This is Info file calc.info, produced by Makeinfo version 1.67 from the input file /ade-src/fsf/calc/calc.texinfo. This file documents Calc, the GNU Emacs calculator. Copyright (C) 1990, 1991 Free Software Foundation, Inc. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided also that the section entitled "GNU General Public License" is included exactly as in the original, and provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that the section entitled "GNU General Public License" may be included in a translation approved by the author instead of in the original English.  File: calc.info, Node: Mathematica Language Mode, Next: Maple Language Mode, Prev: Eqn Language Mode, Up: Language Modes Mathematica Language Mode ------------------------- The `d M' (`calc-mathematica-language') command selects the conventions of Mathematica, a powerful and popular mathematical tool from Wolfram Research, Inc. Notable differences in Mathematica mode are that the names of built-in functions are capitalized, and function calls use square brackets instead of parentheses. Thus the Calc formula `sin(2 x)' is entered and displayed `Sin[2 x]' in Mathematica mode. Vectors and matrices use curly braces in Mathematica. Complex numbers are written `3 + 4 I'. The standard special constants in Calc are written `Pi', `E', `I', `GoldenRatio', `EulerGamma', `Infinity', `ComplexInfinity', and `Indeterminate' in Mathematica mode. Non-decimal numbers are written, e.g., `16^^7fff'. Floating-point numbers in scientific notation are written `1.23*10.^3'. Subscripts use double square brackets: `a[[i]]'.  File: calc.info, Node: Maple Language Mode, Next: Compositions, Prev: Mathematica Language Mode, Up: Language Modes Maple Language Mode ------------------- The `d W' (`calc-maple-language') command selects the conventions of Maple, another mathematical tool from the University of Waterloo. Maple's language is much like C. Underscores are allowed in symbol names; square brackets are used for subscripts; explicit `*'s for multiplications are required. Use either `^' or `**' to denote powers. Maple uses square brackets for lists and curly braces for sets. Calc interprets both notations as vectors, and displays vectors with square brackets. This means Maple sets will be converted to lists when they pass through Calc. As a special case, matrices are written as calls to the function `matrix', given a list of lists as the argument, and can be read in this form or with all-capitals `MATRIX'. The Maple interval notation `2 .. 3' has no surrounding brackets; Calc reads `2 .. 3' as the closed interval `[2 .. 3]', and writes any kind of interval as `2 .. 3'. This means you cannot see the difference between an open and a closed interval while in Maple display mode. Maple writes complex numbers as `3 + 4*I'. Its special constants are `Pi', `E', `I', and `infinity' (all three of `inf', `uinf', and `nan' display as `infinity'). Floating-point numbers are written `1.23*10.^3'. Among things not currently handled by Calc's Maple mode are the various quote symbols, procedures and functional operators, and inert (`&') operators.  File: calc.info, Node: Compositions, Next: Syntax Tables, Prev: Maple Language Mode, Up: Language Modes Compositions ------------ There are several "composition functions" which allow you to get displays in a variety of formats similar to those in Big language mode. Most of these functions do not evaluate to anything; they are placeholders which are left in symbolic form by Calc's evaluator but are recognized by Calc's display formatting routines. Two of these, `string' and `bstring', are described elsewhere. *Note Strings::. For example, `string("ABC")' is displayed as `ABC'. When viewed on the stack it will be indistinguishable from the variable `ABC', but internally it will be stored as `string([65, 66, 67])' and can still be manipulated this way; for example, the selection and vector commands `j 1 v v j u' would select the vector portion of this object and reverse the elements, then deselect to reveal a string whose characters had been reversed. The composition functions do the same thing in all language modes (although their components will of course be formatted in the current language mode). The one exception is Unformatted mode (`d U'), which does not give the composition functions any special treatment. The functions are discussed here because of their relationship to the language modes. * Menu: * Composition Basics:: * Horizontal Compositions:: * Vertical Compositions:: * Other Compositions:: * Information about Compositions:: * User-Defined Compositions::  File: calc.info, Node: Composition Basics, Next: Horizontal Compositions, Prev: Compositions, Up: Compositions Composition Basics .................. Compositions are generally formed by stacking formulas together horizontally or vertically in various ways. Those formulas are themselves compositions. TeX users will find this analogous to TeX's "boxes." Each multi-line composition has a "baseline"; horizontal compositions use the baselines to decide how formulas should be positioned relative to one another. For example, in the Big mode formula 2 a + b 17 + ------ c the second term of the sum is four lines tall and has line three as its baseline. Thus when the term is combined with 17, line three is placed on the same level as the baseline of 17. Another important composition concept is "precedence". This is an integer that represents the binding strength of various operators. For example, `*' has higher precedence (195) than `+' (180), which means that `(a * b) + c' will be formatted without the parentheses, but `a * (b + c)' will keep the parentheses. The operator table used by normal and Big language modes has the following precedences: _ 1200 (subscripts) % 1100 (as in n%) - 1000 (as in -n) ! 1000 (as in !n) mod 400 +/- 300 !! 210 (as in n!!) ! 210 (as in n!) ^ 200 * 195 (or implicit multiplication) / % \ 190 + - 180 (as in a+b) | 170 < = 160 (and other relations) && 110 || 100 ? : 90 !!! 85 &&& 80 ||| 75 := 50 :: 45 => 40 The general rule is that if an operator with precedence `n' occurs as an argument to an operator with precedence `m', then the argument is enclosed in parentheses if `n < m'. Top-level expressions and expressions which are function arguments, vector components, etc., are formatted with precedence zero (so that they normally never get additional parentheses). For binary left-associative operators like `+', the righthand argument is actually formatted with one-higher precedence than shown in the table. This makes sure `(a + b) + c' omits the parentheses, but the unnatural form `a + (b + c)' keeps its parentheses. Right-associative operators like `^' format the lefthand argument with one-higher precedence. The `cprec' function formats an expression with an arbitrary precedence. For example, `cprec(abc, 185)' will combine into sums and products as follows: `7 + abc', `7 (abc)' (because this `cprec' form has higher precedence than addition, but lower precedence than multiplication). A final composition issue is "line breaking". Calc uses two different strategies for "flat" and "non-flat" compositions. A non-flat composition is anything that appears on multiple lines (not counting line breaking). Examples would be matrices and Big mode powers and quotients. Non-flat compositions are displayed exactly as specified. If they come out wider than the current window, you must use horizontal scrolling (`<' and `>') to view them. Flat compositions, on the other hand, will be broken across several lines if they are too wide to fit the window. Certain points in a composition are noted internally as "break points". Calc's general strategy is to fill each line as much as possible, then to move down to the next line starting at the first break point that didn't fit. However, the line breaker understands the hierarchical structure of formulas. It will not break an "inner" formula if it can use an earlier break point from an "outer" formula instead. For example, a vector of sums might be formatted as: [ a + b + c, d + e + f, g + h + i, j + k + l, m ] If the `m' can fit, then so, it seems, could the `g'. But Calc prefers to break at the comma since the comma is part of a "more outer" formula. Calc would break at a plus sign only if it had to, say, if the very first sum in the vector had itself been too large to fit. Of the composition functions described below, only `choriz' generates break points. The `bstring' function (*note Strings::.) also generates breakable items: A break point is added after every space (or group of spaces) except for spaces at the very beginning or end of the string. Composition functions themselves count as levels in the formula hierarchy, so a `choriz' that is a component of a larger `choriz' will be less likely to be broken. As a special case, if a `bstring' occurs as a component of a `choriz' or `choriz'-like object (such as a vector or a list of arguments in a function call), then the break points in that `bstring' will be on the same level as the break points of the surrounding object.  File: calc.info, Node: Horizontal Compositions, Next: Vertical Compositions, Prev: Composition Basics, Up: Compositions Horizontal Compositions ....................... The `choriz' function takes a vector of objects and composes them horizontally. For example, `choriz([17, a b/c, d])' formats as `17a b / cd' in normal language mode, or as a b 17---d c in Big language mode. This is actually one case of the general function `choriz(VEC, SEP, PREC)', where either or both of SEP and PREC may be omitted. PREC gives the "precedence" to use when formatting each of the components of VEC. The default precedence is the precedence from the surrounding environment. SEP is a string (i.e., a vector of character codes as might be entered with `" "' notation) which should separate components of the composition. Also, if SEP is given, the line breaker will allow lines to be broken after each occurrence of SEP. If SEP is omitted, the composition will not be breakable (unless any of its component compositions are breakable). For example, `2 choriz([a, b c, d = e], " + ", 180)' is formatted as `2 a + b c + (d = e)'. To get the `choriz' to have precedence 180 "outwards" as well as "inwards," enclose it in a `cprec' form: `2 cprec(choriz(...), 180)' formats as `2 (a + b c + (d = e))'. The baseline of a horizontal composition is the same as the baselines of the component compositions, which are all aligned.  File: calc.info, Node: Vertical Compositions, Next: Other Compositions, Prev: Horizontal Compositions, Up: Compositions Vertical Compositions ..................... The `cvert' function makes a vertical composition. Each component of the vector is centered in a column. The baseline of the result is by default the top line of the resulting composition. For example, `f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))' formats in Big mode as f( a , 2 ) bb a + 1 ccc 2 b There are several special composition functions that work only as components of a vertical composition. The `cbase' function controls the baseline of the vertical composition; the baseline will be the same as the baseline of whatever component is enclosed in `cbase'. Thus `f(cvert([a, cbase(bb), ccc]), cvert([a^2 + 1, cbase(b^2)]))' displays as 2 a + 1 a 2 f(bb , b ) ccc There are also `ctbase' and `cbbase' functions which make the baseline of the vertical composition equal to the top or bottom line (rather than the baseline) of that component. Thus `cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) + cvert([cbbase(a / b)])' gives a a - - + a + b b - b There should be only one `cbase', `ctbase', or `cbbase' function in a given vertical composition. These functions can also be written with no arguments: `ctbase()' is a zero-height object which means the baseline is the top line of the following item, and `cbbase()' means the baseline is the bottom line of the preceding item. The `crule' function builds a "rule," or horizontal line, across a vertical composition. By itself `crule()' uses `-' characters to build the rule. You can specify any other character, e.g., `crule("=")'. The argument must be a character code or vector of exactly one character code. It is repeated to match the width of the widest item in the stack. For example, a quotient with a thick line is `cvert([a + 1, cbase(crule("=")), b^2])': a + 1 ===== 2 b Finally, the functions `clvert' and `crvert' act exactly like `cvert' except that the items are left- or right-justified in the stack. Thus `clvert([a, bb, ccc]) + crvert([a, bb, ccc])' gives: a + a bb bb ccc ccc Like `choriz', the vertical compositions accept a second argument which gives the precedence to use when formatting the components. Vertical compositions do not support separator strings.  File: calc.info, Node: Other Compositions, Next: Information about Compositions, Prev: Vertical Compositions, Up: Compositions Other Compositions .................. The `csup' function builds a superscripted expression. For example, `csup(a, b)' looks the same as `a^b' does in Big language mode. This is essentially a horizontal composition of `a' and `b', where `b' is shifted up so that its bottom line is one above the baseline. Likewise, the `csub' function builds a subscripted expression. This shifts `b' down so that its top line is one below the bottom line of `a' (note that this is not quite analogous to `csup'). Other arrangements can be obtained by using `choriz' and `cvert' directly. The `cflat' function formats its argument in "flat" mode, as obtained by `d O', if the current language mode is normal or Big. It has no effect in other language modes. For example, `a^(b/c)' is formatted by Big mode like `csup(a, cflat(b/c))' to improve its readability. The `cspace' function creates horizontal space. For example, `cspace(4)' is effectively the same as `string(" ")'. A second string (i.e., vector of characters) argument is repeated instead of the space character. For example, `cspace(4, "ab")' looks like `abababab'. If the second argument is not a string, it is formatted in the normal way and then several copies of that are composed together: `cspace(4, a^2)' yields 2 2 2 2 a a a a If the number argument is zero, this is a zero-width object. The `cvspace' function creates vertical space, or a vertical stack of copies of a certain string or formatted object. The baseline is the center line of the resulting stack. A numerical argument of zero will produce an object which contributes zero height if used in a vertical composition. There are also `ctspace' and `cbspace' functions which create vertical space with the baseline the same as the baseline of the top or bottom copy, respectively, of the second argument. Thus `cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)' displays as: a - a b - a a b + - + - a b b - a b - b  File: calc.info, Node: Information about Compositions, Next: User-Defined Compositions, Prev: Other Compositions, Up: Compositions Information about Compositions .............................. The functions in this section are actual functions; they compose their arguments according to the current language and other display modes, then return a certain measurement of the composition as an integer. The `cwidth' function measures the width, in characters, of a composition. For example, `cwidth(a + b)' is 5, and `cwidth(a / b)' is 5 in normal mode, 1 in Big mode, and 11 in TeX mode (for `{a \over b}'). The argument may involve the composition functions described in this section. The `cheight' function measures the height of a composition. This is the total number of lines in the argument's printed form. The functions `cascent' and `cdescent' measure the amount of the height that is above (and including) the baseline, or below the baseline, respectively. Thus `cascent(X) + cdescent(X)' always equals `cheight(X)'. For a one-line formula like `a + b', `cascent' returns 1 and `cdescent' returns 0. For `a / b' in Big mode, `cascent' returns 2 and `cdescent' returns 1. The only formula for which `cascent' will return zero is `cvspace(0)' or equivalents.  