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Mercury 2.00 manual =================== (C) Copyright 1990 Roger Schlafly. All Rights Reserved. Portions of the software (C) Copyright 1987-88 Borland International. All Rights Reserved. Contents -------- 1. Introduction Features History Shareware policy Disclaimer Files Hardware Requirements Installation 2. Problems Editing Numbers Operators Constants 3. Solving Solutions Graphs Reports 4. Feature Reference Functions Statements Commands 1. Introduction Introduction ------------ Mercury is a program for solving equations on the IBM PC. It is easy to use, interactive, and powerful. With it, you can * evaluate mathematical expressions. * solve for the roots of an equation. * solve a system of equations. * maximize or minimize a function, with or without constraints. * evaluate derivatives and definite integrals. * plot one or more functions. * print a report or a graph. Mercury has a built-in editor, pull-down menus, online help, and all the conveniences necessary to make it accessible to computer novices. For a quick demonstration of Mercury's capabilities, run MERCURY and then load and solve some of the sample problems (*.EKA). If you get stuck, just press the F1 key for help. Mercury is an outgrowth of Eureka, a product of Borland International of Scotts Valley, California. When Borland decided to concentrate on spreadsheet and database products, Eureka was spun off. Mercury was developed from that code base, and is being released in this form to satisfy customer requests for new versions of Eureka. It is labelled version 2.0 because it is similar to what version 2.0 of Eureka would have been. Eureka was also available on the Apple Macintosh. No version of Mercury for the Macintosh is planned at this time. Mercury is available as Shareware. Shareware is a name for programs which are distributed freely or at nominal charge, but which are copyrighted and have restrictions on their usage. Mercury may be copied freely for examination purposes. Distributors may charge a nominal fee, provided that they adhere to the conditions in VENDOR.DOC. Commercial and institutional use is prohibited without a registered license. Such a license may be obtained by sending $49 to Real Software PO Box 1680 Soquel, CA 95073 USA phone or fax: 408-476-3550 CompuServe: 76646,323 Payment may be by check, MasterCard, or Visa. See REGISTER.DOC for an order form. Those who register a license will get the latest version of the software, an installation program, and a printed manual. They will be eligible for technical support, and will be on the mailing list for future information and updates. Site licenses and other special arrangements are available. Comments, bug reports, and suggestions are especially welcome. Disclaimer ---------- The author has made his best effort in producing this software, but hereby disclaims all warranties. He will not be liable for any damages exceeding the price paid for the license to use the software. The person using the software bears all risk as to the quality and performance of the software. Mercury is not a Borland product. Borland will not provide any technical support. Eureka is a registered trademark of Borland International. Borland copyrights and trademarks are used with permission. Mercury is a trademark of the author. Files ----- Mercury consists of these files. MERCURY.EXE main program MERCURY.HLP help file *.BGI graphics drivers *.CHR fonts *.EKA sample problems MANUAL user manual (this file) README quick instructions REGISTER.DOC registration form VENDOR.DOC vendor information Auxilliary files: *.EKA more sample problems *.BGI more graphics drivers *.BIN external functions *.C source to external functions MATHL.H tools for making external functions *.OBJ tools for making external functions *.LIB tools for making external functions If you downloaded Mercury from an online service, you may find the auxilliary files archived separately. The extra files are not needed by most people, and are stored separately to save on connect time. The extra drivers are for uncommon hardware only, and the tools for making external functions require Borland's Turbo C++. Registered customers also get INST.COM, an installation program for configurable options. Hardware Requirements --------------------- Mercury requires an IBM PC or clone with at least 