Question

考虑(a-b)/(c-d)操作，其中a，b，c和d是浮点数（即double类型C ++）。 (a-b)和(c-d)都是（sum - correction）对，如Kahan summation algorithm中所示。简而言之，这些（sum - correction）对的具体内容是sum包含相对于correction中的值较大的值。更准确地说，correction包含由于数值限制（sum类型中53位尾数）而在求和期间不适合double的内容。

考虑到数字的上述特征，计算(a-b)/(c-d)的数值最精确的方法是什么？

额外问题：最好将结果也设为（sum - correction），就像在Kahan求和算法中一样。所以要查找(e-f)=(a-b)/(c-d)，而不仅仅是e=(a-b)/(c-d)。

Answer 1

The div2 algorithm of Dekker (1971) is a good approach.

It requires a mul12(p,q) algorithm which can exactly computes a pair u+v = p*q. Dekker uses a method known as Veltkamp splitting, but if you have access to an fma function, then a much simpler method is

u = p*q
v = fma(p,q,-u)

the actual division then looks like (I've had to change some of the signs since Dekker uses additive pairs instead of subtractive):

r   = a/c
u,v = mul12(r,c)
s   = (a - u - v - b + r*d)/c

The the sum r+s is an accurate approximation to (a-b)/(c-d).

UPDATE: The subtraction and addition are assumed to be left-associative, i.e.

s = ((((a-u)-v)-b)+r*d)/c

This works because if we let rr be the error in the computation of r (i.e. r + rr = a/c exactly), then since u+v = r*c exactly, we have that rr*c = a-u-v exactly, so therefore (a-u-v-b)/c gives a fairly good approximation to the correction term of (a-b)/c.

The final r*d arises due to the following:

(a-b)/(c-d) = (a-b)/c * c/(c-d) = (a-b)/c *(1 + d/(c-d)) 
            = [a-b + (a-b)/(c-d) * d]/c

Now r is also a fairly good initial approximation to (a-b)/(c-d) so we substitute that inside the [...], so we find that (a-u-v-b+r*d)/c is a good approximation to the correction term of (a-b)/(c-d)

Answer 2

对于微小的修正，可能会想到

(a - b) / (c - d) = a/b (1 - b/a) / (1 - c/d) ~ a/b (1 - b/a + c/d)

什么是数字上最精确的方法来划分总和或差异？

2 个答案: