daveg@csvax.caltech.edu (David Gillespie) (06/06/90)
Posting-number: Volume 13, Issue 42
Submitted-by: daveg@csvax.caltech.edu (David Gillespie)
Archive-name: gmcalc/part16
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Xof an angle or complex number. If the input is an HMS form, it is interpreted
Xas degrees-minutes-seconds; otherwise, the input is interpreted according
Xto the current angular mode. It is best to use Radians mode when operating
Xon complex numbers.@refill
X
X@kindex I S
X@pindex calc-arcsin
X@tindex arcsin
XWith the Inverse flag, @code{calc-sin} computes an arcsine. This is also
Xavailable as the @code{calc-arcsin} command or @code{arcsin} algebraic
Xfunction. The returned argument is converted to degrees, radians, or HMS
Xnotation depending on the current angular mode.
X
X@kindex H S
X@pindex calc-sinh
X@tindex sinh
X@kindex H I S
X@pindex calc-arcsinh
X@tindex arcsinh
XWith the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
Xsine, also available as @code{calc-sinh} [@code{sinh}]. With the
XHyperbolic and Inverse flags, it computes the hyperbolic arcsine
X(@code{calc-arcsinh}) [@code{arcsinh}].
X
X@kindex C
X@pindex calc-cos
X@tindex cos
X@kindex I C
X@pindex calc-arccos
X@tindex arccos
X@kindex H C
X@pindex calc-cosh
X@tindex cosh
X@kindex H I C
X@pindex calc-arccosh
X@tindex arccosh
X@kindex T
X@pindex calc-tan
X@tindex tan
X@kindex I T
X@pindex calc-arctan
X@tindex arctan
X@kindex H T
X@pindex calc-tanh
X@tindex tanh
X@kindex H I T
X@pindex calc-arctanh
X@tindex arctanh
XThe shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine
Xof an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}]
Xcomputes the tangent, along with all the various inverse and hyperbolic
Xvariants of these functions.
X
X@pindex calc-arctan2
X@tindex arctan2
XThe @code{calc-arctan2} [@code{arctan2}] command takes two numbers from
Xthe stack and computes the arc tangent of their ratio. The result is in
Xthe full range from -180 (exclusive) to +180 (inclusive) degrees, or the
Xanalogous range in radians. A similar result would be obtained with
X@kbd{/} followed by @kbd{I T}, but the value would only be in the range
Xfrom -90 to 90 degrees since the division loses information about the
Xsigns of the two components, and an error might result from an explicit
Xdivision by zero which @code{arctan2} would avoid. By (arbitrary)
Xdefinition, @samp{arctan2(0,0)=0}.
X
X@pindex calc-sincos
X@tindex sincos
X@tindex arcsincos
XThe @code{calc-sincos} [@code{sincos}] command computes the sine and
Xcosine of a number, returning them as a vector of the form
X@samp{[@var{cos}, @var{sin}]}.
XWith the Inverse flag [@code{arcsincos}], this command takes a two-element
Xvector as an argument and computes @code{arctan2} of the elements.
X(This command does not accept the Hyperbolic flag.)@refill
X
X@node Branch Cuts, Random Numbers, Trigonometric and Hyperbolic Functions, Scientific Functions
X@section Branch Cuts and Principal Values
X
X@cindex Branch cuts
X@cindex Principal values
XAll of the logarithmic, trigonometric, and hyperbolic functions are defined
Xfor complex numbers as well as for reals. This section describes the values
Xreturned in cases where the general result is a family of possible values.
XCalc follows section 12.5.3 of Steele's @cite{Common Lisp, the Language} in
Xthese matters. This section will describe each function briefly; for a
Xmore detailed discussion, consult Steele's book. Except as noted below,
Xthese branch cuts also exactly match those implemented by the HP-28C
Xcalculator.
X
XNote: The current implementations of these functions with complex arguments
Xare designed with proper behavior around the branch cuts in mind, @emph{not}
Xefficiency or accuracy. You may need to increase the floating precision
Xand wait a while to get suitable answers from them.
X
XFor @samp{sqrt(a+bi)}: When @samp{a<0} and @samp{b} is small but positive
Xor zero, the result is close to the @samp{+i} axis. For @samp{b} small and
Xnegative, the result is close to the @samp{-i} axis. The result always lies
Xin the right half of the complex plane.
X
XFor @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
XThe imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
X
XFor @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
X
XFor @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
X
XFor @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
Xor equivalently by @samp{pi/2 - arcsin(z)}.
X
XFor @samp{arctan(z)}: This is defined by
X@samp{-i*ln((1+i*z) * sqrt(1/(1+z^2)))}. The HP-28C differs in that the
Xbranch cuts are continuous on the opposite sides.@refill
X
XFor @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
X
XFor @samp{arccosh(z)}: This is defined by
X@samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}.
X
XFor @samp{arctanh(z)}: This is defined by @samp{ln((1+z) * sqrt(1/(1-z^2)))}.
XAgain this differs from the HP-28C in that the values exactly on the branch
Xcuts are continuous with the opposite sides.
X
X@node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions
X@section Random Numbers
X
X@kindex k r
X@pindex calc-random
X@tindex random
XThe @kbd{k r} (@code{calc-random}) [@code{random}] command produces
Xrandom numbers of various sorts.
X
XGiven a positive numeric prefix argument @samp{M}, it produces a random
Xinteger @samp{N} in the range @samp{0 <= N < M}. Each of the @samp{M}
Xvalues appears with equal probability.@refill
X
XWith no numeric prefix argument, the @kbd{k r} command takes its argument
Xfrom the stack instead. Once again, if this is a positive integer @samp{M}
Xthe result is a random integer less than @samp{M}. However, note that
Xwhile numeric prefix arguments are limited to six digits or so, an @samp{M}
Xtaken from the stack can be arbitrarily large.
X
XIf the value on the stack is a real number @samp{M}, the result is a
Xrandom real number @samp{N} in the range @samp{0 <= N < M} or
X@samp{M < N <= 0}, according to the sign of @samp{M}.
X
XIf @samp{M} is zero, the result is a Gaussian-distributed random real
Xnumber; the distribution has a mean of zero and a variance of one.
XThe algorithm used generates random numbers in pairs; thus, every other
Xcall to this function will be especially fast.
X
XIf @samp{M} is an interval form, the lower and upper bounds specify the
Xacceptable limits of the random numbers. If both bounds are integers,
Xthe result is a random integer in the specified range. If either bound
Xis floating-point, the result is a random real number in the specified
Xrange.
X
X@pindex calc-rrandom
XThe @code{calc-rrandom} command produces a random real number between
Xzero and one. It is equivalent to @samp{random(1.0)}.
X
X@kindex k a
X@pindex calc-random-again
XThe @kbd{k a} (@code{calc-random-again}) command produces another random
Xnumber, re-using the most recent value of @samp{M}.
X
X@node Combinatorial Functions, , Random Numbers, Scientific Functions
X@section Combinatorial Functions
X
XCommands relating to combinatorics and number theory begin with the
X@kbd{k} key prefix.