640K of RAM. About 500K of free RAM is needed. In addition, extended memory (EMS memory) is used if present. If a math coprocessor is available, it will be used. A math coprocessor is recommended for serious number crunchers. Plots require a graphics monitor and a CGA, EGA, VGA, or Hercules card. You can print reports and graphs on any of the popular printers. Select Printer and Model from the Options menu for the list of supported printers. Installation ------------ There are no special installation requirements. The simplest procedure is to make a new directory on your hard disk, and to copy all of the files there. You may wish to put the Mercury directory on your path, or to copy some of the files to your RAM disk. The program INST may be used to change the edit key bindings, and to make a few other installation choices. Various other configuration options are available from within Mercury, and these may be saved to disk. See Options, below. Mercury tries to detect automatically what hardware is present, but there some situations where you may want to override the detection logic. If you have a CGA monitor, it may or may not have the "snow" problem. If so, you may notice random snow on the screen as characters are written there. If so, you can eliminate it by typing set CGASNOWCHK=Y at the DOS prompt before loading Mercury. There must not be any spaces surrounding the equals sign. Mercury automatically detects the math coprocessor, but occasionally Mercury fails because the chip is defective. It can be turned off with set 87=N at the DOS prompt. Mercury also detects various graphics cards, but some cannot be detected. See the SCREEN statement in Chapter 4. If you have a graphics printer, you will want to select it at the Printer menu. The graphics card and printer selection can be saved to the MERCURY.CFG file, and then it will be automatic each time Mercury is loaded. Errors ------ Error messages are pretty much self-explanatory. If an error is detected while parsing the problem file, Mercury puts you in the Edit window with the cursor at the point of the error. The error message at the top of the window, and disappears as soon as you start editing. Other errors are displayed in a special box which is placed on the screen. After you acknowledge the error by pressing the Escape key, you are put back in the menu system. 2. Problems A problem is entered into Mercury by either loading an existing one from the disk, or by creating a new one in the Edit window. This chapter explains how to use the editor, and how to create a problem for solving. Editing ------- Mercury includes a complete text editor for creating and modifying problem files. Usage of the editor is similar to WordStar in nondocument mode, or to Borland's SideKick. The problem files are ordinary ascii files, and can be edited with any other text 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, a default you can change with the INST program. 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.) No Word-Wrap Unlike a more standard word processing program, the editor does not have word-wrap. You must press Enter at the end of each line. AutoIndent 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. Zoom 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. Here is a summary of the editing commands. Many of these commands can be changed with the INST installation program. Cursor Movement Commands: Character left Ctrl-S or Left arrow Character right Ctrl-D or Right arrow Word left Ctrl-A Word right Ctrl-F 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 Page down Ctrl-C Left of line Ctrl-QS or Home Right 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-PgUp Bottom of file Ctrl-QC or Ctrl-PgDn Move to block begin Ctrl-QB Move to block end Ctrl-QK Move to previous point Ctrl-QP Move to marker 0 Ctrl-Q0 Move to marker 1 Ctrl-Q1 Move to marker 2 Ctrl-Q2 Move to 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 character left Ctrl-H or Backspace Delete character Ctrl-G or Del Delete word right Ctrl-T Block commands: Set block begin Ctrl-KB or F7 Set block end Ctrl-KK or F8 Mark word Ctrl-KT Hide block Ctrl-KH Copy block Ctrl-KC Move block Ctrl-KV Delete block Ctrl-KY Read block Ctrl-KR Write block Ctrl-KW Print block Ctrl-KP Miscellaneous Commands: Quit edit, no save Ctrl-KD, Ctrl-KQ, or Esc Save and edit Ctrl-KS or F2 