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113 lines
3.1 KiB
Plaintext
113 lines
3.1 KiB
Plaintext
NAME
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mod - compute the remainder for an integer quotient
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SYNOPSIS
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mod(x, y, rnd)
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x % y
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TYPES
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If x is a matrix or list, the returned value is a matrix or list v of
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the same structure for which each element v[[i]] = mod(x[[i]], y, rnd).
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If x is an xx-object or x is not an object and y is an xx-object,
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this function calls the user-defined function xx_mod(x, y, rnd);
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the types of arguments and returned value are as required by the
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definition of xx_mod().
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If neither x nor y is an object, or x is not a matrix or list:
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x number (real or complex)
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y real
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rnd integer, defaults to config("mod")
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return number
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DESCRIPTION
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If x is real or complex and y is zero, mod(x, y, rnd) returns x.
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If x is complex, mod(x, y, rnd) returns
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mod(re(x), y, rnd) + mod(im(x), y, rnd) * 1i.
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In the following it is assumed x is real and y is nonzero.
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If x/y is an integer mod(x, y, rnd) returns zero.
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If x/y is not an integer, mod(x, y, rnd) returns one of the two numbers
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r for which for some integer q, x = q * v + r and abs(r) < abs(y).
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Which of the two numbers is returned is controlled by rnd.
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If bit 4 of rnd is set (e.g. if 16 <= rnd < 32) abs(r) <= abs(y)/2;
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this uniquely determines r if abs(r) < abs(y)/2. If bit 4 of rnd is
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set and abs(r) = abs(y)/2, or if bit 4 of r is not set, the result for
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r depends on rnd as in the following table:
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(Blank entries indicate that the description would be complicated
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and probably not of much interest.)
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rnd & 15 sign of r parity of q
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0 sgn(y)
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1 -sgn(y)
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2 sgn(x)
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3 -sgn(x)
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4 +
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5 -
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6 sgn(x/y)
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7 -sgn(x/y)
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8 even
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9 odd
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10 even if x/y > 0, otherwise odd
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11 odd if x/y > 0, otherwise even
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12 even if y > 0, otherwise odd
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13 odd if y > 0, otherwise even
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14 even if x > 0, otherwise odd
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15 odd if x > 0, otherwise even
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This dependence on rnd is consistent with quo(x, y, rnd) and
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appr(x, y, rnd) in that for any real x and y and any integer rnd,
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x = y * quo(x, y, rnd) + mod(x, y, rnd).
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mod(x, y, rnd) = x - appr(x, y, rnd)
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If y and rnd are fixed and mod(x, y, rnd) is to be considered as
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a canonical residue of x modulo y, bits 1 and 3 of rnd should be
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zero: if 0 <= rnd < 32, it is only for rnd = 0, 1, 4, 5, 16, 17,
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20, or 21, that the set of possible values for mod(x, y, rnd)
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form an interval of length y, and for any x1, x2,
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mod(x1, y, rnd) = mod(x2, y, rnd)
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is equivalent to:
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x1 is congruent to x2 modulo y.
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This is particularly relevant when working with the ring of
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integers modulo an integer y.
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EXAMPLE
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> print mod(11,5,0), mod(11,5,1), mod(-11,5,2), mod(-11,-5,3)
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1 -4 -1 4
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> print mod(12.5,5,16), mod(12.5,5,17), mod(12.5,5,24), mod(-7.5,-5,24)
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2.5 -2.5 2.5 2.5
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> A = list(11,13,17,23,29)
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> print mod(A,10,0)
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list (5 elements, 5 nonzero):
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[[0]] = 1
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[[1]] = 3
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[[2]] = 7
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[[3]] = 3
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[[4]] = 9
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LIMITS
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none
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LIBRARY
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void modvalue(VALUE *x, VALUE *y, VALUE *rnd, VALUE *result)
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NUMBER *qmod(NUMBER *y, NUMBER *y, long rnd)
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SEE ALSO
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quo, quomod, //, %
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