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Newsgroups: comp.sources.misc From: daveg@synaptics.com (David Gillespie) Subject: v24i090: gnucalc - GNU Emacs Calculator, v2.00, Part42/56 Message-ID: <1991Nov1.183709.20885@sparky.imd.sterling.com> X-Md4-Signature: 89017c936cce42a4c6e60716bbf12c30 Date: Fri, 1 Nov 1991 18:37:09 GMT Approved: kent@sparky.imd.sterling.com Submitted-by: daveg@synaptics.com (David Gillespie) Posting-number: Volume 24, Issue 90 Archive-name: gnucalc/part42 Environment: Emacs Supersedes: gmcalc: Volume 13, Issue 27-45 ---- Cut Here and unpack ---- #!/bin/sh # do not concatenate these parts, unpack them in order with /bin/sh # file calc.texinfo continued # if test ! -r _shar_seq_.tmp; then echo 'Please unpack part 1 first!' exit 1 fi (read Scheck if test "$Scheck" != 42; then echo Please unpack part "$Scheck" next! exit 1 else exit 0 fi ) < _shar_seq_.tmp || exit 1 if test ! -f _shar_wnt_.tmp; then echo 'x - still skipping calc.texinfo' else echo 'x - continuing file calc.texinfo' sed 's/^X//' << 'SHAR_EOF' >> 'calc.texinfo' && lump-sum argument to these functions, but the computation will be rather slow in the four-argument case.@refill X @kindex H b # @tindex nperl The @kbd{H b #} [@code{nperl}] command does the same computation using @code{pvl}. By exchanging @var{payment} and @var{amount} you can also get the solution for @code{fvl}. For example, @code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a bank account earning 8%, it will take nine years to grow to $2000.@refill X @kindex b T @pindex calc-fin-rate @tindex rate The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes the rate of return on an investment. This is also an inverse of @code{pv}: @code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of @var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) = @var{amount}}.@refill X @kindex I b T @tindex H b T @tindex rateb @tindex ratel The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}] commands solve the analogous equations with @code{pvb} or @code{pvl} in place of @code{pv}. Also, @code{rate} and @code{rateb} can accept an optional fourth argument just like @code{pv} and @code{pvb}. To redo the above example from a different perspective, @code{ratel(9, 2000, 1000) = .08}, which says you will need an interest rate of 8% in order to double your account in nine years.@refill X @kindex b I @pindex calc-fin-irr @tindex irr The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the analogous function to @code{rate} but for net present value. Its argument is a vector of payments. Thus @code{irr(@var{payments})} computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0}; this rate is known as the @dfn{internal rate of return}. X @kindex I b I @tindex irrb The @kbd{I b I} [@code{irrb}] command computes the internal rate of return assuming payments occur at the beginning of each period. X @node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions @subsection Depreciation Functions X @noindent The functions in this section calculate @dfn{depreciation}, which is the amount of value that a possession loses over time. These functions are characterized by three parameters: @var{cost}, the original cost of the asset; @var{salvage}, the value the asset will have at the end of its expected ``useful life''; and @var{life}, the number of years (or other periods) of the expected useful life. X There are several methods for calculating depreciation that differ in the way they spread the depreciation over the lifetime of the asset. X @kindex b S @pindex calc-fin-sln @tindex sln The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the ``straight-line'' depreciation. In this method, the asset depreciates by the same amount every year (or period). For example, @samp{sln(12000, 2000, 5)} returns 2000. The asset costs $12000 initially and will be worth $2000 after five years; it loses $2000 per year. X @kindex b Y @pindex calc-fin-syd @tindex syd The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation is higher during the early years of the asset's life. Since the depreciation is different each year, @kbd{b Y} takes a fourth @var{period} parameter which specifies which year is requested, from 1 to @var{life}. If @var{period} is outside this range, the @code{syd} function will return zero. X @kindex b D @pindex calc-fin-ddb @tindex ddb The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an accelerated depreciation using the double-declining balance method. It also takes a fourth @var{period} parameter. X For symmetry, the @code{sln} function will accept a @var{period} parameter as well, although it will ignore its value except that the return value will as usual be zero if @var{period} is out of range. X For example, pushing the vector @cite{[1,2,3,4,5]} (using @kbd{v x 5}) and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$), ddb(12000,2000,5,$)] RET} produces a matrix that allows us to compare the three depreciation methods: X @group @example [ [ 2000, 3333, 4800 ] X [ 2000, 2667, 2880 ] X [ 2000, 2000, 1728 ] X [ 2000, 1333, 592 ] X [ 2000, 667, 0 ] ] @end example @end group X @noindent (Values have been rounded to nearest integers in this figure.) We see that @code{sln} depreciates by the same amount each year, @kbd{syd} depreciates more at the beginning and less at the end, and @kbd{ddb} weights the depreciation even more toward the beginning. X Summing columns with @kbd{V R : +} yields @cite{[10000, 10000, 10000]}; the total depreciation in any method is (by definition) the difference between the cost and the salvage value. X @node Definitions of Financial Functions, , Depreciation Functions, Financial Functions @subsection Definitions X @noindent For your reference, here are the actual formulas used to compute Calc's financial functions. X Calc will not evaluate a financial function unless the @var{rate} or @var{n} argument is known. However, @var{payment} or @var{amount} can be a variable. Calc expands these functions according to the formulas below for symbolic arguments only when you use the @kbd{a "} (@code{calc-expand-formula}) command, or when taking derivatives or integrals or solving equations involving the functions. X @ifinfo These formulas are shown using the conventions of ``Big'' display mode (@kbd{d B}); for example, the formula for @code{fv} written linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}. X @example X n X (1 + rate) - 1 fv(rate, n, pmt) = pmt * --------------- X rate X X n X ((1 + rate) - 1) (1 + rate) fvb(rate, n, pmt) = pmt * ---------------------------- X rate X X n