Martin Ziegler Issued on 2014-04-22 Solutions due: 2014-04-28
Algebraic Complexity Theory SS 2014, Exercise Sheet #2
p0·q0 p1·q0+p0·q1
p2·q0+p1·q1+p0·q2 p2·q1+p1·q2
p2·q2
(∗)
=
1 x0 x20 x30 x40 1 x1 x21 x31 x41 1 x2 x22 x32 x42 1 x3 x23 x33 x43 1 x4 x24 x34 x44
−1
·
(p0+p1·x0+p2·x20)·(q0+q1·x0+q2·x20) (p0+p1·x1+p2·x21)·(q0+q1·x1+q2·x21) (p0+p1·x2+p2·x22)·(q0+q1·x2+q2·x22) (p0+p1·x3+p2·x23)·(q0+q1·x3+q2·x23) (p0+p1·x4+p2·x24)·(q0+q1·x4+q2·x24)
EXERCISE 3:
Recall Karatsuba’s Algorithm for polynomial multiplication usingO(nlog23)⊆O(n1.585)arithmetic operations. Now fix an algebraAover the infinite fieldF.
a) Verify the above identity (*) for any pairwise distinct x0,x1, . . . ,xd∈Fand arbitary p0+p1· X+p2·X2,q0+q1·X+q2·X2∈A[X].
b) Choose xj= j, say, and conclude that two quadratic polynomials overAcan me multiplied using 5 — instead of 9 — multiplications inA(and arbitrary many additions inAas well as multiplications by constants fromF).
c) Derive an algorithm for multiplying two polynomials overAof degree n usingO(nlog35)⊆ O(n1.465)arithmetic operations and constants fromF.
d) Generalize a) and b) in order to obtain an algorithm multiplying p∈A[X]of deg(p)≤k and q∈A[X]of deg(q)≤ℓusing k+ℓ+1 multiplications inA(and arbitrary many additions in Aas well as multiplications by constants fromF).
Can you identify Karatsuba as a special case?
e) Derive, for any fixedε>0, an algorithm multiplying two polynomials overA of degree at most n usingO(n1+ε)arithmetic operations and constants fromF.
EXERCISE 4:
Formalize the following algorithms as straight-line programs and analyze their costs:
a) Compute the determinant of a given 3×3–matrix using Sarrus’ Rule.
b) Compute the determinant of a given n×n–matrix using Laplace’s Expansion.
c) Compute the determinant of a given n×n–matrix using Leibnitz’ Formula.
d) Compute the determinant of a given 3×3–matrix via its LU–decomposition/Gaussian Elimi- nation.