New file F3 Tab Ctrl-I or Tab Tab mode toggle Ctrl-OT Auto indent on/off Ctrl-OI Restore line Ctrl-QL Find Ctrl-QF Find & replace Ctrl-QA Repeat last find Ctrl-L Control key prefix Ctrl-P Abort operation Ctrl-U Restore error message Ctrl-QW Search Options: B Backwards search G Global search n n = any number. Search n times N Replace without asking U Ignore upper/lower case W Search for whole words only S Soundex Problems -------- A problem is a text (ascii) file consisting of comments, definitions, equations, and statements. See the sample files for examples of problems. Comments are not read by the Mercury solver, and serve merely for clarity or documentation. A comment starts with a semicolon (;) and runs until the end of the line. Alternatively, comments can be enclosed in curly braces ({}) and may occupy several lines. Definitions may be either variable initializations or function definitions and are characterized by the := assignment operator. If the left-hand-side is a name, it is taken to be a variable and the right-hand-side is an expression which evaluates to the initial value for that variable. Such initial values can be critical to the iterative solver. If the left-hand-side is a function, as in Square(x) := x^2, then the right-hand-side serves as the definition for that function. In this case, Square is then a function of one argument, and Square(5) = 25. Equations have a mathematical expression on each side of an equals sign (=). An equation may also be an inequality constraint, in which case the equals sign is replaced by <, <=, >, or >=. Statements are special directives to Mercury. They are instructions either to do something other than solve the equations, or to control the solution process. The allowed statements are listed below, and examples are in the problem files. Numbers ------- The range of numbers is plus or minus 1e-307 to 1e+307. When using the exact solver, a question mark indicates that the next digit is uncertain. For example, 1.23? indicates a number that might be anything between 1.225 and 1.235, as these are the numbers which round to 1.23. Imaginary numbers are indicated with the suffix i. For example, x = 2 + 3i is the complex number with real part 2 and imaginary part 3. Percentages may be expressed with the suffix %. For example, 5.1% is the same as .051. Examples -------- A number of example problem files (*.EKA) are on the program disk. Examination of these is strongly recommended, as these demonstrate most of Mercury's features. See Chapter 5. 3. Solving Solving ------- Mercury features three solver technologies. 1. Direct solver. This method consists of applying the rules of high school algebra to manipulate the equations until solutions are obtained for all the variables. For example, 2 x + 3 y = 8 4 x + 5 y = 14 can be solved directly by rearranging the first equation to y = (8 - 2 x)/3 and substituting for y in the second equation. See the LEVEL statement for user control over what manipulations are allowed. 2. Iterative solver. The most difficult systems of equations cannot be solved analytically. The best we can do is find some approximate solution and then apply some iterative procedure to improve the accuracy. In such case it is advisable to provide starting values for the variables. 3. Exact solver. Mercury's most innovative feature is its ability to use exact methods in many cases. Solution Window --------------- When you apply Solve to a problem file, the result is in the Solution Window. The window is editable like the Edit window, and can be scrolled and zoomed for easy viewed. The contents can be saved in a report, if that is desired. The solution lists all of the variables, with their final values. Usually, the value is just a number, displayed to however many digits is specified in the DIGITS statements or menu options. In some cases, a variable was solved directly and symbolically in terms of other variables. If such an expression was used to calculate the value, then the expression is shown. If no value was ever assigned to the variable, then its value is given as "UNDEFINED" or "NO SOLUTION". This occurs if the variable is used only as a dummy variable, such as in a function definition or as a variable of integration. Occasionally, a variable value will be given as NAN, which means Not-A-Number. This means the variable could not be computed