fvl(rate, n, pmt) = pmt * (1 + rate) X X -n X 1 - (1 + rate) pv(rate, n, pmt) = pmt * ---------------- X rate X X -n X (1 - (1 + rate) ) (1 + rate) pvb(rate, n, pmt) = pmt * ----------------------------- X rate X X -n pvl(rate, n, pmt) = pmt * (1 + rate) X X -1 -2 -3 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate) X X -1 -2 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate) X X -n X (amt - x * (1 + rate) ) * rate pmt(rate, n, amt, x) = ------------------------------- X -n X 1 - (1 + rate) X X -n X (amt - x * (1 + rate) ) * rate pmtb(rate, n, amt, x) = ------------------------------- X -n X (1 - (1 + rate) ) (1 + rate) X X amt * rate nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate) X pmt X X amt * rate nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate) X pmt * (1 + rate) X X amt nperl(rate, pmt, amt) = - log(---, 1 + rate) X pmt X X 1/n X pmt ratel(n, pmt, amt) = ------ - 1 X 1/n X amt X X cost - salv sln(cost, salv, life) = ----------- X life X X (cost - salv) * (life - per + 1) syd(cost, salv, life, per) = -------------------------------- X life * (life + 1) / 2 X X book * 2 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far X life @end example @end ifinfo @tex \turnoffactive $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$ $$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$ $$ \code{fvl}(r, n, p) = p (1 + r)^n $$ $$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$ $$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$ $$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$ $$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$ $$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$ $$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$ $$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over X (1 - (1 + r)^{-n}) (1 + r) } $$ $$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$ $$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$ $$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$ $$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$ $$ \code{sln}(c, s, l) = { c - s \over l } $$ $$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$ $$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$ @end tex X In @code{pmt} and @code{pmtb}, @cite{x=0} if omitted. X These functions accept any numeric objects, including error forms, intervals, and even (though not very usefully) complex numbers. The above formulas specify exactly the behavior of these functions with all sorts of inputs. X Note that if the first argument to the @code{log} in @code{nper} is negative, @code{nper} leaves itself in symbolic form rather than returning a (financially meaningless) complex number. X @samp{rate(num, pmt, amt)} solves the equation @samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R} (@code{calc-find-root}), with the interval @samp{[.01% .. 100%]} for an initial guess. The @code{rateb} function is the same except that it uses @code{pvb}. Note that @code{ratel} can be solved directly; its formula is shown in the above list. X Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0} for @samp{rate}. X If you give a fourth argument to @code{nper} or @code{nperb}, Calc will also use @kbd{H a R} to solve the equation using an initial guess interval of @samp{[0 .. 100]}. X A fourth argument to @code{fv} simply sums the two components calculated from the above formulas for @code{fv} and @code{fvl}. The same is true of @code{fvb}, @code{pv}, and @code{pvb}. X The @kbd{ddb} function is computed iteratively; the ``book'' value starts out equal to @var{cost}, and decreases according to the above formula for the specified number of periods. If the book value would decrease below @var{salvage}, it only decreases to @var{salvage} and the depreciation is zero for all subsequent periods. The @code{ddb} function returns the amount the book value decreased in the specified period. X The Calc financial function names were borrowed mostly from Microsoft Excel and Borland's Quattro. The @code{ratel} function corresponds to @samp{@@CGR} in Borland's Reflex. The @code{nper} and @code{nperl} functions correspond to @samp{@@TERM} and @samp{@@CTERM} in Quattro, respectively. Beware that the Calc functions may take their arguments in a different order than your favorite spreadsheet. X @node Binary Functions, , Financial Functions, Arithmetic @section Binary Number Functions X @noindent The commands in this chapter all use two-letter sequences beginning with the @kbd{b} prefix. X @cindex Binary numbers The ``binary'' operations actually work regardless of the currently displayed radix, although their results make the most sense in a radix like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @kbd{d 6} commands, respectively). You may also wish to enable display of leading zeros with @kbd{d z}. @xref{Radix Modes}. X @cindex Word size for binary operations The Calculator maintains a current @dfn{word size} @cite{w}, an arbitrary positive or negative integer. For a positive word size, all of the binary operations described here operate modulo @cite{2^w}. In particular, negative arguments are converted to positive integers modulo @cite{2^w} by all binary functions.@refill X If the word size is negative, binary operations produce 2's complement integers from @c{$-2^{-w-1}$} @cite{-(2^(-w-1))} to @c{$2^{-w-1}-1$} @cite{2^(-w-1)-1} inclusive. Either mode accepts inputs in any range; the sign of @cite{w} affects only the results produced. X @kindex b c @pindex calc-clip @tindex clip The @kbd{b c} (@code{calc-clip}) [@code{clip}] command can be used to clip a number by reducing it modulo @cite{2^w}. The commands described in this chapter automatically clip their results to the current word size. Note that other operations like addition do not use the current word size, since integer addition generally is not ``binary.'' (However, @pxref{Simplification Modes}, @code{calc-bin-simplify-mode}.) For example, with a word size of 8 bits @kbd{b c} converts a number to the range 0 to 255; with a word size of -8 @kbd{b c} converts to the range -128 to 127.