X
X@kindex k g
X@pindex calc-gcd
X@tindex gcd
XThe @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the
XGreatest Common Divisor of two integers. (For non-integer arguments the
Xoperation is left in symbolic form.)@refill
X
X@kindex k l
X@pindex calc-lcm
X@tindex lcm
XThe @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the
XLeast Common Multiple of two integers. The product of the LCM and GCD
Xof two integers is equal to the product of the integers.@refill
X
X@kindex k G
X@pindex calc-extended-gcd
X@tindex egcd
XThe @kbd{k G} (@code{calc-extended-gcd}) [@code{egcd}] command computes
Xthe GCD of two integers @samp{x} and @samp{y} and returns a vector
X@samp{[g, a, b]} where @samp{g = gcd(x,y) = a x + b y}.
X
X@kindex !
X@pindex calc-factorial
X@tindex fact
XThe @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the
Xfactorial of the number at the top of the stack. If the number is an
Xinteger, the result is an exact integer. If the number is an
Xinteger-valued float, the result is a floating-point approximation. If
Xthe number is a non-integral real number, the generalized factorial is used,
Xas defined by the Euler Gamma function. Please note that computation of
Xlarge factorials can be slow; using floating-point format will help
Xsince fewer digits must be maintained.@refill
X
X@pindex calc-gamma
X@tindex gamma
XThe @code{calc-gamma} [@code{gamma}] command computes the Euler Gamma
Xfunction explicitly, according to the rule @samp{gamma(x+1) = fact(x)}.
X
X@kindex k d
X@pindex calc-double-factorial
X@tindex dfact
XThe @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command
Xcomputes the ``double factorial'' of an integer. For an even integer,
Xthis is the product of even integers from 2 to @samp{N}. For an odd
Xinteger, this is the product of odd integers from 3 to @samp{N}. If
Xthe argument is an integer-valued float, the result is a floating-point
Xapproximation.@refill
X
X@kindex k b
X@pindex calc-choose
X@tindex choose
XThe @kbd{k b} (@code{calc-choose}) [@code{choose}] command computes the
Xbinomial coefficient @samp{N}-choose-@samp{M}, where @samp{M} is the number
Xon the top of the stack and @samp{N} is second-to-top. If both arguments
Xare integers, the result is an exact integer. Otherwise, the result is a
Xfloating-point approximation. The binomial coefficient is defined for all
Xreal numbers by @samp{N! / M! (N-M)!}.
X
X@kindex H k b
X@pindex calc-perm
X@tindex perm
XThe @kbd{H k b} (@code{calc-perm}) [@code{perm}] command computes the
Xnumber-of-permutations function @samp{N! / M!}.
X
X@kindex k p
X@pindex calc-prime-test
X@cindex Primes
XThe @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
Xthe top of the stack is prime. For integers less than eight million, the
Xanswer is always exact and reasonably fast. For larger integers, a
Xprobabilistic method is used (See Knuth vol. II, section 4.5.4, algorithm P).
XThe number is first checked against small prime factors (up to 13). Then,
Xany number of iterations of the algorithm are performed. Each step either
Xdiscovers that the number is non-prime, or substantially increases the
Xcertainty that the number is prime. After a few steps, the chance that
Xa number was mistakenly described as prime will be less than one percent.
X(Indeed, this is the worst case; in practice even a single iteration is
Xquite reliable.) After the @kbd{k p} command, the number will be reported
Xas definitely prime or non-prime if possible, or otherwise ``probably''
Xprime with a certain probability of error.
X
XThe normal @kbd{k p} command performs one iteration of the primality
Xtest. Pressing @kbd{k p} repeatedly for the same integer will perform
Xadditional iterations. Also, @kbd{k p} with a numeric prefix performs
Xthe specified number of iterations.
X
X@kindex k f
X@pindex calc-prime-factors
X@tindex prfac
XThe @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command
Xattempts to decompose an integer into its prime factors. For numbers up
Xto 25 million, the answer is exact although it may take some time. The
Xresult is a vector of the prime factors in increasing order. For larger
Xinputs, prime factors above 5000 may not be found, in which case the
Xlast number in the vector will be an unfactored integer greater than 25
Xmillion.
X
X@kindex k n
X@pindex calc-next-prime
X@tindex nextprime
XThe @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds
Xthe next prime above a given number. Essentially, it searches by calling
X@code{calc-prime-test} on successive integers until it finds one that
Xpasses the test. This is quite fast for integers less than eight million,
Xbut once the probabilistic test comes into play the search may be rather
Xslow. Ordinarily this command stops for any prime that passes one iteration
Xof the primality test. With a numeric prefix argument, a number must pass
Xthe specified number of iterations before the search stops. (This only
Xmatters when searching above eight million.) You can always use additional
X@kbd{k p} commands to increase your certainty that the number is indeed
Xprime.
X
X@kindex I k n
X@pindex calc-prev-prime
X@tindex prevprime
XThe @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command
Xanalogously finds the next prime less than a given number.
X
X@kindex k t
X@pindex calc-totient
X@tindex totient
XThe @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the
XEuler ``totient'' function, the number of integers less than @samp{n} which
Xare relatively prime to @samp{n}.
X
X@kindex k m
X@pindex calc-moebius
X@tindex moebius
XThe @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the
XMoebius ``mu'' function. If the input number is a product of @samp{k}
Xdistinct factors, this is @samp{(-1)^k}. If the input number has any
Xduplicate factors (i.e., can be divided by the same prime more than once),
Xthe result is zero.
X
X@node Binary Functions, Matrix Functions, Scientific Functions, Top
X@chapter Binary Number Functions
X
XThe commands in this chapter all use two-letter sequences beginning with
Xthe @kbd{b} prefix.
X
X@cindex Binary numbers
XThe ``binary'' operations actually work regardless of the currently
Xdisplayed radix, although their results make the most sense in a radix
Xlike 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @kbd{d 6}
Xcommands, respectively). You may also wish to enable display of leading
Xzeros with @kbd{d z}. @xref{Radix Modes}.
X
X@cindex Word size for binary operations
XThe Calculator maintains a current @dfn{word size} @samp{w}, an
Xarbitrary positive or negative integer. For a positive word size, all
Xof the binary operations described here operate modulo @samp{2^w}. In
Xparticular, negative arguments are converted to positive integers modulo
X@samp{2^w} by all binary functions.@refill
X
XIf the word size is negative, binary operations produce 2's complement
Xintegers from @samp{-(2^(-w-1))} to @samp{2^(-w-1)-1} inclusive. Either
Xmode accepts inputs in any range; the sign of @samp{w} affects only
Xthe results produced.
