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Expand


Description

  • Expand expands out products and positive integer powers.
  • Expand[transformations] expands out products and positive integer powers and applies transformations at each level of expand procedure.

Examples


Expand a polynomial expressions:

println Expand >> '(x_n + y_n)*(f_m - r_m)'.t
   > x_{n}*f_{m}+f_{m}*y_{n}-r_{m}*y_{n}-x_{n}*r_{m}

println Expand >> '(1 + x)**4'.t
   > x**4+1+4*x**3+6*x**2+4*x


println Expand >> '(x + y)/z'.t
   > x/z+y/z


Expand relabels dummies when necessary:

println Expand >> '(A_m^m + 1)**3'.t
   > 3*A_{m}^{m}*A_{a}^{a}+A_{m}^{m}*A_{a}^{a}*A_{b}^{b}+1+3*A_{b}^{b}


Expand does not go inside functions and denominators; ExpandAll does:

println Expand >> 'f[(x + y)**2]'.t
   > f[(x + y)**2]
println ExpandAll >> 'f[(x + y)**2]'.t
   > f[x**2 + 2*x*y + y**2]


Details

Expand[transformations] will additionally apply transformations during expand procedure:

println Expand['k_a*k^a = 0'.t] >> '(k_a + t_a)*(k^a + t^a)'.t
   > 2*k_a*t^a + t_a*t^a

Passing additional transformations can significantly improve the performance when expanding huge expressions. For example, when a huge expression involves many metric tensors, one can pass EliminateMetrics in order to reduce the number of processed terms. Consider a random example:

//create random generator, which generates
// random tensors consisting of metric and A_m
RandomTensor randomTensor = new RandomTensor();
randomTensor.clearNamespace()
randomTensor.addToNamespace('g_mn'.t, 'A_m'.t)

//loop to warm up JVM
for (def i in 1..1000) {
    def a, b
    //next random tensor
    def t = randomTensor.nextTensorTree(4, 3, 8, '_a'.si)
    def simplify = EliminateMetrics & 'A_a*A^a = 1'.t & 'd^i_i = 10'.t

    //this will typically 10 times faster
    timing {
        a = Expand[simplify] >> t
    }
    //then this
    timing {
        b = (Expand & simplify) >> t
    }

    assert a == b
    println ''
}
The sample output will looks like:
Time: 10 ms.
Time: 1015 ms.

Time: 7 ms.
Time: 6566 ms.

Time: 66 ms.
Time: 983 ms.
...

See also