@refill X @kindex b w @pindex calc-word-size The default word size is 32 bits. All operations except the shifts and rotates allow you to specify a different word size for that one operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the top of stack to the range 0 to 255 regardless of the current word size. To set the word size permanently, use @kbd{b w} (@code{calc-word-size}). This command displays a prompt with the current word size; press @key{RET} immediately to keep this word size, or type a new word size at the prompt. X When the binary operations are written in symbolic form, they take an optional second (or third) word-size parameter. When a formula like @samp{and(a,b)} is finally evaluated, the word size current at that time will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of @i{-8} will always be used. A symbolic binary function will be left in symbolic form unless the all of its argument(s) are integers or integer-valued floats. X If either or both arguments are modulo forms for which @cite{M} is a power of two, that power of two is taken as the word size unless a numeric prefix argument overrides it. The current word size is never consulted when modulo-power-of-two forms are involved. X @kindex b a @pindex calc-and @tindex and The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise AND of the two numbers on the top of the stack. In other words, for each of the @cite{w} binary digits of the two numbers (pairwise), the corresponding bit of the result is 1 if and only if both input bits are 1: @samp{and(2#1100, 2#1010) = 2#1000}. X @kindex b o @pindex calc-or @tindex or The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise inclusive OR of two numbers. A bit is 1 if either of the input bits, or both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}. X @kindex b x @pindex calc-xor @tindex xor The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise exclusive OR of two numbers. A bit is 1 if exactly one of the input bits is 1: @samp{xor(2#1100, 2#1010) = 2#0110}. X @kindex b d @pindex calc-diff @tindex diff The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))}, so that @samp{diff(2#1100, 2#1010) = 2#0100}. X @kindex b n @pindex calc-not @tindex not The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise NOT of a number. A bit is 1 if the input bit is 0 and vice-versa. X @kindex b l @pindex calc-lshift-binary @tindex lsh The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a number left by one bit, or by the number of bits specified in the numeric prefix argument. A negative prefix argument performs a logical right shift, in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)} is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}. Bits shifted ``off the end,'' according to the current word size, are lost. X @kindex H b l The @kbd{H b l} command also does a left shift, but it takes two arguments from the stack (the value to shift, and, at top-of-stack, the number of bits to shift). This version interprets the prefix argument just like the regular binary operations, i.e., as a word size. The Hyperbolic flag has a similar effect on the rest of the binary shift and rotate commands. X @kindex b r @kindex H b r @pindex calc-rshift-binary @tindex rsh The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a number right by one bit, or by the number of bits specified in the numeric prefix argument: @samp{rsh(a,n) = lsh(a,-n)}. X @kindex b L @kindex H b L @pindex calc-lshift-arith @tindex ash The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a number left. It is analogous to @code{lsh}, except that if the shift is rightward (the prefix argument is negative), an arithmetic shift is performed as described below. X @kindex b R @kindex H b R @pindex calc-rshift-arith @tindex rash The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs an ``arithmetic'' shift to the right, in which the leftmost bit (according to the current word size) is duplicated rather than shifting in zeros. This corresponds to dividing by a power of two where the input is interpreted as a signed, twos-complement number. (The distinction between the @samp{rsh} and @samp{rash} operations is totally independent from whether the word size is positive or negative.) With a negative prefix argument, this performs a standard left shift. X @kindex b t @kindex H b t @pindex calc-rotate-binary @tindex rot The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a number one bit to the left. The leftmost bit (according to the current word size) is dropped off the left and shifted in on the right. With a numeric prefix argument, the number is rotated that many bits to the left or right. X @xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that pack and unpack binary integers into sets. (For example, @kbd{b u} unpacks the number @samp{2#11001} to the set of bit-numbers @samp{[0, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1'' bits in a binary integer. X Another interesting use of the set representation of binary integers is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to unpack; type @kbd{31 TAB -} to replace each bit-number in the set with 31 minus that bit-number; type @kbd{b p} to pack the set back into a binary integer. X @node Scientific Functions, Matrix Functions, Arithmetic, Top @chapter Scientific Functions X @noindent The functions described here perform trigonometric and other transcendental calculations. They generally produce floating-point answers correct to the full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse) flag keys must be used to get some of these functions from the keyboard. X @kindex P @pindex calc-pi @cindex @code{pi} variable @vindex pi @kindex H P @cindex @code{e} variable @vindex e @kindex I P @cindex @code{gamma} variable @vindex gamma @cindex Gamma constant, Euler's @cindex Euler's gamma constant @kindex H I P @cindex @code{phi} variable @cindex Phi, golden ratio @cindex Golden ratio One miscellanous command is shift-@kbd{P} (@code{calc-pi}), which pushes the value of @c{$\pi$} @cite{pi} (at the current precision) onto the stack. With the Hyperbolic flag, it pushes the value @cite{e}, the base of natural logarithms. With the Inverse flag, it pushes Euler's constant @c{$\gamma$} @cite{gamma} (about 0.577). With both Inverse and Hyperbolic, it pushes the ``golden ratio'' @c{$\phi$} @cite{phi} (about 1.618). (At present, Euler's constant is not available to unlimited precision; Calc knows only the first 100 digits.) In Symbolic mode, these commands push the actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi}, respectively, instead of their values; @pxref{Symbolic Mode}.