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3micalc 0m1mversion 10m.1m1 0mA complex-number expression parser by Martin W.Scott User Guide icalc User Guide icalc 1mIntroduction 0m3micalc 0mis a terminal-based calculator; the 1mi 0mstands for 1mimaginary0m, denoting the fact that the calculator works with complex numbers. There are many calculator programs for the Amiga, but to my knowledge, none with this ability. 1mComplex Numbers 0mA complex number is made up of two components: a real part, and an imaginary part. The imaginary part is just a real number multiplied by the special number i, which is defined to be the square root of -1. Thus, i*i = -1. This may all seem very artificial, but it's not; complex numbers exist as surely as real numbers do (or don't). It's just that our notion of a real number is based upon our much experience of measurement and arithmetic. If fact, given an arbitrary length of string, we probably couldn't calculate its length as a real number, but only find a rational approximation to it. However, it is not the purpose of this guide to teach the basics of complex numbers - there are plenty of books that do that. But even if you've never heard of complex numbers before, you may still find 3micalc 0museful, because real numbers are just a subset of complex numbers, and can thus be manipulated by this program as in an ordinary calculator-type program. Don't be put off - I hardly ever need to use complex numbers, I just like the facility to be there for the odd occasion when I do use them. 1mExpressions 0m3micalc 0maccepts expressions in the usual form for computer expressions, i.e. multiplication must be explicitly shown by '*', and function arguments enclosed within parentheses. The exception to this is the method of inputing imaginary parts of complex numbers. If you want to enter the number 3 + 4i, you can type either 3 + 4i or 3 + 4*i but 3mnot 0m3 + i4 since this would give an ambiguous grammar - is it i*4 or the variable "i4"? In almost all circumstances, the algebraic "4i" is preferred, but there is one exception, exponentation "^". If you want 3*i^3, you must type 3*i^3 icalc -2- version 1.1 icalc User Guide icalc and not 3i^3 The former gives the correct result, -3i, whilst the latter gives -27i. This is because the grammar sees "3i" as a number in its own right, and not an implicit multiplication. See the appendices for the complete list of commands, operators, constants and builtin functions that 3micalc 0msupports. Expressions are separated by newlines or semi-colons ';'. You may also split an expression over multiple lines using a backslash '\' to indicate continuation to the next input line. Comments are introduced by '#', and the rest of the input line is ignored. You may also place comments after backslashes. 1mSample Expressions 0mx=1-i # an assignment statement assigns to variable x the value 1-i. # and now, two statements separated by a semi-colon sqr(x); x*sqr(x); displays the values -2i and -2-2i. The next example demonstrates a problem inherent in machine calculations: 2*sin(x)*cos(x) - sin(2*x) displays the value 4.4408920985e-16 + 4.4408920985e-16 i. The answer should be 0, and indeed is very close to that. The inaccuracy results because each term has a value as close to the actual value as the computer's internal representation of numbers can hold. 1mStarting icalc 0m3micalc 0mmay be started from either the CLI or Workbench. It will process the the file "s:icalc.init" if it exists, before doing anything else. It is recommended that you use NEWCON: or ConMan to provide editing and history facilities. CLI Usage The synopsis for icalc is icalc [file-list] where file-list is a list of input files. 3micalc 0mwill read (process) the files in the list and exit. If a file-list is icalc -3- version 1.1 icalc User Guide icalc specified, 3micalc 0mwill assume that you do not wish to use the program interactively. If you do, you should specify one of the files as a dash "-"; this is taken to mean "use standard input". If no file-list is given, 3micalc 0mstarts an interactive session. You can obviously redirect output to a file if you wish (and redirect input from a file, but this is generally unnecessary). For example, if you wanted to use definitions contained in a file called "stat.icalc", and then type some expressions in at the terminal, you would use icalc stat.icalc - to start the program. Because the program makes extensive use of recursion, I recommend you use a stack size of about 20K when running 3micalc. 0mWorkbench Usage To start 3micalc 0mfrom the Workbench, simply double-click its icon. You may also pass arguments (definition/command files) to 3micalc 0min the usual Workbench manner (shift-select etc.). 