X
X@kindex b c
X@pindex calc-clip
X@tindex clip
XThe @kbd{b c} (@code{calc-clip})
X[@code{clip}] command can be used to clip a number by reducing it modulo
X@samp{2^w}. The commands described in this chapter automatically clip
Xtheir results to the current word size. Note that other operations like
Xaddition do not use the current word size, since integer addition
Xgenerally is not ``binary.'' (However, @pxref{Simplification Modes},
X@code{calc-bin-simplify-mode}.)@refill
X
X@kindex b w
X@pindex calc-word-size
XThe default word size is 32 bits. All operations except the shifts and
Xrotates allow you to specify a different word size for that one
Xoperation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
Xtop of stack to the range 0 to 255 regardless of the current word size.
XTo set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
XBy default this command displays the current word size; with a numeric
Xprefix, it sets the word size to that value.
X
XWhen the binary operations are written in symbolic form, they take an
Xoptional second (or third) word-size parameter. When a formula like
X@samp{and(a,b)} is finally evaluated, the word size current at that time
Xwill be used, but when @samp{and(a,b,-8)} is evaluated, a word size of -8
Xwill always be used. A symbolic binary function will be left in symbolic
Xform unless the all of its argument(s) are integers or integer-valued floats.
X
XIf either or both arguments are modulo forms for which @samp{M} is a
Xpower of two, that power of two is taken as the word size unless a
Xnumeric prefix argument overrides it. The current word size is never
Xconsulted when modulo-power-of-two forms are involved.
X
X@kindex b a
X@pindex calc-and
X@tindex and
XThe @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise
XAND of the two numbers on the top of the stack. In other words, for each
Xof the @samp{w} binary digits of the two numbers (pairwise), the corresponding
Xbit of the result is 1 if and only if both input bits are 1.
X
X@kindex b o
X@pindex calc-or
X@tindex or
XThe @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise
Xinclusive OR of two numbers. A bit is 1 if either of the input bits, or
Xboth, are 1.
X
X@kindex b x
X@pindex calc-xor
X@tindex xor
XThe @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise
Xexclusive OR of two numbers. A bit is 1 if exactly one of the input bits
Xis 1.
X
X@kindex b d
X@pindex calc-diff
X@tindex diff
XThe @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise
Xdifference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))}.
X
X@kindex b n
X@pindex calc-not
X@tindex not
XThe @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise
XNOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
X
X@kindex b l
X@pindex calc-lshift-binary
X@tindex lsh
XThe @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a
Xnumber left by one bit, or by the number of bits specified in the numeric
Xprefix argument. A negative prefix argument performs a logical right shift,
Xin which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
Xis short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
XBits shifted ``off the end,'' according to the current word size, are lost.
X
X@kindex b r
X@pindex calc-rshift-binary
X@tindex rsh
XThe @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a
Xnumber right by one bit, or by the number of bits specified in the numeric
Xprefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
X
X@kindex b s
X@pindex calc-shift-binary
X@tindex ash
XThe @kbd{b s} (@code{calc-shift-binary}) [@code{ash}] command shifts a
Xnumber left in the same way as @code{lsh}. However, if shifting right,
Xit performs an ``arithmetic'' shift, in which the leftmost bit (according
Xto the current word size) is duplicated rather than shifting in zeros.
XThis corresponds to dividing by two where the input is interpreted as a
Xsigned, twos-complement number. (The distinction between the @samp{lsh}
Xand @samp{ash} operations is totally independent from whether the word
Xsize is positive or negative.)
X
X@kindex b R
X@pindex calc-rotate-binary
X@tindex rot
XThe @kbd{b R} (@code{calc-rotate-binary}) [@code{rot}] command rotates a
Xnumber one bit to the left. The leftmost bit (according to the current
Xword size) is dropped off the left and shifted in on the right. With a
Xnumeric prefix argument, the number is rotated the specified number of
Xbits to the left or right.
X
X@node Matrix Functions, Algebra, Binary Functions, Top
X@chapter Vector and Matrix Functions
X
XMany of the commands described here begin with the @kbd{v} prefix.
X(For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.)
XThe commands usually apply to both plain vectors and matrices; some
Xapply only to matrices or only to square matrices. If the argument
Xhas the wrong dimensions the operation is left in symbolic form.
X
X@menu
X* Building Vectors::
X* Extracting Elements::
X* Vector and Matrix Arithmetic::
X* Reducing and Mapping::
X* Vector and Matrix Formats::
X@end menu
X
X@node Building Vectors, Extracting Elements, Matrix Functions, Matrix Functions
X@section Building Vectors
X
XVectors are entered and displayed using @samp{[a,b,c]} notation.
XMatrices are vectors of which all elements are vectors of equal length.
X(Though none of the standard Calc commands use this concept, a
Xthree-dimensional matrix or rank-3 tensor could be defined as a
Xvector of matrices, and so on.) Vectors and matrices can be added,
Xsubtracted, multiplied, and divided; @pxref{Basic Arithmetic}.@refill
X
X@kindex |
X@pindex calc-concat
XThe @kbd{|} (@code{calc-concat}) command ``concatenates'' two vectors
Xtogether. For example, after @kbd{[ 1 , 2 ] [ 3 , 4 ] |}, the stack
Xwill contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
Xare matrices, the rows of the first matrix are concatenated with the
Xrows of the second. (In other words, two matrices are just two vectors of
Xrow-vectors as far as @code{calc-concat} is concerned.)
X
XIf either argument to @kbd{|} is a scalar (a non-vector), it is treated
Xlike a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
Xproduces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
Xmatrix and the other is a plain vector, the vector is treated as a
Xone-row matrix.
X
X@kindex v p (vectors)
X@pindex calc-pack (vectors)
XThe @kbd{v p} (@code{calc-pack}) command collects several elements from
Xthe stack into a matrix, complex number, HMS form, or error form.
XBy itself, it collects two
Xstack elements into a complex number according to the current Polar
XMode. With a numeric prefix argument of -1 or -2, it builds a
Xrectangular or polar number (respectively) regardless of the polar mode.
XWith an argument of -3 it builds an HMS form from the top three elements
Xon the stack. With an argument of -4 it builds an error form from the
Xtop two elements on the stack. With an argument of -5 it builds a
Xmodulo form.
XWith a positive numeric prefix, it collects the specified number of
Xstack elements into a vector.@refill
X
XThe same effect can be had by pressing @kbd{[} or @kbd{(} to push an
Xincomplete vector or complex number on the stack, using @code{calc-roll-down}
Xto sneak the incomplete object up past a certain number of elements, and
Xthen pressing the matching @kbd{]} or @kbd{)}.
X
X@kindex v u (vectors)
X@pindex calc-unpack (vectors)
XThe @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
Xnumber, HMS form, or other composite object on the top of the stack and
X``unpacks'' it, pushing each of its elements onto the stack as separate
Xobjects. Thus, it is the ``inverse'' of @code{calc-pack}. If the value
Xat the top of the stack is a formula, @code{calc-unpack} unpacks it by
Xpushing each of the arguments of the top-level operator onto the stack.
X
X@kindex v d
X@pindex calc-diag
X@tindex diag
XThe @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
Xsquare matrix. The optional numeric prefix gives the number of rows
Xand columns in the matrix. If the value at the top of the stack is a
Xvector, the elements of the vector are used as the diagonal elements; the
Xprefix, if specified, must match the size of the vector. If the value on
Xthe stack is a scalar, it is used for each element on the diagonal, and
Xthe prefix argument is required.
