Martin Ziegler Issued on 2014-05-27 Solutions due: 2014-06-02
Algebraic Complexity Theory SS 2014, Exercise Sheet #6
EXERCISE 11:
Devise straight-line programs overS1= C,C,+,×,÷
of lengthO(N·polylog N)for multiplying the following structured N×N–matrices, given by theirO(N)parameters, to a given N–vector:
a) A circulant matrix C(c0, . . . ,cN−1):= cj−i mod N
0≤i,j<N
b) A Toeplitz matrix T(t−N+1, . . . ,t0, . . . ,tN−1):= tj−i
0≤i,j<N
c) A Hankel matrix H(h−N+1, . . .,h0, . . . ,hN−1):= hj+i−N
0≤j<N;1≤i≤N
d) A Vandermonde matrix V(v1, . . .,vN):= vij
1≤i≤N;0≤j<N
e) A generalized Hilbert matrix Gp(z1, . . .,zN; w1, . . . ,wN):= (zj−wi)−p
1≤i,j≤N
where p∈Nis fixed and zj6=wi. (For p=1 this is known as Trummer’s Problem. . . )
c0 c1 c2 . . . cN−2 cN−1
cN−1 c0 c1 c2 . .. cN−2
cN−2 cN−1 c0 c1 . .. ... ... . .. ... ... . .. c2
c2 c3 . .. cN−1 c0 c1
c1 c2 . . . cN−2 cN−1 c0
,
h−N+1 h−N+2 . . . h−2 h−1 h0
h−N+2 h−N+3 . . . h−1 h0 h1
h−N+3 h−N+4 . . . h0 h1 h2
... . .. . .. ... . .. ... h−2 h−1 h0 h1 . .. ...
h−1 h0 h1 h2 . . . hN−2
h0 h1 h2 . . . hN−2 hN−1
t0 t1 t2 . . . tN−2 tN−1
t−1 t0 t1 t2 . . . tN−2
t−2 t−1 t0 t1 . .. ... ... . .. . .. ... ... ...
t−N+2 t−N+3 . . . t−1 t0 t1
t−N+1 t−N+2 . . . t−2 t−1 t0
,
1 v1 v21 · · · vN−11 1 v2 v22 · · · vN−12
... ... ... . .. ... 1 vN v2N · · · vN−1N
EXERCISE 12:
The 2D Gravitation/Coulomb potential u of a charge/mass distribution f :R2→Rsatisfies Pois- son’s equation∆u= f .
a) Verify that the fundamental solution u(x,y) =ln(x2+y2)/2 is a solution onR2\ {(0,0)}with
‘force’∇u(x,y) = (Re,Im)x−iy1 .
b) Devise a subquadratic-time algorithm for computing, given the coordinates(xn,yn)of N par- ticles in the plane, their N mutual forces∑N1≤m6=n∇u(xn−xm,yn−ym), 1≤n≤N.