3micalc 0mwill then open a window on the Workbench screen for an interactive session. You can specify the window to be opened by modifying the WINDOW tooltype (click once on the 3micalc 0micon, and select "Info" from the Workbench menu). By default, this is a NEWCON: window. You may also set the default tool of a definition file's project icon to be 3micalc. 0mThen simply double clicking the project icon will launch 3micalc 0mand read the definition file. Remember, however, to set the project icon's stack to 20000, or you may get a stack overflow error. Exiting icalc To end an interactive session of 3micalc 0m(from either CLI or Workbench) use the commands "quit" or "exit". You can also type ^\ (press the control key and '\' simultaneously). 1mVariables 0mVariables in 3micalc 0mcan be named arbitrarily, as long as no name conflicts with a pre-defined symbol name, such as sin or PI. The standard conventions apply: the first character must be a letter or underscore, followed by a string of letters, digits and underscores. Case is significant. Variables are introduced by using them; If a variable is used before it has been initialized (i.e. has had a value assigned to it), 3micalc 0mwill warn the user and use the value zero. Assignments can appear anywhere in an expression, and are in fact expressions in their own right (as in the programming language C). So, you can do things like apple = banana = apricot = 34 to assign the value 34 to these variables. Since icalc -4- version 1.1 icalc User Guide icalc assignments are expressions, their value being the assignment value, you can also do apple = sqr(banana = 12) which assigns the value 12 to "banana", and 144 to "apple". You can of course assign new values to existing variables. 1mConstants and ans 0m3micalc 0mcomes with several predefined constants, e.g. PI, GAMMA etc, which by convention are all in upper-case. You cannot assign a new value to a constant. There is a special constant, "ans", which isn't really a constant at all, but is made so to prevent assignments to it. "ans" contains the value of the previously evaluated expression, and so is useful when splitting a calculation up into smaller chunks. 1 + 2 + 3 6 ans * 12 72 You can also use it for recursive formulas. For example, the logistic equation xnext = rx(1-x) for parameter r = 3.5, initial x = 0.4 could be iterated as follows: r = 3.5; 0.4 # initialize r and ans 3.5 0.4 r*ans*(1-ans) 0.84 r*ans*(1-ans) 0.4704 r*ans*(1-ans) 0.87193344 r*ans*(1-ans) 0.390829306734 and so on. Of course, you needn't use "ans" here; you could just use assignments to "x", but it illustrates the point. Note that once the necessary variables have been initialised, the same expression is evaluated repeatedly. Thus, if you are using a window with history capabilities (such as NEWCON:) you can save a lot of typing. There are other ways to perform repeated calculations in 3micalc. 0mThese are explained in the section "Special Functions". icalc -5- version 1.1 icalc User Guide icalc 1mBuiltin Functions 0m3micalc 0mcomes with many builtin functions, for example sin, cos, sqrt, exp, ln etc. There are some special functions for complex numbers too, namely Im, Re, arg, norm. All functions take real or complex arguments, which can be any expression. For example, asin(sin(1)) returns the value 1 as expected. But ... asin(sin(1-i)) returns 1.484116283319 - 0.831157682449 i; this is because the inverse of sin is a many-valued function. asin returns a value in the PVR (principal value range) as do all many-valued functions in 3micalc. 0mThe PVR is defined (for this program) as the interval [-PI,PI). When working with complex numbers, you should always bear in mind that almost all the trigonometric functions are many-valued, because they are defined in terms of the exponential and logarithmic functions. Another thing to bear in mind if you intend working with reals is that things like ln(-1) return a value, 3.14159265359 i (PI*i) in this case. 1mUser0m-1mdefined Functions 0mUser-defined functions give 3micalc 0mits flexibility and power. Any function that can be stated as a normal expression (or list of expressions) may easily be defined and used like a builtin function. Functions are defined in the following manner: func <name> ( <parameter-list> ) = <expr> so to define the function f(z) = sin(sqrt(z)), simply type func f(z) = sin(sqrt(z)) Note that z is 3mlocal 0mto the function, not a global variable. Thus, if z has been defined as a variable before defining f(z), it is unaltered when you call f(z). There are a number of sample definition files included with 3micalc. 0mYou may find it instructive to examine them. Multi-parameter functions may also be defined. For example, the volume of a right-circular cone is given by the formula volume = 1/3*PI*sqr(r)*h where r is the radius of the base, and h is the height of the cone. We can define a function to compute the volume as follows: icalc -6- version 1.1 icalc User Guide icalc func conevol(r,h) = 1/3*PI*sqr(r)*h conevol(1,3) 3.14159265 It is also possible (though of limited use) to declare functions that take no parameters. Examples of multi-parameter functions are given in the sample definition files. As with most programming languages, don't write circular definitions; 3micalc 0mdoes not detect them, and will swiftly run out of stack space. Recursive functions fall into this category, as there is no way to specify terminal cases. 1mSpecial Functions 0mCertain builtin functions in 3micalc 0mare called 3mspecial 0mbecause of the way they use their arguments. Normally, when you `call' a function, its arguments are first evaluated and then the function is called. With special functions however, the arguments are not evaluated; this means that the expressions themselves (and not just their values) can be used by the function in a number of ways. Some special functions take integers as arguments; if you supply an expression with a non-integral value, this value will be stripped of it's imaginary component, and the remaining real number rounded to the nearest integer. Currently, the special functions are as follows. Sum(var,upto,expr) This command calculates finite sums. The variable `var' is the index of summation, with an optional assignment. While the value of the index is less than or equal to the value of the upto argument, the value of expr is added to an internal summation variable, which is initially zero. Examples will help to clarify usage. To evaluate the sum of the first 100 natural numbers (naively) we could do Sum(n=1,100,n) which gives the answer as 5050 (which is 100/2*(100+1)). Note the initial assignment of 1 to n. If no initial value is specified, a value of zero is assumed (and the user is notified of this). When a summation is complete, the index variable holds a value 1 greater than the value of upto. We can also use Sum() to provide approximations to infinite sums. Sum(n=1, 100, 1/sqr(n)) 1.6349839 Using the fact that n will now hold the value 101, we can continue the summation to 200 by the following: icalc -7- version 1.1 icalc User Guide icalc ans + Sum(n, n+99, 1/sqr(n)) 1.63994655 and continue to obtain closer answers by repeating the last line, ans + Sum(n, n+99, 1/sqr(n)) 1.64160628 One last example, to evaluate the sum of the all even numbers upto 100. Sum(n=1, 50, 2*n) 2550 Prod(var,upto,expr) This command calculates finite products, in an analogous way to the Sum() special function. Examples will demonstrate its use. Prod(n=1, 10, n) calculates 10!, which is 3628800. We can also use Prod() to investigate infinite products, although care should be taken. Consider Wallis's product for PI/2: PI 2×2×4×4×6×6×... -- = --------------- 2 1×3×3×5×5×7×... Here the product is 2/1 * 2/3 * 4/3 * 4/5 * ... and so we must find a method of producing the numerators and denominators from an index. It is easiest to evaluate the product in pairs of terms. Thus, for each term our numerator will be sqr(2*n), and our denominator will be (2*n-1)*(2*n+1) = sqr(2*n)-1. Note that sqr(2*n) appears in both the numerator and denominator. We can therefore optimise the calculation by using a temporary variable to hold this value. We proceed as follows: Prod(n=1, 100, (t = sqr(2*n))/(t-1)) 1.56689375 ans * Prod(n, n+99, (t = sqr(2*n))/(t-1)) 1.56883895 ans * Prod(n, n+99, (t = sqr(2*n))/(t-1)) 1.56949005 every(var,upto,expr) and vevery(var,upto,expr) These functions perform more general iterations than Sum() and Prod(). They return the value of the last expression evaluated, ie. expr evaluated when var = upto. The difference between every() and vevery() is that the latter produces verbose output - it prints the value of the index and the result of the expression for every iteration. icalc -8- version 1.1 icalc User Guide icalc Example: evaluate sin(sin(...(sin(z))...)) with sin() applied 10 times. Here we'll take z to be 1-i. last = 1-i; every(n=1, 10, last = sin(last)) 0.51856952 - 0.01039404 i Although every() and vevery() are useful in some situations, most iterations are more easily performed by using Sum() and Prod(). multi(expr1, expr2, ... , exprN) This special function takes an arbitrary number of arguments, and evaluates each of them from left to right. It returns the value of it's last argument, exprN. multi() was added to make the addition of certain user-functions possible. For example, the one-variable statistical analysis suite given in the file "stat.icalc" uses multi() to update the values of many variables with one function call. If output has not been switched off with the silent command, a call to multi() will print the result of it's last argument. In order to print the results of other arguments, the builtin function print() is supplied. print() takes one argument, prints it to the display, and returns its argument as a result. In this way, expressions can appear to return more than one result. An example: write a function that converts Cartesian coordinates into polar form, returning the radius and angle in degrees, as well as setting the variables r and theta to these values. func rec2pol(x,y) = multi(r = print(sqrt(x*x+y*y)), \ theta = DEG*atan(y/x) ) rec2pol(3,4) 5 53.13010235 Examples of the many uses of these special functions can be found in the example icalc script file "stat.icalc" and the startup file "icalc.init". 