X
XTo build a constant square matrix, e.g., a 3x3 matrix filled with ones,
Xuse @kbd{0 M-3 v d 1 +}, i.e., build a zero matrix first and then add a
Xconstant value to that matrix. (Another alternative would be to use
X@code{calc-build-vector} and @code{calc-arrange-vector}; see below.)
X
X@kindex v i
X@pindex calc-ident
XThe @kbd{v i} (@code{calc-ident}) function builds an identity matrix of
Xthe specified size. It is a convenient form of @code{calc-diag} where the
Xdiagonal element is always one.
X
X@kindex v x
X@pindex calc-index
X@tindex index
XThe @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
Xof consecutive integers from 1 to @var{n}, where @var{n} is the numeric
Xprefix argument. If you do not provide a prefix argument, you will be
Xprompted to enter a suitable number.
X
X@kindex v b
X@pindex calc-build-vector
X@tindex cvec
XThe @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
Xvector of @var{n} copies of the value on the top of the stack, where @var{n}
Xis the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
Xcan also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}.
X
X@kindex v a
X@pindex calc-arrange-vector
X@tindex arrange
X@cindex Arranging a matrix
X@cindex Reshaping a matrix
X@cindex Flattening a matrix
XThe @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] function
Xrearranges a vector to have a certain number of columns and rows. The
Xnumeric prefix argument specifies the number of columns; if you do not
Xprovide an argument, you will be prompted for the number of columns.
XThe vector or matrix on the top of the stack is @dfn{flattened} into a
Xplain vector. If the number of columns is nonzero, this vector is
Xthen formed into a matrix by taking successive groups of @var{n} elements.
XIf the number of columns does not evenly divide the number of elements
Xin the vector, the last row will be short and the result will not be
Xsuitable for use as a matrix. For example, with the matrix
X@samp{[[1, 2], [3, 4]]} on the stack, @kbd{v a 4} produces
X@samp{[[1, 2, 3, 4]]} (a 1x4 matrix), @kbd{v a 1} produces
X@samp{[[1], [2], [3], [4]]} (a 4x1 matrix), @kbd{v a 2} produces
X@samp{[[1, 2], [3, 4]]} (the original 2x2 matrix), @kbd{v a 3} produces
X@samp{[[1, 2, 3], [4]]} (not a matrix), and @kbd{v a 0} produces
Xthe flattened list @samp{[1, 2, 3, 4]}.
X
X@kindex v s
X@kindex I v s
X@pindex calc-sort
X@tindex sort
X@tindex rsort
XThe @kbd{v s} (@code{calc-sort}) [@code{sort}] function sorts the elements of
Xa vector into increasing order. Non-real vector elements are sorted
Xaccording to a kind of lexicographic ordering. The @kbd{I v s} [@code{rsort}]
Xfunction sorts a vector into decreasing order.
X
X@kindex v h
X@pindex calc-histogram
XThe @kbd{v h} (@code{calc-histogram}) [@code{histogram}] function builds a
Xhistogram of a vector of numbers. Vector elements are assumed to be
Xintegers or real numbers in the range [0..@var{n}) for some ``number of
Xbins'' @var{n}, which is the numeric prefix argument given to the
Xcommand. The result is a vector of @var{n} counts of how many times
Xeach value appeared in the original vector. Non-integers in the input
Xare rounded down to integers. Any vector elements outside the specified
Xrange are ignored. (You can tell if elements have been ignored by checking
Xthat the elements in the result vector add up to the length of the input
Xvector.)
X
XWith the Hyperbolic flag, @kbd{v h} pulls two vectors from the stack.
XThe second-to-top vector is the list of numbers as before. The top
Xvector is an equal-sized list of ``weights'' to attach to the elements
Xof the data vector. For example, if the first data element is 4.2 and
Xthe first weight is 10, then 10 will be added to bin 4 of the result
Xvector. Without the hyperbolic flag, every element has a weight of one.
X
X@node Extracting Elements, Vector and Matrix Arithmetic, Building Vectors, Matrix Functions
X@section Extracting Vector and Matrix Elements
X
X@kindex v r
X@pindex calc-mrow
X@tindex mrow
XThe @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
Xthe matrix on the top of the stack, or one element of the plain vector on
Xthe top of the stack. The row or element is specified by the numeric
Xprefix argument; the default is to prompt for the row or element number.
XThe matrix or vector is replaced by the specified row or element in the
Xform of a vector or scalar, respectively.
X
X@tindex mrrow
XGiven a negative prefix argument, @kbd{v r} instead deletes one row or
Xelement from the matrix or vector on the top of the stack. Thus
X@kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r}
Xreplaces the matrix with the same matrix with its second row removed.
XIn algebraic form this function is called @code{mrrow}.
X
X@tindex getdiag
XGiven a prefix argument of zero, @kbd{v r} extracts the diagonal elements
Xof a square matrix in the form of a vector. In algebraic form this
Xfunction is called @code{getdiag}.
X
X@kindex v c
X@pindex calc-mcol
X@tindex mcol
X@tindex mrcol
XThe @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is
Xthe analogous operation on columns of a matrix. Given a plain vector
Xit extracts (or removes) one element, just like @kbd{v r}.@refill
X
XTo extract a matrix element at a given row and column, use @kbd{v r} to
Xextract the row as a vector, then @kbd{v c} to extract the column element
Xfrom that vector.
X
X@kindex v l
X@pindex calc-vlength
X@tindex vlen
XThe @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
Xlength of a vector. The length of a non-vector is considered to be zero.
XNote that matrices are just vectors of vectors for the purposes of this
Xcommand.@refill
X
X@node Vector and Matrix Arithmetic, Reducing and Mapping, Extracting Elements, Matrix Functions
X@section Vector and Matrix Arithmetic
X
XBasic arithmetic operations like addition and multiplication are defined
Xfor vectors and matrices as well as for numbers. Division of matrices, in
Xthe sense of multiplying by the inverse, is supported. (Division by a
Xmatrix actually uses LU-decomposition for greater accuracy and speed.)
X@xref{Basic Arithmetic}.
X
XThe following functions are applied element-wise if their arguments are
Xvectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
X@code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
X@code{float}, @code{frac}. @xref{Function Index}.@refill
X
X@kindex v t
X@pindex calc-transpose
X@tindex trn
XThe @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
Xthe transpose of the matrix at the top of the stack. If the argument
Xis a plain vector, it is treated as a row vector and transposed into
Xa one-column matrix.
X
X@kindex V J
X@pindex calc-conj-transpose
XThe @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes
Xthe conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
X
X@kindex A (vectors)
X@pindex calc-abs (vectors)
X@tindex abs (vectors)
XThe @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the
XFrobenius norm of a vector or matrix argument. This is the square
Xroot of the sum of the squares of the absolute values of the
Xelements of the vector or matrix.@refill
X
X@kindex v n
X@pindex calc-rnorm
X@tindex rnorm
XThe @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes
Xthe row norm, or infinity-norm, of a vector or matrix. For a plain
Xvector, this is the maximum of the absolute values of the elements.