accurately, and only crude bounds were obtained. For example, a result "x = NAN(2.1,7.5)" indicates that x may be anything between 2.1 and 7.5. Sometimes such imprecise results represent the best that can be said about an ill-posed problem, but sometimes it represents a failure on the part of Mercury. Such results should be used with caution. When a variable appears to have the value of a simple fraction, it is noted in the solution. For example, solving 3 x = 1 gives x = 0.333333333333333 { 1 / 3 }. The fraction is reduced, meaning that the numerator and denominator have no nontrivial factors. The logic which recognizes these fractions is particularly effective with the Exact solver. In some cases, multiple roots for some variable are found. These are listed after the list of variable values. If the problem file has a LIST statement, then the list of the abscissae and ordinate appear in the solution window. The solution always has a list of residuals. These are extremely important for assessing the validity and accuracy of a solution. A residual is a quantity that should be zero if the equation were solved perfectly. It is the absolute value of the difference between the left and right hand sides of the equation. For example, solving SIN(x) = .5 might give as a residual { 1E-16 } SIN(x)-.5=0 The number in braces, 1E-16, is the residual. It is the absolute value of SIN(x)-.5 using the solved value of x. Residuals of this size are quite common, as internally Mercury calculates to 16-digit precision, and there is no machine representable x for which SIN(x) is exactly 0.5. The equation to the right of the braces is often the same as one of the input equations, but sometimes Mercury rearranges some terms. It might be a simple rearrangement as above, but it may also the result of symbolically solving for one of variables and substituting it into other equations. In any case, the equations shown with the residuals are the actual equations used by the solver. Some control of the symbolic substitutions is allowed by the LEVEL statement or the Level option under the Solver menu. Occasionally, you may see variables name T1, T2, ... These are temporary variables introduced by Mercury in the course of solving. Most commonly, they are used to enforce inequality constraints. The constraint EXP(x + y) + y * z <= 10 may be replaced with EXP(x + y) + y * z + T1 = 10 T1 >= 0 The latter has some technical advantages, even though there is an extra equation and an extra variable. Values for the extra variable are not shown. Finally, the solution ends with the largest residual, the method, and the processor. The largest residual is simply the maximum of the previous residuals. Assuming your problem is not scaled too badly, then anything less than about 1E-10 should be considered a success. The method is Exact, Direct, or Iterative. The exact and direct solvers are more reliable, and Mercury tries them first. This logic can be altered with the METHOD statement. The processor is a 8087, 80287, 80387, or emulator. Reports ------- A report is an unformatted ascii file holding the problem and the solution. It can be saved to disk or printed to keep a record of the solution. To save to disk, press Go in the Report menu. (This menu item doesn't become active until you have solved a problem.) Normally, the report has the same name as the problem but with the .EKA suffix replaced with .RPT. You can use other file names if you please. Another way to keep a record of problems and solutions is to have a Log file. Just turn Capture On in the Report menu. Then all problems and solutions will be appended to a log file. The reports and logs are ordinary ascii files, and can be printed. If you have a Postscript printer, you will have to select a mode in which it will accept an ascii file, or load a utility for printing ascii files. (No such utility is provided with Mercury.) Graphics -------- Mercury may be used to generate graphs. These visually show the behavior of functions. A graphics monitor is required. To get a graph, you must first define a function of one variable in your problem file. An example from the file BUMPY.EKA is bumpy(x) := x^2 SIN(x) which defines a function called "bumpy". The "x" is a dummy variable. The has values func(0) = 0^2 SIN(0), func(1) = 1^2 SIN(1), etc. To graph the function from -15 to +15, use the