Sum

Sum 是求和的函式，內建Loop，寫成: 

]Sum(k = a to b, f(k)) = Σ(k = a to b, f(k)) //其中a、b 為上下限，f(k) 為被加函式。

]

]Σ(k = 1 to 3, 2x) = 2x||x = 1|| + 2x||x = 2|| + 2x||x = 3|| = 2 + 4 + 6 = 12

]

]Σ(k = 2 to 5, a(k)||a(k) = {1, 10, 4, 6, 0.8}||) = a(2) + a(3) + a(4) + a(5) = 10 + 4 + 6 + 0.8 = 20.8

 

針對被加函式，若被加函式可拆為常數與變數相成，則常數可提出。

]Σ(k = a to b, C * f(k)) = C * Σ(k = a to b, f(k))

 

Sum 函式裡，並沒有規定上下線必須擺在前或後，從小到大、從大到小都是允許的。

]Σ(k = a to b, f(k)) = Σ(k = b to a, f(k))

 

上下線拆解規則: 

]rule.. Σ(k = a to b||a < b||, f(k)) = stick(c, a) * Σ(k = a to c, f(k)) + stick(c, b) * Σ(k = c to b, f(k)) - stick(c, a) * stick(b, c) * which(0.5, a, b, c) //特別注意a 必須小於b。

 

拆解規則很長，基本上可以用理解的: 

]rule.. Σ(k = a to b||a < b||, f(k))

]if (a < b < c) then.. 

]ans = Σ(k = a to c, f(k)) - Σ(k = b to c + 1, f(k))

]if (a < c < b) then..

]ans = Σ(k = a to c, f(k)) + Σ(k = c + 1 to b, f(k))

]if (c < a < b) then.. 

]ans = -Σ(k = c to a - 1, f(k)) + Σ(k = c to b, f(k))