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Kron_delta

Function: kron_delta (<x>, <y>) Represents the Kronecker delta function.

kron_delta simplifies to 1 when <x> and <y> are identical or demonstrably equivalent, and it simplifies to 0 when <x> and <y> are demonstrably not equivalent. Otherwise, it is not certain whether <x> and <y> are equivalent, and kron_delta simplifies to a noun expression. kron_delta implements a cautious policy with respect to floating point expressions: if the difference <x> - <y> is a floating point number, kron_delta simplifies to a noun expression when <x> is apparently equivalent to <y>.

Specifically, kron_delta(<x>, <y>) simplifies to 1 when is(x = y) is true. kron_delta also simplifies to 1 when sign(abs(<x> - <y>)) is zero and <x> - <y> is not a floating point number (neither an ordinary float nor a bigfloat). kron_delta simplifies to 0 when sign(abs(<x> - <y>)) is pos.

Otherwise, sign(abs(<x> - <y>)) is something other than pos or zero, or it is zero and <x> - <y> is a floating point number. In these cases, kron_delta returns a noun expression.

kron_delta is declared to be symmetric. That is, kron_delta(<x>, <y>) is equal to kron_delta(<y>, <x>).

Examples:

The arguments of kron_delta are identical. kron_delta simplifies to 1.

          (%i1) kron_delta (a, a);
          (%o1)                           1
          (%i2) kron_delta (x^2 - y^2, x^2 - y^2);
          (%o2)                           1
          (%i3) float (kron_delta (1/10, 0.1));
          (%o3)                           1

The arguments of kron_delta are equivalent, and their difference is not a floating point number. kron_delta simplifies to 1.

          (%i1) assume (equal (x, y));
          (%o1)                     [equal(x, y)]
          (%i2) kron_delta (x, y);
          (%o2)                           1

The arguments of kron_delta are not equivalent. kron_delta simplifies to 0.

          (%i1) kron_delta (a + 1, a);
          (%o1)                           0
          (%i2) assume (a > b)$
          (%i3) kron_delta (a, b);
          (%o3)                           0
          (%i4) kron_delta (1/5, 0.7);
          (%o4)                           0

The arguments of kron_delta might or might not be equivalent. kron_delta simplifies to a noun expression.

          (%i1) kron_delta (a, b);
          (%o1)                   kron_delta(a, b)
          (%i2) assume(x >= y)$
          (%i3) kron_delta (x, y);
          (%o3)                   kron_delta(x, y)

The arguments of kron_delta are equivalent, but their difference is a floating point number. kron_delta simplifies to a noun expression.

          (%i1) 1/4 - 0.25;
          (%o1)                          0.0
          (%i2) 1/10 - 0.1;
          (%o2)                          0.0
          (%i3) 0.25 - 0.25b0;
          Warning:  Float to bigfloat conversion of 0.25
          (%o3)                         0.0b0
          (%i4) kron_delta (1/4, 0.25);
                                            1
          (%o4)                  kron_delta(-, 0.25)
                                            4
          (%i5) kron_delta (1/10, 0.1);
                                            1
          (%o5)                  kron_delta(--, 0.1)
                                            10
          (%i6) kron_delta (0.25, 0.25b0);
          Warning:  Float to bigfloat conversion of 0.25
          (%o6)               kron_delta(0.25, 2.5b-1)

kron_delta is symmetric.

          (%i1) kron_delta (x, y);
          (%o1)                   kron_delta(x, y)
          (%i2) kron_delta (y, x);
          (%o2)                   kron_delta(x, y)
          (%i3) kron_delta (x, y) - kron_delta (y, x);
          (%o3)                           0
          (%i4) is (equal (kron_delta (x, y), kron_delta (y, x)));
          (%o4)                         true
          (%i5) is (kron_delta (x, y) = kron_delta (y, x));
          (%o5)                         true

(%o1)                                true
(%i2)