statements: PLOT bumpy GBOUNDS -15, +15 To assimilate this information into the Mercury solver, you must do a Solve, even though you are not really solving for anything in this case. Once the problem has been solved, the View, Print, and Write commands become available under the Graph menu. Select View to see the graph on the screen. If the graph looks a little ragged, perhaps not enough points were used to plot it. The default is 30. Increasing it to 100 will give a much smoother graph. This can be done by either adding the GPOINTS 100 statement to the problem file and solving it again, or by selecting GPoints from the Graph menu. It is also possible to have "parametric" plots, where both X and Y are functions of some other independent variable. Merely use the XAXIS statement to specify the X function. The GBOUNDS statement then refers to the independent variable. These graphs can be printed on graphics printers. This includes most printers, except for daisy-wheel printers. Selecting Print will print the graph to whatever printer has been selected under the Options/Printer menu. You can also select Write, which prints the graph to a file. Such a file can be sent to a printer at a later time. The primary usefulness of this feature is to export the graph in Lotus or Postscript format, where it can be imported by another program. The various options in the Graph menu are really the defaults, and are overridden by the corresponding statements in the problem file. If there are statements in the problem file setting graph options, then the menu may not accurately reflect the options used to draw the graph. Nevertheless, once a graph has been created, these options can be modified from the graph menu, and then View will show the revised graph. 4. Feature Reference This chapter gives an alphabetical lookup for the Mercury built-in functions and statements. Operators --------- + add. - subtract. * multiply. a space may also be used. / divide. ^ power. Eg, 2^3 = 8. ~ complex conjugate. < less than. <= less than. > greater than. >= greater than. = equals. := variable initialization or function definition. : equation separator. % percent, eg, 17% = .17. These have the usual mathematical precedence rules, with ^ having higher precedence than * and /, which in turn are higher than + and -. Constants --------- These two constants are built-in. INF plus infinity PI 3.1415926535... You can also use the greek letter by typing Alt-P. Functions --------- Functions take one or more arguments. The arguments are separated by commas and enclosed in parentheses. Here is a list of the built-in functions. Additional functions can be defined within a problem file. ABS(x) The absolute value of x. You can also use |x|. ACOS(x) The arc cosine of x, inverse to COS(x). The result is in radians. ASIN(x) The arc sine of x, inverse to SIN(x). The result is in radians. ATAN(x) The arc tangent of x, inverse to TAN(x). The result is in radians. BESSJ(n,x) The Bessel function of the first kind. The first argument must be an integer. BESSY(n,x) The Bessel function of the second kind, sometimes also called the Weber function. The first argument must be an integer. For each n, it is a solution to Bessel's differential equation with a singularity at zero. BINOM(n,k) The number of combinations, choosing k of n items. Given by n! / (k! (n-k)!). CEIL(x) The smallest integer greater than or equal to x. Eg, CEIL(4.7) = 5. COS(x) The cosine of x. The angle x is measured in radians. COSH(x) The hyperbolic cosine of x, (EXP(x) + EXP(-x))/2. CTERM(rate,fv,pv) The number of periods required for a given present value (pv) to increase to a given future value (fv) if it grows at the given interest rate. The function is compatible with Quattro/Lotus. DERIV(f(x),x) The derivative of the function f(x) with respect to x. In most cases, the derivative is computed symbolically. DLGAMMA(n,x) The n-th derivative of LOGGAMMA, n = 0,1,2,... ERF(x) The error function of x. It is related to NCUM(x). ERF(x) = 2 NCUM(x SQRT(2)) - 1 Eg, ERF(-INF) = -1, ERF(0) = 0, ERF(+INF) = +1. ERFC(x) The complementary error function of x. See ERF and NCUM. ERFC(x) = 1 - ERF(x) Eg, ERFC(-INF) = +1, ERFC(0) = 0, ERFC(+INF) = -1. EXP(x) The exponential function of x, e^x, e = 2.71828... It is the inverse to LN(x). FACT(x) x factorial, sometimes written x!. Eg, FACT(5) = 5*4*3*2*1 = 