NAME
    mod - compute the remainder for an integer quotient

SYNOPSIS
    mod(x, y [,rnd])
    x % y

TYPES
    If x is a matrix or list, the returned value is a matrix or list v of
    the same structure for which each element v[[i]] = mod(x[[i]], y, rnd).

    If x is an xx-object or x is not an object and y is an xx-object,
    this function calls the user-defined function xx_mod(x, y, rnd);
    the types of arguments and returned value are as required by the
    definition of xx_mod().

    If neither x nor y is an object, or x is not a matrix or list:

    x		number (real or complex)
    y		real
    rnd		integer, defaults to config("mod")

    return	number

DESCRIPTION
    The expression:

    	x % y

    is equivalent to call:

    	mod(x, y)

    The function:

	mod(x, y, rnd)

    is equivalent to:

    	config("mod", rnd), x % y

    except that the global config("mod") value does not change.

    If x is real or complex and y is zero, mod(x, y, rnd) returns x.

    If x is complex, mod(x, y, rnd) returns
		mod(re(x), y, rnd) + mod(im(x), y, rnd) * 1i.

    In the following it is assumed x is real and y is nonzero.

    If x/y is an integer mod(x, y, rnd) returns zero.

    If x/y is not an integer, mod(x, y, rnd) returns one of the two
    values of r for which for some integer q exists such that x = q * y + r
    and abs(r) < abs(y).  Which of the two values or r that is returned is
    controlled by rnd.

    If bit 4 of rnd is set (e.g. if 16 <= rnd < 32) abs(r) <= abs(y)/2;
    this uniquely determines r if abs(r) < abs(y)/2.  If bit 4 of rnd is
    set and abs(r) = abs(y)/2, or if bit 4 of r is not set, the result for
    r depends on rnd as in the following table:

	     rnd & 15	   sign of r		parity of q

		0	     sgn(y)
		1	    -sgn(y)
		2	     sgn(x)
		3	    -sgn(x)
		4	      +
		5	      -
		6	     sgn(x/y)
		7	    -sgn(x/y)
		8				   even
		9				   odd
	       10				even if x/y > 0, otherwise odd
	       11				odd if x/y > 0, otherwise even
	       12				even if y > 0, otherwise odd
	       13				odd if y > 0, otherwise even
	       14				even if x > 0, otherwise odd
	       15				odd if x > 0, otherwise even

		NOTE: Blank entries in the table above indicate that the
		     description would be complicated and probably not of
		     much interest.

    The C language method of modulus and integer division is:

	    config("quomod", 2)
	    config("quo", 2)
	    config("mod", 2)

    This dependence on rnd is consistent with quo(x, y, rnd) and
    appr(x, y, rnd) in that for any real x and y and any integer rnd,

	    x = y * quo(x, y, rnd) + mod(x, y, rnd).
	    mod(x, y, rnd) = x - appr(x, y, rnd)

    If y and rnd are fixed and mod(x, y, rnd) is to be considered as
    a canonical residue of x % y, bits 1 and 3 of rnd should be
    zero: if 0 <= rnd < 32, it is only for rnd = 0, 1, 4, 5, 16, 17,
    20, or 21, that the set of possible values for mod(x, y, rnd)
    form an interval of length y, and for any x1, x2,

	    mod(x1, y, rnd) = mod(x2, y, rnd)

    is equivalent to:

	    x1 is congruent to x2 modulo y.

    This is particularly relevant when working with the ring of
    integers modulo an integer y.

EXAMPLE
    ; print mod(11,5,0), mod(11,5,1), mod(-11,5,2), mod(-11,-5,3)
    1 -4 -1 4

    ; print mod(12.5,5,16), mod(12.5,5,17), mod(12.5,5,24), mod(-7.5,-5,24)
    2.5 -2.5 2.5 2.5

    ; A = list(11,13,17,23,29)
    ; print mod(A,10,0)

    list (5 elements, 5 nonzero):
	[[0]] = 1
	[[1]] = 3
	[[2]] = 7
	[[3]] = 3
	[[4]] = 9

LIMITS
    none

LINK LIBRARY
    void modvalue(VALUE *x, VALUE *y, VALUE *rnd, VALUE *result)
    NUMBER *qmod(NUMBER *y, NUMBER *y, long rnd)

SEE ALSO
    quo, quomod, //, %

## Copyright (C) 1999-2006  Landon Curt Noll
##
## Calc is open software; you can redistribute it and/or modify it under
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## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
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## Under source code control:	1995/09/18 02:09:31
## File existed as early as:	1995
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