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174 CHAPTER 7. CALCULUS
(7.14) t 1 = ∏ r i−1
i . Furthermore every factor of t 2 arises from the r i , and is not
repeated. Hence we can choose
t 1 = gcd(r, r ′ ) and t 2 = r/t 1 . (7.19)
Having done this, we can solve for the coefficients in s 1 and s 2 , and the resulting
equations are linear in the unknown coefficients. More precisely, the equations
become
q = s ′ r t ′
1 − s
1t 2
1 + s 2 t 1 , (7.20)
t i t 1
where the polynomial divisions are exact, and the linearity is now obvious. The
programmer should note that s 1 /t 1 is guaranteed to be in lowest terms, but
s 2 /t 2 is not (and indeed will be 0 if there is no logarithmic term).
7.2.4 The Trager–Rothstein Algorithm
Whether we use the method of section 7.2.2 or 7.2.3, we have to integrate
the logarithmic part. (7.8)–(7.11) shows that this may, but need not, require
algebraic numbers. How do we tell? The answer is provided by the following
∑
observation 3 [Rot76, Tra76]: if we write the integral of the logarithmic part as
ci log v i , we can determine the equation satisfied by the c i , i.e. the analogue
of (7.12), by purely rational computations.
So write
∫
s2
= ∑ c i log v i , (7.21)
t 2
where we can assume 4 :
1.
s 2
t2
is in lowest terms;
2. the v i are polynomials (using log f g
= log f − log g);
3. the v i are square-free (using log ∏ f i i = ∑ i log f i );
4. the v i are relatively prime (using c log pq +d log pr = (c+d) log p+c log q +
d log r);
5. the c i are all different (using c log p + c log q = c log pq);
6. the c i generate the smallest possible extension of the original field of coefficients.
(7.21) can be rewritten as
s 2
t 2
= ∑ c i
v ′ i
v i
. (7.22)
3 As happens surprisingly often in computer algebra, this was a case of simultaneous discovery.
4 We are using “standard” properties of the log operator here without explicit justification:
they are justified in Section 7.3, and revisited in Section 8.6.