XFor a matrix, this is the maximum of the row-absolute-value-sums,
Xi.e., of the sums of the absolute values of the elements along the
Xvarious rows.
X
X@kindex V N
X@pindex calc-cnorm
X@tindex cnorm
XThe @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes
Xthe column norm, or one-norm, of a vector or matrix. For a plain
Xvector, this is the sum of the absolute values of the elements.
XFor a matrix, this is the maximum of the column-absolute-value-sums.
XGeneral @samp{k}-norms for @samp{k} other than one or infinity are
Xnot provided.
X
X@kindex V C
X@pindex calc-cross
X@tindex cross
XThe @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the
Xright-handed cross product of two vectors, each of which must have
Xexactly three elements.
X
X@kindex V I
X@pindex calc-inv
X@tindex inv
XThe @kbd{V I} (@code{calc-inv}) [@code{inv}] command computes the
Xinverse of a square matrix. If the matrix is singular, the inverse
Xoperation is left in symbolic form. Matrix inverses are recorded so
Xthat once an inverse (or determinant) of a particular matrix has been
Xcomputed, the inverse and determinant of the matrix can be recomputed
Xquickly in the future.
X
XIf the argument to @kbd{V I} is a plain number @samp{x}, this
Xcommand simply computes @samp{1/x}. The same operation is also
Xavailable on the @kbd{&} key.
X
X@kindex V D
X@pindex calc-mdet
X@tindex det
XThe @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
Xdeterminant of a square matrix.
X
X@kindex V L
X@pindex calc-mlud
X@tindex lud
XThe @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the
XLU decomposition of a matrix. The result is a list of three matrices
Xwhich, when multiplied together left-to-right, form the original matrix.
XThe first is a permutation matrix that arises from pivoting in the
Xalgorithm, the second is lower-triangular with ones on the diagonal,
Xand the third is upper-triangular.
X
X@kindex V T
X@pindex calc-mtrace
X@tindex tr
XThe @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the
Xtrace of a square matrix. This is defined as the sum of the diagonal
Xelements of the matrix.
X
X@node Reducing and Mapping, Vector and Matrix Formats, Vector and Matrix Arithmetic, Matrix Functions
X@section Reducing and Mapping Vectors
X
XThe commands in this section allow for more general operations on the
Xelements of vectors.
X
X@kindex V A
X@pindex calc-apply
X@tindex apply
XThe @kbd{V A} (@code{calc-apply}) command applies a given operator to
Xthe elements of a vector. For example, applying the hypothetical
Xfunction @code{f} to the vector @samp{[1, 2, 3]} would produce the
Xfunction call @samp{f(1, 2, 3)}. Applying the @code{+} function to the
Xvector @samp{[a, b]} produces @samp{a + b}. Applying @code{+} to
Xthe vector @samp{[a, b, c]} is an error, since the @code{+} function
Xaccepts exactly two arguments. (While @code{calc-apply} is useful in
Xsome cases, you will usually find that @code{calc-reduce} or @code{calc-map},
Xdescribed below, are closer to what you want.)
X
XThe @kbd{V A} command prompts you to press the key corresponding to the
Xdesired operator. Press @kbd{?} for a list of all available operators.
XOr, press @samp{x} at the operator prompt to select any algebraic
Xfunction by name to use as the operator. This can include functions you
Xhave defined yourself using the @kbd{Z F} command. (@xref{Algebraic
XDefinitions}.) If you give a name for which no function has been
Xdefined, the result is left in symbolic form, as in @samp{f(1, 2,
X3)}.@refill
X
XIt is also possible to apply any formula to a vector by treating that
Xformula as a function. When prompted for the operator to use, press
X@kbd{'} (the apostrophe) and type your formula as an algebraic entry.
XYou will then be prompted for the argument list, which defaults to a
Xlist of all variables that appear in the formula, sorted into alphabetic
Xorder. For example, suppose you enter the formula @samp{x + 2y^x}.
XThe default argument list would be @samp{(x y)}, which means that if
Xthis function is applied to the arguments @samp{[3, 10]}, the result would
Xbe @samp{3 + 2*10^3}. If you plan to use a certain formula in this
Xway often, you might consider defining it as a function with @kbd{Z F}.
X
XIf you press @kbd{$} at the operator prompt, the effect is similar to
Xthe apostrophe except that the relevant formula is taken from top-of-stack
Xinstead. The actual vector argument of the @code{calc-apply} command is
Xthen taken from the second-to-top stack position. You will still be
Xprompted for an argument list.
X
X@cindex Generic functions
X@cindex Lambda expressions
X@tindex lambda
XThe algebraic notation for a ``generic function'' is @samp{lambda(a1,a2,a3,x)}
Xwhere the @code{a}s are argument variables, and @code{x} is a formula
Xinvolving those variables. For example, when you use the apostrophe to
Xenter the formula @samp{x + 2y^x} as an operator with argument list
X@samp{(x y)}, Calc builds the generic function @samp{lambda(x, y, x + 2y^x)}
Xand gives that to the @code{apply} operation instead of a function name.
XYou can enter a lambda expression by hand if you wish, either directly
Xwith @kbd{'}, or on the stack with @kbd{$}. (The word @code{lambda}
Xderives from Lisp notation and the theory of functions. Note that there
Xis no actual Calc function called @code{lambda}; the whole point is that
Xthe @code{lambda} expression is used in its symbolic form, not evaluated
Xfor an answer.)
X
X@kindex V R
X@pindex calc-reduce
X@tindex reduce
XThe @kbd{V R} (@code{calc-reduce}) command applies a given binary operator
Xacross all the elements of a vector. A binary operator is a function such
Xas @code{+} or @code{max} which takes two arguments. For example, reducing
X@code{+} over a vector computes the sum of the elements of the vector.
XReducing @code{-} computes the first element minus each of the remaining
Xelements. Reducing @code{max} computes the maximum element, and so on.
XIn general, reducing @code{f} over the vector @samp{[a, b, c, d]} produces
X@samp{f(f(f(a, b), c), d)}.
X
X@kindex V M
X@pindex calc-map
X@tindex map
XThe @kbd{V M} (@code{calc-map}) command applies a given operator elementwise
Xto one or more vectors. For example, mapping @code{A} [@code{abs}] produces
Xa vector of the absolute values of the elements in the input vector. Mapping
X@code{+} pops two vectors from the stack, which must be of equal length, and
Xproduces a vector of the pairwise sums of the elements. If either argument
Xis a non-vector, it is duplicated for each element of the other vector. For
Xexample, @kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
XWith the 2 listed first, it would have computed a vector of powers of two.