File: calc.info, Node: User-Defined Compositions, Prev: Information about Compositions, Up: Compositions User-Defined Compositions ......................... The `Z C' (`calc-user-define-composition') command lets you define the display format for any algebraic function. You provide a formula containing a certain number of argument variables on the stack. Any time Calc formats a call to the specified function in the current language mode and with that number of arguments, Calc effectively replaces the function call with that formula with the arguments replaced. Calc builds the default argument list by sorting all the variable names that appear in the formula into alphabetical order. You can edit this argument list before pressing if you wish. Any variables in the formula that do not appear in the argument list will be displayed literally; any arguments that do not appear in the formula will not affect the display at all. You can define formats for built-in functions, for functions you have defined with `Z F' (*note Algebraic Definitions::.), or for functions which have no definitions but are being used as purely syntactic objects. You can define different formats for each language mode, and for each number of arguments, using a succession of `Z C' commands. When Calc formats a function call, it first searches for a format defined for the current language mode (and number of arguments); if there is none, it uses the format defined for the Normal language mode. If neither format exists, Calc uses its built-in standard format for that function (usually just `FUNC(ARGS)'). If you execute `Z C' with the number 0 on the stack instead of a formula, any defined formats for the function in the current language mode will be removed. The function will revert to its standard format. For example, the default format for the binomial coefficient function `choose(n, m)' in the Big language mode is n ( ) m You might prefer the notation, C n m To define this notation, first make sure you are in Big mode, then put the formula choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])]) on the stack and type `Z C'. Answer the first prompt with `choose'. The second prompt will be the default argument list of `(C m n)'. Edit this list to be `(n m)' and press . Now, try it out: For example, turn simplification off with `m O' and enter `choose(a,b) + choose(7,3)' as an algebraic entry. C + C a b 7 3 As another example, let's define the usual notation for Stirling numbers of the first kind, `stir1(n, m)'. This is just like the regular format for binomial coefficients but with square brackets instead of parentheses. choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")]) Now type `Z C stir1 ', edit the argument list to `(n m)', and type . The formula provided to `Z C' usually will involve composition functions, but it doesn't have to. Putting the formula `a + b + c' onto the stack and typing `Z C foo ' would define the function `foo(x,y,z)' to display like `x + y + z'. This "sum" will act exactly like a real sum for all formatting purposes (it will be parenthesized the same, and so on). However it will be computationally unrelated to a sum. For example, the formula `2 * foo(1, 2, 3)' will display as `2 (1 + 2 + 3)'. Operator precedences have caused the "sum" to be written in parentheses, but the arguments have not actually been summed. (Generally a display format like this would be undesirable, since it can easily be confused with a real sum.) The special function `eval' can be used inside a `Z C' composition formula to cause all or part of the formula to be evaluated at display time. For example, if the formula is `a + eval(b + c)', then `foo(1, 2, 3)' will be displayed as `1 + 5'. Evaluation will use the default simplifications, regardless of the current simplification mode. There are also `evalsimp' and `evalextsimp' which simplify as if by `a s' and `a e' (respectively). Note that these "functions" operate only in the context of composition formulas (and also in rewrite rules, where they serve a similar purpose; *note Rewrite Rules::.). On the stack, a call to `eval' will be left in symbolic form. It is not a good idea to use `eval' except as a last resort. It can cause the display of formulas to be extremely slow. For example, while `eval(a + b)' might seem quite fast and simple, there are several situations where it could be slow. For example, `a' and/or `b' could be polar complex numbers, in which case doing the sum requires trigonometry. Or, `a' could be the factorial `fact(100)' which is unevaluated because you have typed `m O'; `eval' will evaluate it anyway to produce a large, unwieldy integer. You can save your display formats permanently using the `Z P' command (*note Creating User Keys::.).  File: calc.info, Node: Syntax Tables, Prev: Compositions, Up: Language Modes Syntax Tables ------------- Syntax tables do for input what compositions do for output: They allow you to teach custom notations to Calc's formula parser. Calc keeps a separate syntax table for each language mode. (Note that the Calc "syntax tables" discussed here are completely unrelated to the syntax tables described in the Emacs manual.) The `Z S' (`calc-edit-user-syntax') command edits the syntax table for the current language mode. If you want your syntax to work in any language, define it in the normal language mode. Type `M-# M-#' to finish editing the syntax table, or `M-# x' to cancel the edit. The `m m' command saves all the syntax tables along with the other mode settings; *note General Mode Commands::.. * Menu: * Syntax Table Basics:: * Precedence in Syntax Tables:: * Advanced Syntax Patterns:: * Conditional Syntax Rules::  File: calc.info, Node: Syntax Table Basics, Next: Precedence in Syntax Tables, Prev: Syntax Tables, Up: Syntax Tables Syntax Table Basics ................... "Parsing" is the process of converting a raw string of characters, such as you would type in during algebraic entry, into a Calc formula. Calc's parser works in two stages. First, the input is broken down into "tokens", such as words, numbers, and punctuation symbols like `+', `:=', and `+/-'. Space between tokens is ignored (except when it serves to separate adjacent words). Next, the parser matches this string of tokens against various built-in syntactic patterns, such as "an expression followed