do another step (Y/N)?',c) UNTIL UPCASE(c[1]) == 'N' RETURN(x) TEXT END To use this subroutine, enter the command: Newton(x) where x is a number between -1.5 and 1.5. ╚ 74 9999 Displaying messages and accepting input Subroutines can write messages on the screen with the command WRITE, for example: WRITE('The final answers are',x,'and',y) Multiple values, including strings, separated by commas, can appear in a single WRITE statement. When WRITE is executed, the subroutine pauses until the user presses a key. Subroutines can also pause to accept input if you use one of the commands INPUT or INPUTS. INPUT asks for a numerical value; INPUTS for a string. Some sample commands are: INPUT('Enter initial value',x) INPUTS('Do you want to continue (Y/N)',c) Each INPUT command has two parts: a message to be displayed and a variable to hold the users input. You can make your subroutine issue an audible sound with the command BEEP on its own line. ╚ 75 9999 Decision and looping operators X(PLORE) supports all the standard looping and conditional statements found in modern programming languages except CASE. Here are some examples: IF x < 3 OR y > 2 WHILE ABS(x) < 1 REPEAT z = z+3 x = f(x) oldx = x w = x*y END x = x-f(x)/g(x) ELSE IF x > 3 UNTIL ABS(x) < 0.01 z = 5 ELSE w = 4 END FOR i = 1 TO 20 FOR x = 0.05 TO 1 STEP 0.05 s = s + f(i/20) s = s + f(x) END END All conditional constructs, except REPEAT, must be terminated by END. The term "then" is not needed after IF. The command EXIT, on its own line, causes the immediate termination of the current subroutine. ╚ 76 9999 Programming and graphics Sometimes you will want to execute a sequence of graphing commands in a subroutine, but X(PLORE) normally stops after each one and waits for the user to press <ENTER> or <ESC>. You can prevent X(PLORE) from stopping by prefacing your list of graphics commands with the command GRAPHICS. All commands can be executed while GRAPHICS is in force, but X(PLORE) will not pause after each graphing command. The effect of GRAPHICS can be turned off with the command TEXT. If you want to stop in the middle of a sequence of graphics commands while GRAPHICS is in force, use the command PAUSE. X(PLORE) will pause until the user presses <ENTER> or <ESC>. While paused, the user can move the crosshairs but not zoom a graph. You can also stop your subroutines for n seconds with the command WAIT(n). While GRAPHICS is in effect, you may wish to require the user to input a value. The usual INPUT command could be used, but its window would obscure part of the graphics screen. Alternatively, you may use the command: INPUT@(x,y,n,v) X(PLORE) will place a box sufficient to hold n characters at location (x,y) on the screen, and wait for you to type a value into the box. The value entered will be placed in the variable v. To input a string instead of a value, use INPUTS@. ╚ 77 9999 Using external files X(PLORE) will save files of commands or data for reloading later. To save all your current formulas and subroutines, press function key F7 and enter the name of the file you want to save. The file name can include path information. To reload a saved file, press F8 and enter the name of the file. See also <A>. To save data, use the command SAVE(data,filename). The filename must be a string value or surrounded by apostrophes. The data can be either a variable or an expression. Use FSAVE to force overwriting an existing file. To reload the saved data into a variable, use the command LOAD(var-name, filename). You can also load any ASCII file into a list of strings with the command LOADTEXT(var-name, filename). To save a graph, either press F7 while the graph is visible, or use the command SAVEGRAPH(filename) to save the last graph. Use FSAVEGRAPH to force overwriting an existing file. To load a graph, which will appear on the screen and become your last graph , use the command LOADGRAPH(filename). To see a list of all the files in a directory, press Shift-F8 and enter the name of the directory. This name can include a mask. After viewing the directory, you will be asked if you wish to make it your current directory. You can also change your current directory by pressing Control-F8 and entering the name of the new current directory. ╚ 9 63 78 9999 Printing X(PLORE) can print text to any printer, and it can print special characters and graphs on printers than can emulate the Epson, IBM or LaserJet printers. All printer operations are entered via the function key F2. After pressing this key, CC will ask you to select your printer type if you have not already done so, and then present you with a menu of choices: 1. Start printing a record of all input and results 2. Stop printing 3. Print all formulas from input window, all subroutines and all variables 4. Print all formulas from input window 5. Print all subroutines 6. Print all variables 7. Skip to top of page 8. Save input window, subroutines and variables to disk 9. Change printer type Option 1 prints all of your entries and results as you enter them. Option 2 cancels option 1. Options 3-6 print all current values. Option 8 prints to an ASCII disk file instead of your printer. See <A> for information about PRINT, PRINTGRAPH and NEWPAGE. ╚ 80 79 9999 Printing variables and graphs, Ejecting pages You can print a graph either by pressing F2 while the graph is visible or by entering the command PRINTGRAPH to print your last graph. You can print a variable or any other expression, useful for lists or matrices, with the command PRINT(data). For example: m = MATRIX(1/(i+j-1),i=1,4,j=1,4) PRINT(m) To force the printer to eject the current page and start a new page, use the command: NEWPAGE ╚ 80 9999 Disk and Memory Management X(PLORE) is a large program. It will run in 512K of memory, but larger problems require 640K. X(PLORE) uses overlays to reduce the amount of code that must be in memory at any time, so it runs best off a hard disk. If you must run X(PLORE) from a floppy disk, you must use a disk of at least 720K capacity. You should make as much memory as possible available before running X(PLORE). Delete all but the most necessary TSR programs before starting X(PLORE). You can free up memory while X(PLORE) is running by using FORGET to delete unnecessary variables <A> and Shift-F9 to transfer your Last Graph information to disk . ╚ 13 63 81 9999 Subroutine Headers and Parameters Each subroutine must begin with one of the lines: PROCEDURE procedure-name(parameter-list) FUNCTION function-name(parameter-list) The procedure-name or function-name can be up to 12 characters long beginning with a letter. It cannot duplicate the name of any of X(PLORE)'s built-in functions or the name of any of your variables or functions from the Input Window. The parameter list is a list of variable names <A>. When you call the subroutine, you specify initial values for these variables, which can then be used in the subroutine. You an also pass a function definition to a subroutine by passing its name as a parameter. You identify a function-name parameter in a subroutine header by preceding it with %. See for an example. ╚ 13 74 82 9999 Algebraic Simplification X(PLORE) is capable of minimal algebraic simplification, which is intended mostly for putting X(PLORE)'s symbolic derivatives into reasonable form. If you evaluate a formula with undefined variables, X(PLORE) will simplify the formula as much as it can. X(PLORE) usually will leave powers of sums, for example (x+3)^5, unexpanded. To change this and force X(PLORE) to expand these binomial expressions, enter the command: EXPANDON To return X(PLORE) to its default mode of simplifying expressions, enter: EXPANDOFF ╚ 13 74 83 9999