@refill X @tindex sqr The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere; @pxref{Basic Arithmetic}. With the Inverse flag [@code{sqr}], this command computes the square of the argument. X @xref{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. X @menu * Logarithmic Functions:: * Trigonometric and Hyperbolic Functions:: * Advanced Math Functions:: * Branch Cuts:: * Random Numbers:: * Combinatorial Functions:: * Probability Distribution Functions:: @end menu X @node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions @section Logarithmic Functions X @noindent @kindex L @kindex I E @pindex calc-ln @tindex ln The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural logarithm of the real or complex number on the top of the stack. With the Inverse flag it computes the exponential function instead, although this is redundant with the @kbd{E} command. X @kindex E @kindex I L @pindex calc-exp @tindex exp The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the exponential, i.e., @cite{e} raised to the power of the number on the stack. The meanings of the Inverse and Hyperbolic flags follow from those for the @code{calc-ln} command. X @kindex H L @kindex H I L @kindex H E @kindex H I E @pindex calc-log10 @tindex log10 @tindex exp10 The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common (base-10) logarithm of a number. The meanings of the Inverse [@code{exp10}] and Hyperbolic flags follow from those for the @kbd{L} command. Note that the common logarithm of a complex number is computed by taking the natural logarithm and dividing by @c{$\ln10$} @cite{ln(10)}. X @kindex B @kindex I B @pindex calc-log @tindex log @tindex alog The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since @c{$2^{10} = 1024$} @cite{2^10 = 1024}. In certain cases like @samp{log(3,9)}, the result will be either @cite{1:2} or @cite{0.5} depending on the current Fraction Mode setting. With the Inverse flag [@code{alog}], this command is similar to @kbd{^} except that the order of the arguments is reversed. X @kindex f I @pindex calc-ilog @tindex ilog The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the integer logarithm of a number to any base. The number and the base must themselves be positive integers. This is the true logarithm, rounded down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @cite{x} in the range from 1000 to 9999. If both arguments are positive integers, exact integer arithmetic is used; otherwise, this is equivalent to @samp{floor(log(x,b))}. X @kindex f E @pindex calc-expm1 @tindex expm1 The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes @c{$e^x - 1$} @cite{exp(x)-1}, but using an algorithm that produces a more accurate answer when the result is close to zero, i.e., when @c{$e^x$} @cite{exp(x)} is close to one. X @kindex f L @pindex calc-lnp1 @tindex lnp1 The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes @c{$\ln(x+1)$} @cite{ln(x+1)}, producing a more accurate answer when @cite{x} is close to zero. X @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions @section Trigonometric/Hyperbolic Functions X @noindent @kindex S @pindex calc-sin @tindex sin The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine of an angle or complex number. If the input is an HMS form, it is interpreted as degrees-minutes-seconds; otherwise, the input is interpreted according to the current angular mode. It is best to use Radians mode when operating on complex numbers.@refill X Calc's ``units'' mechanism includes angular units like @code{deg}, @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated all the time, the @kbd{u s} (@code{calc-simplify-units}) command will simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless of the current angular mode. @xref{Basic Operations on Units}. X Also, the symbolic variable @code{pi} is not ordinarily recognized in arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but the @kbd{a s} (@code{calc-simplify}) command recognizes many such formulas when the current angular mode is radians @emph{and} symbolic mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}. @xref{Symbolic Mode}. Beware, this simplification occurs even if you have stored a different value in the variable @samp{pi}; this is one reason why changing built-in variables is a bad idea. Arguments of the form @cite{x} plus a multiple of @c{$\pi/2$} @cite{pi/2} are also simplified. Calc includes similar formulas for @code{cos} and @code{tan}.@refill X The @kbd{a s} command knows all angles which are integer multiples of @c{$\pi/12$} @cite{pi/12}, @c{$\pi/10$} @cite{pi/10}, or @c{$\pi/8$} @cite{pi/8} radians. In degrees mode, analogous simplifications occur for integer multiples of 15 or 18 degrees, and for arguments plus multiples of 90 degrees. X @kindex I S @pindex calc-arcsin @tindex arcsin With the Inverse flag, @code{calc-sin} computes an arcsine. This is also available as the @code{calc-arcsin} command or @code{arcsin} algebraic function. The returned argument is converted to degrees, radians, or HMS notation depending on the current angular mode. X @kindex H S @pindex calc-sinh @tindex sinh @kindex H I S @pindex calc-arcsinh @tindex arcsinh With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic sine, also available as @code{calc-sinh} [@code{sinh}]. With the Hyperbolic and Inverse flags, it computes the hyperbolic arcsine (@code{calc-arcsinh}) [@code{arcsinh}]. X @kindex C @pindex calc-cos @tindex cos @kindex I C @pindex calc-arccos @tindex arccos @kindex H C @pindex calc-cosh @tindex cosh @kindex H I C @pindex calc-arccosh @tindex arccosh @kindex T @pindex calc-tan @tindex tan @kindex I T @pindex calc-arctan @tindex arctan @kindex H T @pindex calc-tanh @tindex tanh @kindex H I T @pindex calc-arctanh @tindex arctanh The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}] computes the tangent, along with all the various inverse and hyperbolic variants of these functions. X @kindex f T @pindex calc-arctan2 @tindex arctan2 The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two numbers from the stack and computes the arc tangent of their ratio. The result is in the full range from @i{-180} (exclusive) to @i{+180} (inclusive) degrees, or the analogous range in radians. A similar result would be obtained with @kbd{/} followed by @kbd{I T}, but the value would only be in the range from @i{-90} to @i{+90} degrees since the division loses information about the signs of the two components, and an error might result from an explicit division by zero which @code{arctan2} would avoid. By (arbitrary) definition, @samp{arctan2(0,0)=0}. X @pindex calc-sincos @tindex sincos @tindex arcsincos The @code{calc-sincos} [@code{sincos}] command computes the sine and cosine of a number, returning them as a vector of the form @samp{[@var{cos}, @var{sin}]}. With the Inverse flag [@code{arcsincos}], this command takes a two-element vector as an argument and computes @code{arctan2} of the elements. (This command does not accept the Hyperbolic flag.)