1mCommands 0m3micalc 0mhas a few commands to make life easier. quit exit not surprisingly, these commands terminate 3micalc. 0msilent stops generation of any further output. However, errors and warnings are still displayed. It is useful for icalc -9- version 1.1 icalc User Guide icalc initialization files containing lots of definitions. verbose restores generation of output. help reminds you what other commands are available. vars lists all currently defined variables and their values. consts lists all predefined constants and their values. builtins lists all built-in functions available, including special functions. functions lists all user-defined functions, with their parameter lists. 1mBugs 0mWhat bugs? Well, I'm sure there must be the odd one or two, but I haven't noticed. Should you find any, please let me know and I'll try to fix them. If there's something you hate about 3micalc 0mand would like changed, again drop me a line (at the address below). 1mCompiling 0m3micalc 0mwas written using Lattice C (version 5.10a) but should compile with no problems with Manx C. I use the version of bison on Fish #136; if you're using a later version or Yacc, you may need to change the makefile. The code is meant to be portable, so Amiga-specific code (there's not much) is #ifdef'd. Also, to implement the WINDOW tooltype, I've used a slightly modified version of Lattice's _main function; since I'm probably not permitted to distribute the modified source, I've included just the compiled module, myumain.o. 1mCredits 0mThanks to the GNU team for bison (it makes these things so much easier to write and modify) and to William Loftus for the Amiga port. Thanks also to Steve Koren for Sksh, and Mike Meyer et al for Mg3; together they make a great development environment. icalc -10- version 1.1 icalc User Guide icalc This is my first bison-based project, and I learnt a lot from "The Unix Programming Environment" by Brian Kernighan and Rob Pike. Their "hoc" taught me a lot about using Yacc. 1mDistribution 0m3micalc 0mis not public domain. You are permitted to distribute it at only a nominal charge to cover costs. All files should be included, with the exception of the source files (in the "src" directory). Under no circumstances should a modified version of 3micalc 0mbe distributed. If you make modifications which you feel may be useful, send a copy to me and I'll (probably) include them with the next release. 1mThe Bottom Line 0mAlthough contributions are not required, they would be most welcome (I'm a poor student). Don't let that inhibit you from sending bug reports, praise, suggestions, neat programs etc. I'd get most pleasure from hearing if anyone actually uses this bloody program! I can be reached at: Martin W. Scott, 23 Drum Brae North, Edinburgh, EH4 8AT SCOTLAND. Thanks for reading this far. The appendices follow. icalc -11- version 1.1 icalc User Guide icalc 1mAppendix 1 0m- 1mOperators 0m(1min increasing order of precedence0m) = assignment + addition - subtraction, unary minus * multiplication / division ^ exponentation ' conjugate: z' = conjugate of z 1mAppendix 2 0m- 1mConstants 0mPI 4*atan(1) (that's one definition) E exp(1) GAMMA Euler's constant, 0.57721566... DEG Number of degrees in one radian PHI Golden ratio 1.61803398... LOG10 ln(10) LOG2 ln(2) 1mAppendix 3 0m- 1mBuiltin functions 0mSingle-argument functions: sin trigonometric functions cos tan asin inverse trigonometric functions acos atan sinh hyperbolic trigonometric functions cosh tanh exp exponential function ln natural logarithm sqr square sqrt square root conj conjugate (see also ' operator) abs absolute value (sqrt(norm(z)) norm not really norm; if z=x+iy, norm(z) = x²+y² arg principal argument Re real part of complex number Im imaginary part of complex number int real and imaginary parts rounded to nearest integer ceil ceil of real and imaginary parts (eg. ceil(1.2) = 2) floor floor of real and imaginary parts (eg. floor(1.2) = 1) print prints and returns the value of its argument prec adjust number of decimal places displayed Special functions (for more details, see the section headed "Special Functions" above): Sum calculate a finite sum Prod calculate a finite product every perform a general iteration quietly vevery perform a general iteration verbosely icalc -12- version 1.1 icalc User Guide icalc multi evaluate many expressions icalc -13- version 1.1