120. FLOOR(x) The largest integer less than or equal to x. Eg, FLOOR(4.7) = 4. FRAC(x) The fractional part of x, x - FLOOR(x). Eg, FRAC(4.7) = 0.7. FV(pmt,rate,nper) The future value of payments (pmt) at the end of nper periods, assuming the given interest rate. The function is compatible with Quattro/Lotus. FVAL(rate,nper,pmt,pv) Gives the future value of a financial with a payment pmt in each of nper periods, with the given present value pv, and the given interest rate. The function is compatible with Quattro Pro. It uses the financial mode specified by the FINANCE statement to determine whether the payments are at the end of the period (the default) or the beginning. HYPOT(x,y) The polar coordinate radius corresponding to cartesian coordinates (x,y). Eg, HYPOT(3,4) = 5. IFPOS(x,y,z) Gives y if x >= 0, else it gives z. Eg, IFPOS(-3,6,7) = 7. IM(x) The imaginary part of x. The COMPLEX option must be in use. INTEG(f(x),x,a,b) INTEG(f(x),x,a,b) gives the numerical integral of f(x) with respect to x from a to b. Improper integrals are allowed, so a can be -INF, b can be +INF, and f can be singular at a or b. The approximation method is adaptive and nonlinear, and often is accurate to 15 digits. INTEG1(f(x),x,a,b) INTEG1 is similar to INTEG, but uses a simpler strategy. It is nonadaptive and linear, and may be preferred in situations where the result is supposed to be a smooth function of a and b. INTEG1 is not nearly as accurate as INTEG on singular functions. LIMIT(f(x),x,a,INF) Estimates the limit of f(x), as x goes from a to infinity. It uses a numerical method to compute the limit of f(x), as x goes from a to infinity. It is an nonlinear extrapolation based on a fairly small number of terms. The limit is only reliable if the function is well-behaved, and this feature should be used with caution. LN(x) The natural logarithm (base e) of x. It is the inverse to EXP(x). LOG10(x) The logarithm of x to the base 10. The inverse is 10^x. LOGGAMMA(x) The logarithm of the gamma function of x. MAX(x,...) The maximum value of the arguments. Eg, MAX(-10,5,7) = 7. MIN(x,...) The minimum value of the arguments. Eg, MIN(-10,5,7) = -10. NCUM(x) The normal cumulative probability function. NCUM(X) is the probability that a normal random variable is below its mean, plus X standard deviations. Eg, NCUM(-INF) = 0, NCUM(0) = .5, NCUM(+INF) = 1. PAYMT(rate,nper,pv,fv) Gives the present value of a financial with a payment pmt in each of nper periods, with the given future value fv, and the given interest rate. The function is compatible with Quattro Pro. It uses the financial mode specified by the FINANCE statement to determine whether the payments are at the end of the period (the default) or the beginning. PMT(pv,rate,nper) Gives the amount of the payments required in each of nper periods to balance the given present value pv and interest rate. The function is compatible with Quattro/Lotus. POLAR(x,y) The polar coordinate angle corresponding to cartesian (x,y). Same as the Quattro/Lotus @ATAN2(x,y) or the Fortran/C atan2(y,x). POLY(x,a0,a1,...,an) The polynomial a0 x^n + ... + an. Eg, the polynomial 2 x^3 - 3 x + 7 can also be written POLY(x,2,0,-3,7). The chief advantage of the latter is notational convenience, and so it can be recognized by the special polynomial solver. POS(x) The positive part of x, ie, x if positive and 0 if negative. PV(pmt,rate,nper) Gives the present value of nper payments of amount pmt, using the given interest rate. The function is compatible with Quattro/Lotus. PVAL(rate,nper,pmt,fv) Gives the present value of a financial with a payment pmt in each of nper periods, with the given future value fv, and the given interest rate. The function is compatible with Quattro Pro. It uses the financial mode specified by the FINANCE statement to determine whether the payments are at the end of the period (the default) or the beginning. RATE(fv,pv,nper) Gives the interest rate required for the present value (pv) to increase to the future value (fv) in nper periods. The function is compatible with Quattro/Lotus. RE(x) The real part of x. The COMPLEX option must be in use. REM(x,y) The remainder of x divided by y. Eg, REM(20,3) = 2. SGN(x) Gives +1 if x > 0, -1 if x < 0. SIN(x) The sine of x. The angle x is measured in radians. SINH(x) The hyperbolic sine of x, (EXP(x) - EXP(-x))/2. SQRT(x) The square root of