@refill X @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions @section Advanced Mathematical Functions X @noindent Calc can compute a variety of less common functions that arise in various branches of mathematics. All of the functions described in this section allow arbitrary complex arguments and, except as noted, will work to arbitrarily large precisions. They can not at present handle error forms or intervals as arguments. X NOTE: These functions are still experimental. In particular, their accuracy is not guaranteed in all domains. It is advisable to set the current precision comfortably higher than you actually need when using these functions. Also, these functions may be impractically slow for some values of the arguments. X @kindex f g @pindex calc-gamma @tindex gamma The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler gamma function. For positive integer arguments, this is related to the factorial function: @samp{gamma(n+1) = fact(n)}. For general complex arguments the gamma function can be defined by the following definite integral: @c{$\Gamma(a) = \int_0^\infty t^{a-1} e^t dt$} @cite{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}. (The actual implementation uses far more efficient computational methods.) X @kindex f G @kindex I f G @kindex H f G @kindex H I f G @pindex calc-inc-gamma @tindex gammaP @tindex gammaQ @tindex gammag @tindex gammaG The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by the integral, @c{$P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)$} @cite{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}. This implies that @samp{gammaP(a,inf) = 1} for any @cite{a} (see the definition of the normal gamma function). X Several other varieties of incomplete gamma function are defined. The complement of @cite{P(a,x)}, called @cite{Q(a,x) = 1-P(a,x)} by some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command. You can think of this as taking the other half of the integral, from @cite{x} to infinity. X @ifinfo The functions corresponding to the integrals that define @cite{P(a,x)} and @cite{Q(a,x)} but without the normalizing @cite{1/gamma(a)} factor are called @cite{g(a,x)} and @cite{G(a,x)}, respectively (where @cite{g} and @cite{G} represent the lower- and upper-case Greek letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}] and @kbd{H I f G} [@code{gammaG}] commands. @end ifinfo @tex \turnoffactive The functions corresponding to the integrals that define $P(a,x)$ and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$ factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively. You can obtain these using the \kbd{H f G} [\code{gammag}] and \kbd{I H f G} [\code{gammaG}] commands. @end tex X @kindex f b @pindex calc-beta @tindex beta The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the Euler beta function, which is defined in terms of the gamma function as @c{$B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)$} @cite{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)}, or by @c{$B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt$} @cite{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}. X @kindex f B @kindex H f B @pindex calc-inc-beta @tindex betaI @tindex betaB The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes the incomplete beta function @cite{I(x,a,b)}. It is defined by @c{$I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)$} @cite{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}. Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding un-normalized version [@code{betaB}]. X @kindex f e @kindex I f e @pindex calc-erf @tindex erf @tindex erfc The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the error function @c{$\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt$} @cite{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}. The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}] is the corresponding integral from @samp{x} to infinity; the sum @c{$\hbox{erf}(x) + \hbox{erfc}(x) = 1$} @cite{erf(x) + erfc(x) = 1}. X @kindex f j @kindex f y @pindex calc-bessel-J @pindex calc-bessel-Y @tindex besJ @tindex besY The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y} (@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel functions of the first and second kinds, respectively. In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter @cite{n} is often an integer, but is not required to be one. Calc's implementation of the Bessel functions currently limits the precision to 8 digits, and may not be exact even to that precision. Use with care!@refill X @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions @section Branch Cuts and Principal Values X @noindent @cindex Branch cuts @cindex Principal values All of the logarithmic, trigonometric, and other scientific functions are defined for complex numbers as well as for reals. This section describes the values returned in cases where the general result is a family of possible values. Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language}, second edition, in these matters. This section will describe each function briefly; for a more detailed discussion (including some nifty diagrams), consult Steele's book. X Note that the branch cuts for @code{arctan} and @code{arctanh} were changed between the first and second editions of Steele. Versions of Calc starting with 2.00 follow the second edition. X The new branch cuts exactly match those of the HP-28/48 calculators. They also match those of Mathematica 1.2, except that Mathematica's @code{arctan} cut is always in the right half of the complex plane, and its @code{arctanh} cut is always in the top half of the plane. Calc's cuts are continuous with quadrants I and III for @code{arctan}, or II and IV for @code{arctanh}. X Note: The current implementations of these functions with complex arguments are designed with proper behavior around the branch cuts in mind, @emph{not} efficiency or accuracy. You may need to increase the floating precision and wait a while to get suitable answers from them. X For @samp{sqrt(a+bi)}: When @cite{a<0} and @cite{b} is small but positive or zero, the result is close to the @cite{+i} axis. For @cite{b} small and negative, the result is close to the @cite{-i} axis. The result always lies in the right half of the complex plane. X For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}. The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}. Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the negative real axis. X The following table describes these branch cuts in another way. If the real and imaginary parts of @cite{z} are as shown, then the real and imaginary parts of @cite{f(z)} will be as shown. Here @code{eps} stands for a small positive value; each occurrence of @code{eps} may stand for a different small value. X @smallexample X z sqrt(z) ln(z) ---------------------------------------- X +, 0 +, 0 any, 0 X -, 0 0, + any, pi X -, +eps +eps, + +eps, + X -, -eps +eps, - +eps, - @end smallexample X For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}. One interesting consequence of this is that @samp{(-8)^1:3} does not evaluate to @i{-2} as you might expect, but to the complex number @cite{(1., 1.732)}. Both of these are valid cube roots of @i{-8} (as is @cite{(1., -1.732)}); Calc chooses a perhaps less-obvious root for the sake of mathematical consistency. X For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}. The branch cuts are on the real axis, less than @i{-1} and greater than 1. X For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))}, or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on the real axis, less than @i{-1} and greater than 1. X For @samp{arctan(z)}: This is defined by @samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the imaginary axis, below @cite{-i} and above @cite{i}. X For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}. The branch cuts are on the imaginary axis, below @cite{-i} and above @cite{i}. X For @samp{arccosh(z)}: This is defined by @samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the real axis less than 1. X For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}. The branch cuts are on the real axis, less than @i{-1} and greater than 1. X The following entries for @code{arcsin}, @code{arccos}, and @code{arctan} assume the current angular mode is radians. The hyperbolic functions operate independently of the angular mode. X @smallexample X z arcsin(z) arccos(z) ------------------------------------------------------- X (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0 X (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps X (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps X <-1, 0 -pi/2, + pi, - X <-1, +eps -pi/2 + eps, + pi - eps, - X <-1, -eps -pi/2 + eps, - pi - eps, + X >1, 0 pi/2, - 0, + X >1, +eps pi/2 - eps, + +eps, - X >1, -eps pi/2 - eps, - +eps, + @end smallexample X @smallexample X z arccosh(z) arctanh(z) ----------------------------------------------------- X (-1..1), 0 0, (0..pi) any, 0 X (-1..1), +eps +eps, (0..pi) any, +eps X (-1..1), -eps +eps, (-pi..0) any, -eps X <-1, 0 +, pi -, pi/2 X <-1, +eps +, pi - eps -, pi/2 - eps X <-1, -eps +, -pi + eps -, -pi/2 + eps X >1, 0 +, 0 +, -pi/2 X >1, +eps +, +eps +, pi/2 - eps X >1, -eps +, -eps +, -pi/2 + eps @end smallexample X @smallexample X z arcsinh(z) arctan(z) ----------------------------------------------------- X 0, (-1..1) 0, (-pi/2..pi/2) 0, any X 0, <-1 -, -pi/2 -pi/2, - X +eps, <-1 +, -pi/2 + eps pi/2 - eps, - X -eps, <-1 -, -pi/2 + eps -pi/2 + eps, - X 0, >1 +, pi/2 pi/2, + X +eps, >1 +, pi/2 - eps pi/2 - eps, + X -eps, >1 -, pi/2 - eps -pi/2 + eps, + @end smallexample X Finally, the following identities help to illustrate the relationship between the complex trigonometric and hyperbolic functions. They are valid everywhere, including on the branch cuts. X @smallexample sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z) cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z) tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z) sinh(i*z) = i*sin(z) cosh(i*z) = cos(z) @end smallexample X The ``advanced math'' functions (gamma, Bessel, etc.) are also defined for general complex arguments, but their branch cuts and principal values are not rigorously specified at present. X @node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions @section Random Numbers X @noindent @kindex k r @pindex calc-random @tindex random The @kbd{k r} (@code{calc-random}) [@code{random}] command produces random numbers of various sorts. X Given a positive numeric prefix argument @cite{M}, it produces a random integer @cite{N} in the range @c{$0 \le N < M$} @cite{0 <= N < M}. Each of the @cite{M} values appears with equal probability.@refill X With no numeric prefix argument, the @kbd{k r} command takes its argument from the stack instead. Once again, if this is a positive integer @cite{M} the result is a random integer less than @cite{M}. However, note that while numeric prefix arguments are limited to six digits or so, an @cite{M} taken from the stack can be arbitrarily large. If @cite{M} is negative, the result is a random integer in the range @c{$M < N \le 0$} @cite{M < N <= 0}. X If the value on the stack is a floating-point number @cite{M}, the result is a random floating-point number @cite{N} in the range @c{$0 \le N < M$} @cite{0 <= N < M} or @c{$M < N \le 0$} @cite{M < N <= 0}, according to the sign of @cite{M}. X If @cite{M} is zero, the result is a Gaussian-distributed random real number; the distribution has a mean of zero and a standard deviation of one. The algorithm used generates random numbers in pairs; thus, every other call to this function will be especially fast. X If @cite{M} is an error form @c{$m$ @code{+/-} $\sigma$} @samp{@var{m} +/- @var{s}} where @var{m} and @c{$\sigma$} @var{s} are both real numbers, the result uses a Gaussian distribution with mean @var{m} and standard deviation @c{$\sigma$} @var{s}. X If @cite{M} is an interval form, the lower and upper bounds specify the acceptable limits of the random numbers. If both bounds are integers, the result is a random integer in the specified range. If either bound is floating-point, the result is a random real number in the specified range. If the interval is open at enter end, the result will be sure not to equal that end value. (This makes a big difference for integer intervals, but for floating-point intervals it's relatively minor: with a precision of 6, @samp{random([1.0..2.0))} will return any of one million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may additionally return 2.00000, but the probability of this happening is extremely small.) X If @cite{M} is a vector, the result is one element taken at random from the vector. All elements of the vector are given equal probabilities. X @vindex RandSeed The sequence of numbers produced by @kbd{k r} is completely random by default, i.e., the sequence is seeded each time you start Calc using the current time and other information. You can get a reproducible sequence by storing a particular ``seed value'' in the Calc variable @code{RandSeed}. Any integer will do for a seed; integers of from 1 to 12 digits are good. If you