by `+' followed by another expression" or "a name followed by `(', zero or more expressions separated by commas, and `)'." A "syntax table" is a list of user-defined "syntax rules", which allow you to specify new patterns to define your own favorite input notations. Calc's parser always checks the syntax table for the current language mode, then the table for the normal language mode, before it uses its built-in rules to parse an algebraic formula you have entered. Each syntax rule should go on its own line; it consists of a "pattern", a `:=' symbol, and a Calc formula with an optional "condition". (Syntax rules resemble algebraic rewrite rules, but the notation for patterns is completely different.) A syntax pattern is a list of tokens, separated by spaces. Except for a few special symbols, tokens in syntax patterns are matched literally, from left to right. For example, the rule, foo ( ) := 2+3 would cause Calc to parse the formula `4+foo()*5' as if it were `4+(2+3)*5'. Notice that the parentheses were written as two separate tokens in the rule. As a result, the rule works for both `foo()' and `foo ( )'. If we had written the rule as `foo () := 2+3', then Calc would treat `()' as a single, indivisible token, so that `foo( )' would not be recognized by the rule. (It would be parsed as a regular zero-argument function call instead.) In fact, this rule would also make trouble for the rest of Calc's parser: An unrelated formula like `bar()' would now be tokenized into `bar ()' instead of `bar ( )', so that the standard parser for function calls would no longer recognize it! While it is possible to make a token with a mixture of letters and punctuation symbols, this is not recommended. It is better to break it into several tokens, as we did with `foo()' above. The symbol `#' in a syntax pattern matches any Calc expression. On the righthand side, the things that matched the `#'s can be referred to as `#1', `#2', and so on (where `#1' matches the leftmost `#' in the pattern). For example, these rules match a user-defined function, prefix operator, infix operator, and postfix operator, respectively: foo ( # ) := myfunc(#1) foo # := myprefix(#1) # foo # := myinfix(#1,#2) # foo := mypostfix(#1) Thus `foo(3)' will parse as `myfunc(3)', and `2+3 foo' will parse as `mypostfix(2+3)'. It is important to write the first two rules in the order shown, because Calc tries rules in order from first to last. If the pattern `foo #' came first, it would match anything that could match the `foo ( # )' rule, since an expression in parentheses is itself a valid expression. Thus the `foo ( # )' rule would never get to match anything. Likewise, the last two rules must be written in the order shown or else `3 foo 4' will be parsed as `mypostfix(3) * 4'. (Of course, the best way to avoid these ambiguities is not to use the same symbol in more than one way at the same time! In case you're not convinced, try the following exercise: How will the above rules parse the input `foo(3,4)', if at all? Work it out for yourself, then try it in Calc and see.) Calc is quite flexible about what sorts of patterns are allowed. The only rule is that every pattern must begin with a literal token (like `foo' in the first two patterns above), or with a `#' followed by a literal token (as in the last two patterns). After that, any mixture is allowed, although putting two `#'s in a row will not be very useful since two expressions with nothing between them will be parsed as one expression that uses implicit multiplication. As a more practical example, Maple uses the notation `sum(a(i), i=1..10)' for sums, which Calc's Maple mode doesn't recognize at present. To handle this syntax, we simply add the rule, sum ( # , # = # .. # ) := sum(#1,#2,#3,#4) to the Maple mode syntax table. As another example, C mode can't read assignment operators like `++' and `*='. We can define these operators quite easily: # *= # := muleq(#1,#2) # ++ := postinc(#1) ++ # := preinc(#1) To complete the job, we would use corresponding composition functions and `Z C' to cause these functions to display in their respective Maple and C notations. (Note that the C example ignores issues of operator precedence, which are discussed in the next section.) You can enclose any token in quotes to prevent its usual interpretation in syntax patterns: # ":=" # := becomes(#1,#2) Quotes also allow you to include spaces in a token, although once again it is generally better to use two tokens than one token with an embedded space. To include an actual quotation mark in a quoted token, precede it with a backslash. (This also works to include backslashes in tokens.) # "bad token" # "/\"\\" # := silly(#1,#2,#3) This will parse `3 bad token 4 /"\ 5' to `silly(3,4,5)'. The token `#' has a predefined meaning in Calc's formula parser; it is not legal to use `"#"' in a syntax rule. However, longer tokens that include the `#' character are allowed. Also, while `"$"' and `"\""' are allowed as tokens, their presence in the syntax table will prevent those characters from working in their usual ways (referring to stack entries and quoting strings, respectively). Finally, the notation `%%' anywhere in a syntax table causes the rest of the line to be ignored as a comment.  