XMapping a user-defined function pops as many arguments from the stack as
Xthe function requires. If you give an undefined name, you will be prompted
Xfor the number of arguments to use.@refill
X
X@tindex add
X@tindex sub
X@tindex mul
X@tindex div
X@tindex pow
X@tindex neg
X@tindex mod
X@tindex vconcat
XIn algebraic formulas, these operations can be expressed as
X@samp{apply(@var{oper}, @var{vec})},
X@samp{reduce(@var{oper}, @var{vec})}, and
X@samp{map(@var{oper}, @var{v1}, @var{v2}, ...)}.
XFor example, @samp{map(gcd, [10,20,30], x)}.
XIf @var{vec}, or all of the @var{v}'s,
Xare non-vectors, the expression is left in symbolic form. Here, @var{oper}
Xis any function name like @code{max} or @code{gcd}, or a @code{lambda}
Xexpression as described above. Operators that are
Xnormally written as symbols have the names @code{add}, @code{sub}, @code{mul},
X@code{div}, @code{pow}, @code{neg}, @code{mod}, and @code{vconcat}.@refill
X
X(Experts will notice that it's not quite proper to use a variable to name
Xa function, since the name @code{gcd} corresponds to the Lisp variable
X@code{var-gcd} but to the function @code{calcFunc-gcd}. Calc automatically
Xmakes this translation, so you don't have to worry about it.)
X
X@menu
X* Map/Reduce Examples::
X* Mapping Matrices::
X* Reducing Matrices::
X@end menu
X
X@node Map/Reduce Examples, Mapping Matrices, Reducing and Mapping, Reducing and Mapping
X@subsection Further Examples
X
X@cindex Statistical operations
XConsider computing the mean and standard deviation of a list of values.
XTo compute the mean, sum the values and divide by the
Xlength of the vector. To automate this process, type the sequence
X@kbd{@key{RET} V R + @key{TAB} v l /} to duplicate the vector, compute
Xits sum, bring the duplicate to the top of the stack and compute its length,
Xthen divide the sum by the length.@refill
X
XTo compute the standard deviation, duplicate twice, compute the mean as
Xabove, then type @kbd{V M - 2 V M ^ V R + @key{TAB} v l 1 - / Q}.
XThe steps are to subtract the mean from the elements of the vector,
Xsquare the elements, sum the elements, compute the length, then divide
Xthe sum of squares by the length minus one to yield the variance; the
Xstandard deviation is the square root of this. Note that some of these
Xoperations could be done more easily with other built-in commands;
Xfor example, plain @kbd{-} will subtract a number from a vector elementwise,
Xand @kbd{A} will compute the square root of the sum of the squares of the
Xelements of a vector. @xref{Keyboard Macros}, for a convenient way to
Xprogram a key sequence like this into the Calculator for easy re-use.
X
XIf you have a matrix and wish to compute the means and standard deviations
Xof all the rows or columns at once, just use the appropriate mapping modes
Xas described in the next two sections.
X
X@cindex Summations (by vector mapping)
XAs another example, suppose you wish to sum the function @samp{x^1.5}
Xover the integers from 5 to 20. First, build a vector of index values
Xwith @kbd{v x 16 @key{RET} 4 +}. Next, map the function @samp{x^1.5}
Xover this vector with @kbd{V M ' x^1.5}. Finally, sum the vector with
X@kbd{V R +}. This same technique can be used to compute any sum or
Xproduct over a reasonably small set of index values.
X
X@node Mapping Matrices, Reducing Matrices, Map/Reduce Examples, Reducing and Mapping
X@subsection Mapping Matrices
X
X@cindex Mapping modes
XWhen matrices are involved in a mapping operation, they can be interpreted
Xin several ways. The current @dfn{mapping mode} controls this
Xinterpretation; you set the mode by typing a two-letter combination
Xstarting with @kbd{m} when Calc asks which operator to use. Once you
Xhave set a mapping mode, that mode stays in effect until you change it.
XThe default mode is map-by-elements. The mapping mode is also used by
Xthe @code{calc-reduce} command, as described in the next section.
XThe mapping mode has no effect if the arguments are plain vectors.
X
XUse @kbd{m e} to select map-by-elements mode. In this mode, the operator
Xis applied to each element of the argument matrix, or to corresponding
Xelements of the argument matrices. For example, mapping by elements
Xthe sine function over a matrix replaces each matrix element by its sine.
XMapping addition by elements adds corresponding matrix elements. If
Xeither argument is a scalar, the same scalar is mapped with each element
Xof the other matrix.
X
X@tindex mapr
XUse @kbd{m r} to select map-by-rows mode. In this mode, the operator is
Xapplied to each row of the argument matrix, or to corresponding rows of
Xthe argument matrices. Rows are presented to the operator as plain vectors.
XFor example, mapping multiplication by rows computes a vector of dot
Xproducts of the rows of the input matrices, since multiplying two plain
Xvectors computes a dot product. If either argument is scalar, or a plain
Xvector of the same length as a row of the other matrix, it is mapped with
Xevery row of the matrix.
X
X@tindex mapc
XUse @kbd{m c} to select map-by-columns mode. Essentially, this transposes
Xall matrix arguments, maps by rows, then transposes the result.
X
X@tindex mapa
XUse @kbd{m a} to select map-across mode. In this mode, the input is a
Xsingle matrix with as many columns as the operator takes arguments.
XThe columns of the matrix are treated as arguments in a regular vector
Xmapping operation. For example, mapping multiplication across matrix
X@samp{[[a, b], [c, d], [e, f]]} produces the vector @samp{[a*b, c*d, e*f]}.
X
X@tindex mapd
XUse @kbd{m d} to select map-down mode. This mode is analogous to map-across,
Xexcept that it treats the rows of the matrix as arguments to the mapping
Xoperation.
X
XAs another example, you could concatenate two matrices horizontally by
Xmapping @samp{|} (vector concatenation) by rows.
X
X@node Reducing Matrices, , Mapping Matrices, Reducing and Mapping
X@subsection Reducing Matrices
X
XThe current @dfn{mapping mode}, as described in the previous section,
Xalso influences reduction operations when the argument is a matrix.
X
XUse @kbd{m e} to select reduce-by-elements mode. In this mode, all the
Xelements of the matrix are reduced into one scalar result. For example,
Xmapping addition by elements over a 3x4 matrix adds all 12 elements
Xtogether. Normal English left-to-right, top-to-bottom order is used;
Xfor example, reducing subtraction by elements over the matrix
X@samp{[[a, b], [c, d]]} produces @samp{(((a - b) - c) - d)}.
X
X@tindex reducer
XUse @kbd{m r} to select reduce-by-rows mode. In this mode, the rows of
Xthe matrix are treated as vectors; the operator is applied to these
Xrow vectors. For example, mapping addition by rows produces a vector
Xof the sums of the various columns of the matrix, because adding two
Xvectors does a standard vector addition. Reducing multiplication by
Xrows is not very useful unless there are only two rows, in which case
Xit computes the dot product of the two rows.
X
X@tindex reducec
XUse @kbd{m c} to select reduce-by-columns mode. In this mode, the columns
Xof the matrix are treated as vectors; the operator is reduced along the
Xcolumn vectors. In other words, it transposes the argument matrix and
Xthen reduces by rows.