later store a different integer into @code{RandSeed}, Calc will switch to a different pseudo-random sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself from the current time. If you store the same integer that you used before back into @code{RandSeed}, you will get the exact same sequence of random numbers as before. X @pindex calc-rrandom The @code{calc-rrandom} command produces a random real number between zero and one. It is equivalent to @samp{random(1.0)}. X @kindex k a @pindex calc-random-again The @kbd{k a} (@code{calc-random-again}) command produces another random number, re-using the most recent value of @cite{M}. With a numeric prefix argument @var{n}, it produces @var{n} more random numbers using that value of @cite{M}. X @kindex k h @pindex calc-shuffle @tindex shuffle The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several random values with no duplicates. The value on the top of the stack specifies the set from which the random values are drawn, and may be any of the @cite{M} formats described above. The numeric prefix argument gives the length of the desired list. (If you do not provide a numeric prefix argument, the length of the list is taken from the top of the stack, and @cite{M} from second-to-top.) X If @cite{M} is a floating-point number, zero, or an error form (so that the random values are being drawn from the set of real numbers) there is little practical difference between using @kbd{k h} and using @kbd{k r} several times. But if the set of possible values consists of just a few integers, or the elements of a vector, then there is a very real chance that multiple @kbd{k r}'s will produce the same number more than once. The @kbd{k h} command produces a vector whose elements are always distinct. (Actually, there is a slight exception: If @cite{M} is a vector, no given vector element will be drawn more than once, but if several elements of @cite{M} are equal, they may each make it into the result vector.) X One use of @kbd{k h} is to rearrange a list at random. This happens if the prefix argument is equal to the number of values in the list: @kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list @samp{[2.5, 1, 1.5, 3, 2]}. As a convenient feature, if the argument @var{n} is negative it is replaced by the size of the set represented by @cite{M}. Naturally, this is allowed only when @cite{M} specifies a small discrete set of possibilities. X To do the equivalent of @kbd{k h} but with duplications allowed, given @cite{M} on the stack and with @var{n} just entered as a numeric prefix, use @kbd{v b} to build a vector of copies of @cite{M}, then use @kbd{V M k r} to ``map'' the normal @kbd{k r} function over the elements of this vector. @xref{Matrix Functions}. X @menu * Random Number Generator:: (Complete description of Calc's algorithm) @end menu X @node Random Number Generator, , Random Numbers, Random Numbers @subsection Random Number Generator X Calc's random number generator uses several methods to ensure that the numbers it produces are highly random. Knuth's @emph{Art of Computer Programming}, Volume II, contains a thorough description of the theory of random number generators and their measurement and characterization. X If @code{RandSeed} has no stored value, Calc calls Emacs' built-in @code{random} function to get a stream of random numbers, which it then treats in various ways to avoid problems inherent in the simple random number generators that many systems use to implement @code{random}. X When Calc's random number generator is first invoked, it ``seeds'' the low-level random sequence using the time of day, so that the random number sequence will be different every time you use Calc. X Since Emacs Lisp doesn't specify the range of values that will be returned by its @code{random} function, Calc exercises the function several times to estimate the range. When Calc subsequently uses the @code{random} function, it takes only 10 bits of the result near the most-significant end. (It avoids at least the bottom four bits, preferably more, and also tries to avoid the top two bits.) This strategy works well with the linear congruential generators that are typically used to implement @code{random}. X If @code{RandSeed} contains an integer, Calc uses this integer to seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A, computing @c{$X_{n-55} - X_{n-24}$} @cite{X_n-55 - X_n-24}). This method expands the seed value into a large table which is maintained internally; the variable @code{RandSeed} is changed from, e.g., 42 to the vector @cite{[42]} to indicate that the seed has been absorbed into this table. When @code{RandSeed} contains a vector, @kbd{k r} and related commands continue to use the same internal table as last time. There is no way to extract the complete state of the random number generator so that you can restart it from any point; you can only restart it from the same initial seed value. A simple way to restart from the same seed is to type @kbd{s r RandSeed} to get the seed vector, @kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed} to reseed the generator with that number. X Calc uses a ``shuffling'' method as described in algorithm 3.2.2B of Knuth. It fills a table with 13 random 10-bit numbers. Then, to generate a new random number, it uses the previous number to index into the table, picks the value it finds there as the new random number, then replaces that table entry with a new value obtained from a call to the base random number generator (either the additive congruential generator or the @code{random} function supplied by the system). If there are any flaws in the base generator, shuffling will tend to even them out. But if the system provides an excellent @code{random} function, shuffling will not damage its randomness. X To create a random integer of a certain number of digits, Calc builds the integer three decimal digits at a time. For each group of three digits, Calc calls its 10-bit random number generator (which returns a value from 0 to 1023); if the random value is 1000 or more, Calc throws it out and tries again until it gets a suitable value. X To create a random floating-point number with precision @var{p}, Calc simply creates a random @var{p}-digit integer and multiplies by @c{$10^{-p}$} @cite{10^-p}. The resulting random numbers should be very clean, but note that relatively small numbers will have few significant random digits. In other words, with a precision of 12, you will occasionally get numbers on the order of @c{$10^{-9}$} @cite{10^-9} or @c{$10^{-10}$} @cite{10^-10}, but those