x. SUM(f(x),x,a,b) The sum of f from a to b, f(a) + f(a + 1) + ... + f(b) If b = +INF, the limit is estimated as with the LIMIT function. TAN(x) The tangent of x, SIN(x)/COS(x). The angle x is measured in radians. TANH(x) The hyperbolic tangent of x, SINH(x)/COSH(x). TERM(pmt,rate,fv) The number of periods required for a given payment stream (pmt) to increase to a given future value (fv) if it grows at the given interest rate. The function is compatible with Quattro/Lotus. Statements ---------- BOUNDS lower,upper Specify bounds for the domains of the variables. The default is BOUNDS -INF,INF. Some solving time can be saved with the statement BOUNDS -1e6,1e6 if you are sure that all the relevant values of your variables are less than a million. CASEFOLD Ignore case in names. Otherwise, the variable x is different from X. CASESTRICT Using CASESTRICT, with no args, causes built-in functions, constants, and statements to be upper case only. Normally such predefined names are not case sensitive, so SIN(x) is the same as sin(x). Using this option reduces the chance of a name conflict. COMPLEX Declares that all the variables are to be considered complex variables. Their solutions have a real and imaginary part. These built-in functions may be used in a complex problem: ABS, COS, COSH, EXP, IM, LN, RE, SIN, SINH, SQRT, SUM, TAN, TANH. DIGITS n Display real numbers to n digits of precision. The default is 18. EPSILON x Forces numbers very close to zero to be displayed as zero. Useful for showing complex numbers which are actually real. Default is EPSILON 0. EXTERNAL function Declares the name to be an externally defined function, to be found on the disk in the problem directory with the file name function.BIN. These functions are written in Turbo C++ or TASM, and compiled into .BIN files. See the samples. FINANCE type Gives the type parameter to the Quattro compatible financial functions, PVAL, PAYMT, and FVAL. The default is type = 0, meaning that the payments are at the end of the period. Use type = 1 if the payments are at the beginning. GBOUNDS xleft, xright The lower (xleft) and upper (xright) bounds for the X-axis in a graph. GPOINTS n The number of points (function values) used to draw a graph. The default is 30. If your graph is too ragged, try increasing the number of points. GSTYLE n Allows graph curves to be thick (GSTYLE 0) or thin (GSTYLE 1). INCLUDE filename The file with the given name is inserted in the problem at that point. The filename must be in quotes, as in INCLUDE "CONST.INC". The default file name extension is .INC. Include files cannot be nested. ITERATIONS n The maximum number of iterations used by the iterative solver. The default is 100. LEGEND Puts a legend on the graph which identifies the function or functions. Recommended when plotting more than one function. LEVEL n Allows user control over what symbolic substitutions are made. 0. no symbolic solving 1. solve X = constant 2. solve X = constant expression 3. solve only when X occurs once, no other variables 4. solve only when X occurs once, other variables possible 5. solve linear equations 6. solve squares, using square roots 7. solve quadratic equations 8. solve powers The default is LEVEL 5, solving linear equations but not using square roots. Strictly speaking, the substitutions involving roots are not mathematically equivalent, as the substitution must arbitrarily choose the positive or negative square root. It is sometimes preferable to use LEVEL 0 to turn off all the symbolic manipulations. Then Mercury effectively weights all equations equally, and the solution can be considered the solution to a least squares problem. LIST function List one or more functions in the solution window. Syntax is the same as for PLOT. MAXIMIZE x The variable x is maximized. METHOD n Specify the solver method, where n = 1, 2, or 3. 