File: calc.info, Node: Precedence in Syntax Tables, Next: Advanced Syntax Patterns, Prev: Syntax Table Basics, Up: Syntax Tables Precedence .......... Different operators are generally assigned different "precedences". By default, an operator defined by a rule like # foo # := foo(#1,#2) will have an extremely low precedence, so that `2*3+4 foo 5 == 6' will be parsed as `(2*3+4) foo (5 == 6)'. To change the precedence of an operator, use the notation `#/P' in place of `#', where P is an integer precedence level. For example, 185 lies between the precedences for `+' and `*', so if we change this rule to #/185 foo #/186 := foo(#1,#2) then `2+3 foo 4*5' will be parsed as `2+(3 foo (4*5))'. Also, because we've given the righthand expression slightly higher precedence, our new operator will be left-associative: `1 foo 2 foo 3' will be parsed as `(1 foo 2) foo 3'. By raising the precedence of the lefthand expression instead, we can create a right-associative operator. *Note Composition Basics::, for a table of precedences of the standard Calc operators. For the precedences of operators in other language modes, look in the Calc source file `calc-lang.el'.  File: calc.info, Node: Advanced Syntax Patterns, Next: Conditional Syntax Rules, Prev: Precedence in Syntax Tables, Up: Syntax Tables Advanced Syntax Patterns ........................ To match a function with a variable number of arguments, you could write foo ( # ) := myfunc(#1) foo ( # , # ) := myfunc(#1,#2) foo ( # , # , # ) := myfunc(#1,#2,#3) but this isn't very elegant. To match variable numbers of items, Calc uses some notations inspired regular expressions and the "extended BNF" style used by some language designers. foo ( { # }*, ) := apply(myfunc,#1) The token `{' introduces a repeated or optional portion. One of the three tokens `}*', `}+', or `}?' ends the portion. These will match zero or more, one or more, or zero or one copies of the enclosed pattern, respectively. In addition, `}*' and `}+' can be followed by a separator token (with no space in between, as shown above). Thus `{ # }*,' matches nothing, or one expression, or several expressions separated by commas. A complete `{ ... }' item matches as a vector of the items that matched inside it. For example, the above rule will match `foo(1,2,3)' to get `apply(myfunc,[1,2,3])'. The Calc `apply' function takes a function name and a vector of arguments and builds a call to the function with those arguments, so the net result is the formula `myfunc(1,2,3)'. If the body of a `{ ... }' contains several `#'s (or nested `{ ... }' constructs), then the items will be strung together into the resulting vector. If the body does not contain anything but literal tokens, the result will always be an empty vector. foo ( { # , # }+, ) := bar(#1) foo ( { { # }*, }*; ) := matrix(#1) will parse `foo(1,2,3,4)' as `bar([1,2,3,4])', and `foo(1,2;3,4)' as `matrix([[1,2],[3,4]])'. Also, after some thought it's easy to see how this pair of rules will parse `foo(1,2,3)' as `matrix([[1,2,3]])', since the first rule will only match an even number of arguments. The rule foo ( # { , # , # }? ) := bar(#1,#2) will parse `foo(2,3,4)' as `bar(2,[3,4])', and `foo(2)' as `bar(2,[])'. The notation `{ ... }?.' (note the trailing period) works just the same as regular `{ ... }?', except that it does not count as an argument; the following two rules are equivalent: foo ( # , { also }? # ) := bar(#1,#3) foo ( # , { also }?. # ) := bar(#1,#2) Note that in the first case the optional text counts as `#2', which will always be an empty vector, but in the second case no empty vector is produced. Another variant is `{ ... }?$', which means the body is optional only at the end of the input formula. All built-in syntax rules in Calc use this for closing delimiters, so that during algebraic entry you can type `[sqrt(2), sqrt(3 RET', omitting the closing parenthesis and bracket. Calc does this automatically for trailing `)', `]', and `>' tokens in syntax rules, but you can use `{ ... }?$' explicitly to get this effect with any token (such as `"}"' or `end'). Like `{ ... }?.', this notation does not count as an argument. Conversely, you can use quotes, as in `")"', to prevent a closing-delimiter token from being automatically treated as optional. Calc's parser does not have full backtracking, which means some patterns will not work as you might expect: foo ( { # , }? # , # ) := bar(#1,#2,#3) Here we are trying to make the first argument optional, so that `foo(2,3)' parses as `bar([],2,3)'. Unfortunately, Calc first tries to match `2,' against the optional part of the pattern, finds a match, and so goes ahead to match the rest of the pattern. Later on it will fail to match the second comma, but it doesn't know how to go back and try the other alternative at that point. One way to get around this would be to use two rules: foo ( # , # , # ) := bar([#1],#2,#3) foo ( # , # ) := bar([],#1,#2) More precisely, when Calc wants to match an optional or repeated part of a pattern, it scans forward attempting to match that part. If it reaches the end of the optional part without failing, it "finalizes" its choice and proceeds. If it fails, though, it backs up and tries the other alternative. Thus Calc has "partial" backtracking. A fully backtracking parser would go on to make sure the rest of the pattern matched before finalizing the choice.  File: calc.info, Node: Conditional Syntax Rules, Prev: Advanced Syntax Patterns, Up: Syntax Tables Conditional Syntax Rules ........................ It is possible to attach a "condition" to a syntax rule. For example, the rules foo ( # ) := ifoo(#1) :: integer(#1) foo ( # ) := gfoo(#1) will parse `foo(3)' as `ifoo(3)', but will parse `foo(3.5)' and `foo(x)' as calls to `gfoo'. Any number of conditions may be attached; all must be true for the rule to succeed. A condition is "true" if it evaluates to a nonzero number. *Note Logical Operations::, for a list of Calc functions like `integer' that perform logical tests. The exact sequence of events is as follows: When Calc tries a rule, it first matches the pattern as usual. It then substitutes `#1', `#2', etc., in the conditions, if any. Next, the conditions are simplified and evaluated in order from left to right, as if by the `a s' algebra command (*note Simplifying Formulas::.). Each result is true if it is a nonzero number, or an expression that can be proven to be nonzero (*note Declarations::.). If the results of all conditions are true, the expression (such as `ifoo(#1)') has its `#'s substituted, and that is the result of the parse. If the result of any condition is false, Calc goes on to try the next rule in the syntax table. Syntax rules also support `let' conditions, which operate in exactly the same way as they do in algebraic rewrite rules. *Note Other Features of Rewrite Rules::, for details. A `let' condition is always true, but as a side effect it defines a variable which can be used in later conditions, and also in the expression after the `:=' sign: foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x) The `dnumint' function tests if a value is numerically an integer, i.e., either a true integer or an integer-valued float. This rule will parse `foo' with a half-integer argument, like `foo(3.5)', to a call like `hifoo(4.)'. The lefthand side of a syntax rule `let' must be a simple variable, not the arbitrary pattern that is allowed in rewrite rules. The `matches' function is also treated specially in syntax rule conditions (again, in the same way as in rewrite rules). *Note Matching Commands::. If the matching pattern contains meta-variables, then those meta-variables may be used in later conditions and in the result expression. The arguments to `matches' are not evaluated in this situation. sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c]) This is another way to implement the Maple mode `sum' notation. In this approach, we allow `#2' to equal the whole expression `i=1..10'. Then, we use `matches' to break it apart into its components. If the expression turns out not to match the pattern, the syntax rule will fail. Note that `Z S' always uses Calc's normal language mode for editing expressions in syntax rules, so we must use regular Calc notation for the interval `[b..c]' that will correspond to the Maple mode interval `1..10'.  File: calc.info, Node: Modes Variable, Next: Calc Mode Line, Prev: Language Modes, Up: Mode Settings The `Modes' Variable ==================== The `m g' (`calc-get-modes') command pushes onto the stack a vector of numbers that describes the various mode settings that are in effect. With a numeric prefix argument, it pushes only the Nth mode, i.e., the Nth element of this vector. Keyboard macros can use the `m g' command to modify their behavior based on the current mode settings. The modes vector is also available in the special variable `Modes'. In other words, `m g' is like `s r Modes RET'. It will not work to store into this variable; in fact, if you do, `Modes' will cease to track the current modes. (The `m g' command will continue to work, however.) In general, each number in this vector is suitable as a numeric prefix argument to the associated mode-setting command. (Recall that the `~' key takes a number from the stack and gives it as a numeric prefix to the next command.) The elements of the modes vector are as follows: 1. Current precision. Default is 12; associated command is `p'. 2. Binary word size. Default is 32; associated command is `b w'. 3. Stack size (not counting the value about to be pushed by `m g'). This is zero if `m g' is executed with an empty stack. 4. Number radix. Default is 10; command is `d r'. 5. Floating-point format. This is the number of digits, plus the constant 0 for normal notation, 10000 for scientific notation, 20000 for engineering notation, or 30000 for fixed-point notation. These codes are acceptable as prefix arguments to the `d n' command, but note that this may lose information: For example, `d s' and `C-u 12 d s' have similar (but not quite identical) effects if the current precision is 12, but they both produce a code of 10012, which will be treated by `d n' as `C-u 12 d s'. If the precision then changes, the float format will still be frozen at 12 significant figures. 6. Angular mode. Default is 1 (degrees). Other values are 2 (radians) and 3 (HMS). The `m d' command accepts these prefixes. 7. Symbolic mode. Value is 0 or 1; default is 0. Command is `m s'. 8. Fraction mode. Value is 0 or 1; default is 0. Command is `m f'. 9. Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0. Command is `m p'. 10. Matrix/scalar mode. Default value is -1. Value is 0 for scalar mode, -2 for matrix mode, or N for NxN matrix mode. Command is `m v'. 11. Simplification mode. Default is 1. Value is -1 for off (`m O'), 0 for `m N', 2 for `m B', 3 for `m A', 4 for `m E', or 5 for `m U'. The `m D' command accepts these prefixes. 12. Infinite mode. Default is -1 (off). Value is 1 if the mode is on, or 0 if the mode is on with positive zeros. Command is `m i'. For example, the sequence `M-1 m g RET 2 + ~ p' increases the precision by two, leaving a copy of the old precision on the stack. Later, `~ p' will restore the original precision using that stack value. (This sequence might be especially useful inside a keyboard macro.) As another example, `M-3 m g 1 - ~ DEL' deletes all but the oldest (bottommost) stack entry. Yet another example: The HP-48 "round" command rounds a number to the current displayed precision. You could roughly emulate this in Calc with the sequence `M-5 m g 10000 % ~ c c'. (This would not work for fixed-point mode, but it wouldn't be hard to do a full emulation with the help of the `Z [' and `Z ]' programming commands. *Note Conditionals in Macros::.)  