X
X@tindex reducea
XUse @kbd{m a} to select reduce-across mode. In this mode, the rows of the
Xmatrix are treated as vectors; each row is reduced in the usual way, and
Xthe results are collected into a vector. For example, reducing
Xmultiplication across a matrix produces a vector of the products of the
Xvarious rows. Reducing addition produces a vector of the sums of the
Xvarious rows, producing the same result as reducing by columns but in a
Xdifferent way.
X
X@tindex reduced
XUse @kbd{m d} to select reduce-down mode. In this mode, the columns of the
Xmatrix are treated as vectors which are reduced individually. This is
Xthe same as transposing the argument matrix and then reducing across.
X
X@node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions
X@section Vector and Matrix Display Formats
X
XCommands for controlling vector and matrix display use the @kbd{v} prefix
Xinstead of the usual @kbd{d} prefix.
X
X@kindex v <
X@pindex calc-matrix-left-justify
X@kindex v =
X@pindex calc-matrix-center-justify
X@kindex v >
X@pindex calc-matrix-right-justify
XThe @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
X(@code{calc-matrix-right-justify}), and @kbd{v =}
X(@code{calc-matrix-center-justify}) commands control whether matrix elements
Xare justified to the left, right, or center of their columns.@refill
X
X@kindex v [
X@pindex calc-vector-brackets
X@kindex v @{
X@pindex calc-vector-braces
X@kindex v (
X@pindex calc-vector-parens
XThe @kbd{v [} (@code{calc-vector-brackets}) command turns the square
Xbrackets that surround vectors and matrices displayed in the stack on
Xand off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
X(@code{calc-vector-parens}) commands use curly braces or parentheses,
Xrespectively, instead of square brackets. For example, @kbd{v @{} might
Xbe used in preparation for yanking a matrix into a buffer running
XMathematica. (In fact, the Mathematica language mode uses this mode;
X@pxref{Mathematica Language Mode}.) Note that, regardless of the
Xdisplay mode, either brackets and braces may be used to enter vectors,
Xand parentheses may never be used for this purpose.@refill
X
X@kindex v ,
X@pindex calc-vector-commas
XThe @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
Xoff in vector and matrix display.@refill
X
X@node Algebra, Units, Matrix Functions, Top
X@chapter Algebra
X
XThe commands for operating on algebraic formulas use the @kbd{a} key
Xprefix. @xref{Editing Stack Entries}, to see how to manipulate formulas
Xusing regular Emacs editing commands.
X
XMany of these commands prompt for a variable name or formula. If you
Xanswer the prompt with a blank line, the variable or formula is taken
Xfrom top-of-stack, and the normal argument for the command is taken
Xfrom second-to-top.
X
XWhen doing algebraic work, you may find several of the Calculator's
Xmodes to be helpful, including algebraic-simplification mode (@kbd{m A}),
Xalgebraic-entry mode (@kbd{m a}), fraction mode (@kbd{m f}), and
Xsymbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
Xof these modes. You may also wish to select ``big'' display mode (@kbd{d B}).
X@xref{Normal Language Modes}.
X
X@menu
X* Algebraic Manipulation::
X* Rewrite Rules::
X* Logical Operations::
X* Calculus::
X@end menu
X
X@node Algebraic Manipulation, Rewrite Rules, Algebra, Algebra
X@section Algebraic Manipulation
X
X@kindex a s
X@pindex calc-simplify
XThe @kbd{a s} (@code{calc-simplify}) command applies various algebraic
Xrules to simplify a formula. This generally entails combining like
Xterms of sums and products. The result will be a formula which is
Xmathematically equivalent to the original one, but somewhat simpler.
X
X@kindex a e
X@pindex calc-simplify-extended
XThe @kbd{a e} (@code{calc-simplify-extended}) command is like @kbd{a s}
Xexcept that it applies some additional simplifications which are not
X``safe'' in all cases. For example, powers are simplified according
Xto the rule @samp{(a^b)^c = a^(b c)}, even though this is not always
Xtrue (consider @samp{sqrt(a^2)} for negative @samp{a}). The symbolic
Xintegrator uses @code{calc-simplify-extended}; one effect of this is
Xthat the integrator's results must be used with caution. Where an
Xintegral table will often attach conditions like ``for positive @samp{a}
Xonly,'' Calc (and most other symbolic integration programs) will simply
Xproduce an unqualified result.
X
X@kindex a x
X@pindex calc-expand
XThe @kbd{a x} (@code{calc-expand}) command expands an expression by
Xapplying the distributive law. It applies to products, quotients, and
Xpowers involving sums. By default, it fully distributes all parts of
Xthe expression. With the a numeric prefix argument, the distributive
Xlaw is applied only the specified number of times, then the partially
Xexpanded expression is left on the stack.
X
XCalc's automatic simplifications will sometimes reverse a partial
Xexpansion. For example, the first step in expanding @samp{(x+1)^3} is
Xto write @samp{(x+1)*(x+1)^2}. If @code{calc-expand} stops there and
Xtries to put this formula onto the stack, though, Calc will automatically
Xsimplify it back to @samp{(x+1)^3} form. The solution is to turn
Xsimplification off first (@pxref{Simplification Modes}), or to run
X@kbd{a x} without a numeric prefix argument so that it expands all
Xthe way in one step.
X
X@kindex a c
X@pindex calc-collect
XThe @kbd{a c} (@code{calc-collect}) command rearranges a formula as a
Xpolynomial in a given variable. For example, expanding @samp{(x+y)^3}
Xwith @kbd{a x} produces a sum of four terms, initially written in
Xarbitrary order. You can use @kbd{a c x} or @kbd{a c y} to arrange
Xthe terms in order of decreasing powers of @code{x} or @code{y},
Xrespectively. You can also give any expression about which to collect:
X@kbd{a c ln(x+1)} will collect together all terms multiplied by @samp{ln(x+1)}
Xor integer powers thereof.
X
X@kindex a b
X@pindex calc-substitute
XThe @kbd{a b} (@code{calc-substitute}) command substitutes occurrences
Xof some variable or sub-expression of an expression with a new
Xsub-expression. For example, substituting @samp{sin(x)} to @samp{cos(y)}
Xin @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
X@samp{2 cos(y)^2 + x cos(y) + sin(2 x)}.
XNote that this is a purely structural substitution; the lone @code{x} and the
X@samp{sin(2 x)} stayed the same because they did not look like
X@samp{sin(x)}. @xref{Rewrite Rules}, for a more general method for
Xdoing substitutions.@refill
X
XThe @code{calc-substitute} command normally prompts for two formulas,
Xthe old one and the new one. If you enter a blank line for the first
Xprompt, all three arguments are taken from the stack (old, then new,
Xthen target expression). If you type an old formula but then enter a
Xblank line for the new one, the new formula is taken from top-of-stack
Xand the target from second-to-top. If you answer both prompts, the
Xtarget is taken from top-of-stack as usual.
