3

I'm not sure if this is exactly what you're looking for, but my go-to volume for these kinds of question is Symbolic Integration I by Manuel Bronstein. Risch's original treatment is sketchy in many places, and Bronstein did a lot of work to flesh out the details and actually implement Risch's methods. From the Foreword by B. F. Caviness:
With the advent ...

2

You have an interesting kind of partitioning problem here. (Maybe that's why you are using lattices.)
One thing that should be noted. If d is a join b for some given a and b which are incomparable, then d is NOT a meet c for any c in the lattice. So when you focus on the first coordinate a, you immediately divide X minus a into three sets: those above a, ...

1

J. H. Davenport, On the integration of algebraic functions. Lecture Notes in Computer Science, 102. Springer-Verlag, Berlin-New York, 1981.
J. H. Davenport, Integration in closed form. Computers in mathematical research (Cardiff, 1986), 119–134, Inst. Math. Appl. Conf. Ser. New Ser., 14, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988.

answered Jun 24 at 12:54

Alexandre Eremenko

63k66 gold badges174174 silver badges297297 bronze badges

1

Let
$$M = \begin{pmatrix} | & | & \cdots & | \\ v_1 & v_2 & \cdots & v_d \\ | & | & \cdots & | \end{pmatrix}$$
and let $e_1, \ldots, e_N$ be the standard unit vectors. Then consider the linear programs indexed by $i = 1, \ldots, N$:
$$\begin{aligned}
&\text{maximize }\langle e_i,Mx\rangle \\
&\text{subject to }\...

1

A very simple and efficient heuristic that I found recently is as follows:
proceed as in the calculation of Minimum Spanning Trees with Kruskal algorithm, i.e. by inserting edges into a Disjoint Set datastructure in order of increasing length.
if adding an edge generates a cycle
add that cycle to the set of covering cycles
restart the algorithm with that ...

Only top voted, non community-wiki answers of a minimum length are eligible

#### Related Tags

algorithms × 1203graph-theory × 249

co.combinatorics × 232

computational-complexity × 174

nt.number-theory × 154

linear-algebra × 101

computer-science × 79

reference-request × 64

gr.group-theory × 64

mg.metric-geometry × 60

computational-geometry × 54

pr.probability × 52

ag.algebraic-geometry × 46

na.numerical-analysis × 42

computational-number-theory × 40

polynomials × 39

approximation-algorithms × 39

discrete-geometry × 37

matrices × 35

linear-programming × 31

combinatorial-optimization × 31

oc.optimization-and-control × 30

ac.commutative-algebra × 29

prime-numbers × 28

permutations × 25