File: calc.info, Node: Calc Mode Line, Prev: Modes Variable, Up: Mode Settings The Calc Mode Line ================== This section is a summary of all symbols that can appear on the Calc mode line, the highlighted bar that appears under the Calc stack window (or under an editing window in Embedded Mode). The basic mode line format is: --%%-Calc: 12 Deg OTHER MODES (Calculator) The `%%' is the Emacs symbol for "read-only"; it shows that regular Emacs commands are not allowed to edit the stack buffer as if it were text. The word `Calc:' changes to `CalcEmbed:' if Embedded Mode is enabled. The words after this describe the various Calc modes that are in effect. The first mode is always the current precision, an integer. The second mode is always the angular mode, either `Deg', `Rad', or `Hms'. Here is a complete list of the remaining symbols that can appear on the mode line: `Alg' Algebraic mode (`m a'; *note Algebraic Entry::.). `Alg[(' Incomplete algebraic mode (`C-u m a'). `Alg*' Total algebraic mode (`m t'). `Symb' Symbolic mode (`m s'; *note Symbolic Mode::.). `Matrix' Matrix mode (`m v'; *note Matrix Mode::.). `MatrixN' Dimensioned matrix mode (`C-u N m v'). `Scalar' Scalar mode (`m v'; *note Matrix Mode::.). `Polar' Polar complex mode (`m p'; *note Polar Mode::.). `Frac' Fraction mode (`m f'; *note Fraction Mode::.). `Inf' Infinite mode (`m i'; *note Infinite Mode::.). `+Inf' Positive infinite mode (`C-u 0 m i'). `NoSimp' Default simplifications off (`m O'; *note Simplification Modes::.). `NumSimp' Default simplifications for numeric arguments only (`m N'). `BinSimpW' Binary-integer simplification mode; word size W (`m B', `b w'). `AlgSimp' Algebraic simplification mode (`m A'). `ExtSimp' Extended algebraic simplification mode (`m E'). `UnitSimp' Units simplification mode (`m U'). `Bin' Current radix is 2 (`d 2'; *note Radix Modes::.). `Oct' Current radix is 8 (`d 8'). `Hex' Current radix is 16 (`d 6'). `RadixN' Current radix is N (`d r'). `Zero' Leading zeros (`d z'; *note Radix Modes::.). `Big' Big language mode (`d B'; *note Normal Language Modes::.). `Flat' One-line normal language mode (`d O'). `Unform' Unformatted language mode (`d U'). `C' C language mode (`d C'; *note C FORTRAN Pascal::.). `Pascal' Pascal language mode (`d P'). `Fortran' FORTRAN language mode (`d F'). `TeX' TeX language mode (`d T'; *note TeX Language Mode::.). `Eqn' "Eqn" language mode (`d E'; *note Eqn Language Mode::.). `Math' Mathematica language mode (`d M'; *note Mathematica Language Mode::.). `Maple' Maple language mode (`d W'; *note Maple Language Mode::.). `NormN' Normal float mode with N digits (`d n'; *note Float Formats::.). `FixN' Fixed point mode with N digits after the point (`d f'). `Sci' Scientific notation mode (`d s'). `SciN' Scientific notation with N digits (`d s'). `Eng' Engineering notation mode (`d e'). `EngN' Engineering notation with N digits (`d e'). `LeftN' Left-justified display indented by N (`d <'; *note Justification::.). `Right' Right-justified display (`d >'). `RightN' Right-justified display with width N (`d >'). `Center' Centered display (`d ='). `CenterN' Centered display with center column N (`d ='). `WidN' Line breaking with width N (`d b'; *note Normal Language Modes::.). `Wide' No line breaking (`d b'). `Break' Selections show deep structure (`j b'; *note Making Selections::.). `Save' Record modes in `~/.emacs' (`m R'; *note General Mode Commands::.). `Local' Record modes in Embedded buffer (`m R'). `LocEdit' Record modes as editing-only in Embedded buffer (`m R'). `LocPerm' Record modes as permanent-only in Embedded buffer (`m R'). `Global' Record modes as global in Embedded buffer (`m R'). `Manual' Automatic recomputation turned off (`m C'; *note Automatic Recomputation::.). `Graph' GNUPLOT process is alive in background (*note Graphics::.). `Sel' Top-of-stack has a selection (Embedded only; *note Making Selections::.). `Dirty' The stack display may not be up-to-date (*note Display Modes::.). `Inv' "Inverse" prefix was pressed (`I'; *note Inverse and Hyperbolic::.). `Hyp' "Hyperbolic" prefix was pressed (`H'). `Keep' "Keep-arguments" prefix was pressed (`K'). `Narrow' Stack is truncated (`d t'; *note Truncating the Stack::.). In addition, the symbols `Active' and `~Active' can appear as minor modes on an Embedded buffer's mode line. *Note Embedded Mode::.  File: calc.info, Node: Arithmetic, Next: Scientific Functions, Prev: Mode Settings, Up: Top Arithmetic Functions ******************** This chapter describes the Calc commands for doing simple calculations on numbers, such as addition, absolute value, and square roots. These commands work by removing the top one or two values from the stack, performing the desired operation, and pushing the result back onto the stack. If the operation cannot be performed, the result pushed is a formula instead of a number, such as `2/0' (because division by zero is illegal) or `sqrt(x)' (because the argument `x' is a formula). Most of the commands described here can be invoked by a single keystroke. Some of the more obscure ones are two-letter sequences beginning with the `f' ("functions") prefix key. *Note Prefix Arguments::, for a discussion of the effect of numeric prefix arguments on commands in this chapter which do not otherwise interpret a prefix argument. * Menu: * Basic Arithmetic:: * Integer Truncation:: * Complex Number Functions:: * Conversions:: * Date Arithmetic:: * Financial Functions:: * Binary Functions::