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Algebraic Complexity Theory SS 2014, Exercise Sheet #6

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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):= (zjwi)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 forcesN1≤m6=n∇u(xnxm,ynym), 1≤nN.

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