1. Exact 2. Direct 3. Iterative The default is METHOD 1, the exact solver. If it fails, the direct solver is tried, and it that also fails, the iterative solver is tried. MINIMIZE x The variable x is minimized. PENALTY n The value of the penalty constant, an arbitrary constant used to enforce constraints when the iterative solver is used to maximize or minimize a variable. The default is 30. Increasing it forces the approximate solutions closer to the constraint manifold at the cost of making the extremum harder to find. Readers of The Hitchhiker's Guide to the Galaxy suggest using the value 42. PLOT function Creates a plot of the named function. To view the graph, select Graph from the main menu. To plot several functions on the same graph, list the names separated by commas. The functions must be defined in the problem file, and be functions of one variable. For example, to plot SIN(x) and x^3-x from -5 to +5, use f(x) := SIN(x) g(x) := x^3 - x GBOUNDS -5, +5 PLOT f, g PRINT Same as PRINTFILE "LPT1". It prints the graph to the DOS device LPT1, which is usually a printer on parallel port 1. The printing can be configured to use another port from the Options menu, and the result saved to a setup file. PRINTFILE filename Print the graph to the given file, whose name is in quotes. These names have special meaning. "PAR1" parallel port 1 "SER1" serial port 1 "PAR2" parallel port 2 "SER2" serial port 2 "LPT1" DOS lpt1 "LPT2" DOS lpt2 "LPT3" DOS lpt3 "LPT4" DOS lpt4 "EPT" IBM postscript port PROGRESS x Show the value of the variable x in the Progress box. Mercury does not solve for all variables at once, so sometimes there might be no value for the variable until the end of the solution process. In such cases, the Progress window shows "x = NAN", meaning "x is not a number". If no PROGRESS statement is given, the Progress window shows the value of the "Residual". This is the sum of squares of quantities which should be zero, and which Mercury is trying to minimize. The smaller the value, the more likely that Mercury is getting close to the solution. REPORT filename Allows naming a report file from within a problem. ROOTSIGN n When solving an equation like x^2 = a, Mercury may take the square root of both sides. This introduces two roots, +SQRT(a) and -SQRT(a), and Mercury does not know which to take. You can specify that you always want the positive root with ROOTSIGN +1, or always the negative root with ROOTSIGN -1. The default is ROOTSIGN 0, which leaves Mercury unable to evaluate the square root. SCREEN n Choose a graphics screen. The default is SCREEN 0, which means Mercury automatically tries to detect your graphics hardware. Some graphics cards and modes cannot be detected, and must be selected with a SCREEN statement. 0. AutoDetect 1. CGA 2. MCGA 3. EGA 64k 4. EGA 256k 5. EGA Mono 6. IBM 8514 7. Hercules 8. ATT 400 line 9. VGA 10. 3270 PC 11. Tandy 1000 12. ATI Super VGA 13. Orchid Super VGA 14. Paradise Super VGA 15. Vega Super VGA 16. Compro ABLE II Super VGA 17. SOTA Super VGA From the main menu, the graphics screen can be selected with Graph/Mode. Such a selection is ineffective after the first graph is drawn, as the graphics has been initialized at that point. We recommend saving the desired graphics mode to the MERCURY.CFG configuration file. (Note: The drivers for some of these graphics modes is not included with some versions of Mercury. Registered customers receive the full set of drivers.) SHOW x Show specified variables in the specified order. Other variables are not displayed in the solution window. SOLVE x Instructs Mercury to solve only for the variable x. By default, it solves for all variables named in the problem. If SOLVE is given with a polynomial function name, as in SOLVE f f(x) := POLY(x,1,0,0,1) then Mercury will apply a special polynomial root finder to get all real and complex roots. START x The starting value for uninitialized variables while using the iterative solver. The default is 1.00. TERMS n The most number of terms used by the INTEG and LIMIT functions. The default is 50. TIME n The maximum time allowed for solution. The default is 200 seconds. Same as MAXTIME in Eureka 1.0. TITLE label Argument is a quoted string, to be used as a graph title. Default is no title. SUBTITLE label Argument is a quoted string, to be used as a graph subtitle. Default is no title. XLABEL label A label for the X-axis of a graph. The label must be in quotes. YLABEL label A label for the Y-axis of a graph. The label must be in quotes. XAXIS function Specifies that the X-axis be a function instead of an independent variable. The function must be defined elsewhere as a function of one variable. The X axis of the graph is a function with the given name. Useful for parametric plots. Eg, to graph (x,y) = (cos(t),sin(t)) as t goes from 0 to 2*PI, use x(t) := cos(t) y(t) := sin(t) GBOUNDS 0, 2 PI XAXIS x PLOT y