numbers will only have two or three random digits since they correspond to small integers times @c{$10^{-12}$} @cite{10^-12}. X To create a random integer in the interval @samp{[0 .. @var{m})}, Calc counts the digits in @var{m}, creates a random integer with three additional digits, then reduces modulo @var{m}. Unless @var{m} is a power of ten the resulting values will be very slightly biased toward the lower numbers, but this bias will be less than 0.1%. (For example, if @var{m} is 42, Calc will reduce a random integer less than 100000 modulo 42 to get a result less than 42. It is easy to show that the numbers 40 and 41 will be only 2380/2381 as likely to result from this modulo operation as numbers 39 and below.) If @var{m} is a power of ten, however, the numbers should be completely unbiased. X The Gaussian random numbers generated by @samp{random(0.0)} use the ``polar'' method described in Knuth section 3.4.1C. This method generates a pair of Gaussian random numbers at a time, so only every other call to @samp{random(0.0)} will require significant calculations. X @node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions @section Combinatorial Functions X @noindent Commands relating to combinatorics and number theory begin with the @kbd{k} key prefix. X @kindex k g @pindex calc-gcd @tindex gcd The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the Greatest Common Divisor of two integers. It also accepts fractions; the GCD of two fractions is defined by taking the GCD of the numerators, and the LCM of the denominators. This definition is consistent with the idea that @samp{a / gcd(a,x)} should yield an integer for any @samp{a} and @samp{x}. For other types of arguments, the operation is left in symbolic form.@refill X @kindex k l @pindex calc-lcm @tindex lcm The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the Least Common Multiple of two integers or fractions. The product of the LCM and GCD of two numbers is equal to the product of the numbers.@refill X @kindex k E @pindex calc-extended-gcd @tindex egcd The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes the GCD of two integers @cite{x} and @cite{y} and returns a vector @cite{[g, a, b]} where @c{$g = \gcd(x,y) = a x + b y$} @cite{g = gcd(x,y) = a x + b y}. X @kindex ! @pindex calc-factorial @tindex fact @tindex ! The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the factorial of the number at the top of the stack. If the number is an integer, the result is an exact integer. If the number is an integer-valued float, the result is a floating-point approximation. If the number is a non-integral real number, the generalized factorial is used, as defined by the Euler Gamma function. Please note that computation of large factorials can be slow; using floating-point format will help since fewer digits must be maintained. The same is true of many of the commands in this section.@refill X @kindex k d @pindex calc-double-factorial @tindex dfact The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command computes the ``double factorial'' of an integer. For an even integer, this is the product of even integers from 2 to @cite{N}. For an odd integer, this is the product of odd integers from 3 to @cite{N}. If the argument is an integer-valued float, the result is a floating-point approximation. This function is undefined for negative even integers. The notation @cite{N!!} is also recognized for double factorials.@refill X @kindex k c @pindex calc-choose @tindex choose The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the binomial coefficient @cite{N}-choose-@cite{M}, where @cite{M} is the number on the top of the stack and @cite{N} is second-to-top. If both arguments are integers, the result is an exact integer. Otherwise, the result is a floating-point approximation. The binomial coefficient is defined for all real numbers by @c{$N! \over M! (N-M)!\,$} @cite{N! / M! (N-M)!}. X @kindex H k c @pindex calc-perm @tindex perm @ifinfo The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the number-of-permutations function @cite{N! / (N-M)!}. @end ifinfo @iftex The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the number-of-perm\-utations function @c{$N! \over (N-M)!\,$} @cite{N! / (N-M)!}. @end iftex X @kindex k b @kindex H k b @pindex calc-bernoulli-number @tindex bern The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command computes a given Bernoulli number. The value at the top of the stack is a nonnegative integer @cite{n} that specifies which Bernoulli number is desired. The @kbd{H k b} command computes a Bernoulli polynomial, taking @cite{n} from the second-to-top position and @cite{x} from the top of the stack. If @cite{x} is a variable or formula the result is a polynomial in @cite{x}; if @cite{x} is a number the result is a number. X @kindex k e @kindex H k e @pindex calc-euler-number @tindex euler The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly computes an Euler number, and @kbd{H k e} computes an Euler polynomial. Bernoulli and Euler numbers occur in the Taylor expansions of several functions. X @kindex k s @kindex H k s @pindex calc-stirling-number @tindex stir1 @tindex stir2 The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command computes a Stirling number of the first kind@c{ $n \brack m$} @asis{}, given two integers @cite{n} and @cite{m} on the stack. The @kbd{H k s} [@code{stir2}] command computes a Stirling number of the second kind@c{ $n \brace m$} @asis{}. These are the number of @cite{m}-cycle permutations of @cite{n} objects, and the number of ways to partition @cite{n} objects into @cite{m} non-empty sets, respectively. X @kindex k p @pindex calc-prime-test @cindex Primes The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on the top of the stack is prime. For integers less than eight million, the answer is always exact and reasonably fast. For larger integers, a probabilistic method is used (See Knuth vol. II, section 4.5.4, algorithm P). The number is first checked against small prime factors (up to 13). Then, any number of iterations of the algorithm are performed. Each step either discovers that the number is non-prime, or substantially increases the SHAR_EOF true || echo 'restore of calc.texinfo failed' fi echo 'End of part 42' echo 'File calc.texinfo is continued in part 43' echo 43 > _shar_seq_.tmp exit 0 exit 0 # Just in case... -- Kent Landfield INTERNET: kent@sparky.IMD.Sterling.COM Sterling Software, IMD UUCP: uunet!sparky!kent Phone: (402) 291-8300 FAX: (402) 291-4362 Please send comp.sources.misc-related mail to kent@uunet.uu.net.