X
X@node Rewrite Rules, Logical Operations, Algebraic Manipulation, Algebra
X@section Rewrite Rules
X
X@cindex Rewrite rules
X@cindex Transformations
X@cindex Pattern matching
X@kindex a r
X@pindex calc-rewrite
XThe @kbd{a r} (@code{calc-rewrite}) command makes substitutions in a
Xformula according to a specified pattern or patterns. It is like a
Xgeneralized version of @code{calc-substitute}. You provide Calc with
Xone or more @dfn{rewrite rules}, which take the form of vectors of
Xtwo or three elements: @samp{[old, new]} or @samp{[old, new, cond]}.
XTo specify a set of rules, make a vector of rule vectors. (Calc will
Xinterpret this structure as a matrix as long as the rules either all
Xinclude or all omit the @samp{cond} part. Whether or not this is true
Xmakes no difference to @code{calc-rewrite} itself.)
X
XWhen you are prompted to enter the rewrite rules, you can either type
Xthe vector directly (as an algebraic entry), or type the name of a
XCalc variable into which you have stored a rules vector, or enter
Xa blank line in which case the rules vector will be taken from the
Xtop of the stack. It is most convenient to store rules you use often
Xin a variable and invoke them by naming the variable. You may also
Xwish to use @kbd{Z V} (@code{calc-permanent-variable}) to save the
Xrules permanently; @pxref{Storing Variables}.
X
XTo match a particular formula @samp{x} with a particular rule vector
X@samp{[old, new]}, Calc compares the structure of @samp{x} with the
Xstructure of @samp{old}. Variables that appear in @samp{old} are
Xtreated as ``meta-variables''; the corresponding positions in @samp{x}
Xmay contain any sub-formulas. For example, the pattern @samp{f(x,y)}
Xwould match the expression @samp{f(12, a+1)} with the meta-variable
X@samp{x} corresponding to 12 and with @samp{y} corresponding to
X@samp{a+1}. However, this pattern would not match @samp{f(12)} or
X@samp{g(12, a+1)}, since there is no assignment of the meta-variables
Xthat will allow the pattern to match these expressions. Notice that
Xif the pattern is a single variable, it will match any expression.
X
XIf a given meta-variable appears more than once in @var{old}, the
Xcorresponding sub-formulas of @samp{x} must be identical. Thus
Xthe pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
X@samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(12, 12.0)}.
X
XThings other than variables must match exactly between the pattern
Xand the target formula. To match a particular variable, use the
Xpseudo-function @samp{quote(v)} in the pattern. For example, the
Xpattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
X@samp{sin(x)+y}.
X
XPattern-matching is completely structural, not taking the algebraic
Xproperties of formulas into account. Thus @samp{f(x+1)} matches
Xthe formula @samp{f(a+1)} but not @samp{f(a)}, @samp{f(a+2)}, or
X@samp{f(1+a)}.@refill
X
XIf the @samp{old} pattern is found to match a given formula, that
Xformula is replaced by @samp{new}, where any occurrences in @samp{new}
Xof meta-variables from the pattern are replaced with the sub-formulas
Xthat they matched. Thus, applying the rule @samp{[f(x,y), g(y+x,x)]}
Xto @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
X
XIf a @samp{cond} part is also present in the rule, this is an
Xadditional condition that must be satisfied before the rule is
Xaccepted. Once @samp{old} has been successfully matched to the
Xtarget expression, @samp{cond} is evaluated (with the meta-variables
Xsubstituted for the values they matched) and simplified with
X@code{calc-simplify}. If the result is a nonzero number, the rule
Xis accepted. If the result is zero or if it is still a symbolic
Xformula, the rule is rejected. @xref{Logical Operations}, for a
Xnumber of functions that return 1 or 0 according to the result of
Xsome test. For example, the formula @samp{n > 0} simplifies to
X1 or 0 if @samp{n} is a replaced by a positive or nonpositive
Xnumber, respectively. Thus, the rule @samp{[f(x,y), g(y+x,x), x+y > 0]}
Xwould be applied to @samp{f(0, 4)} but not to @samp{f(-3, 2)} or
X@samp{f(12, a+1)}. (In the former case, the condition can be shown
Xnot to be satisfied; in the latter case, the condition merely cannot
Xbe shown to be satisfied, but that is enough to reject the rule.)
X
XWhen @code{calc-rewrite} is used, it takes an expression from the
Xtop of the stack and attempts to match any of the specified rules
Xto any part of the expression, starting with the whole expression
Xand then, if that fails, trying deeper and deeper sub-expressions.
XFor each part of the expression, the rules are tried in the
Xspecified order. The first rule to match the first sub-expression
Xwins; it replaces the matched sub-expression according to the
X@samp{new} part of the rule. By default, only one substitution
Xis made per @code{calc-rewrite} command.
X
XGiven a positive numeric prefix argument, @code{calc-rewrite} will
Xperform up to that many substitutions before stopping. First, the
Xtop-level formula is matched and substituted repeatedly until it
Xno longer matches the pattern; then, sub-formulas are tried, and
Xso on. If the limit still has not been reached, the top-level
Xformula is check again (in case a substitution of one of its
Xarguments causes it again to match). This continues until no
Xfurther matches can be made, or until the limit is reached.
X
XGiven a negative numeric prefix argument, @code{calc-rewrite} will
Xmatch and substitute the top-level expression up to that many times,
Xbut will not attempt to match any sub-expressions.
X
XA prefix argument of zero is equivalent to a prefix of 25. This is
Xmeant to be a convenient limit that is usually enough to finish the
Xjob, but not enough to be too slow in case the rule transformations
Xget into an infinite loop.
X
XAs a larger example, suppose we wish to substitute the pattern
X@samp{sin(x)^2 + cos(x)^2} with 1 in a given expression. The rule
X@samp{[sin(x)^2 + cos(x)^2, 1]} would work, but only for the most
Xliteral matches. This would miss the formula @samp{2 sin(a)^2 + 2
Xcos(a)^2}, for example, because our pattern does not account for both
Xterms being multiplied by an equal factor. A more flexible solution
Xwould be to use the rule @samp{[cos(x)^2, 1 - sin(x)^2]}, followed by
Xalgebraic simplification if necessary.@refill
X
XSimilarly, the pattern @samp{[f(x, x+1), g(x)]} is not a very reliable
Xrule since it fails to match in cases like @samp{f(2, 3)}. A better
Xrule would be @samp{[f(x,y), g(x), y=x+1]}, since the algebraic
Xsimplifier is applied to conditions but not to patterns.
X
XAs another example, we could define our own factorial function with the
Xrules @samp{[[f(0), 1], [f(n), n*f(n-1), n>0]]}. Enter this rules
Xvector and store it in a variable: @kbd{s factrules}. Now, given a
Xfactorial formula like @samp{f(5)} on the stack, type @samp{C-u 0 a r
Xfactrules} to apply these rules repeatedly. After six applications,
X@code{calc-rewrite} will stop with 120 on the stack. If we expect to
SHAR_EOF
echo "End of part 16"
echo "File calc.texinfo is continued in part 17"
echo "17" > s2_seq_.tmp
exit 0