Lecture 6 1 Multipoint evaluation of a polynomial
... over all 0 ≤ j < 2k−i−1 for a fixed i + 1, we spend 2k−i−1 · O(M(2i )) = O(M(n)) operations (by Observation 1). Now summing over all 0 ≤ i ≤ k − 1, we can bound the complexity of this (pre)computation step by O(M(n) log n) operations over R. Algorithm 1 Multipoint evaluation (Pre)computation step: C ...
... over all 0 ≤ j < 2k−i−1 for a fixed i + 1, we spend 2k−i−1 · O(M(2i )) = O(M(n)) operations (by Observation 1). Now summing over all 0 ≤ i ≤ k − 1, we can bound the complexity of this (pre)computation step by O(M(n) log n) operations over R. Algorithm 1 Multipoint evaluation (Pre)computation step: C ...
1 Factorization of Polynomials
... – Irreducible polynomials in F [x] of degree 1: – Irreducible polynomials in F [x] of degree 2: – Irreducible polynomials in F [x] of degree 3: – Irreducible polynomials in F [x] of degree 4+: – No simple characterization in general for high degree polynomials. (The conditions in Gallian are necess ...
... – Irreducible polynomials in F [x] of degree 1: – Irreducible polynomials in F [x] of degree 2: – Irreducible polynomials in F [x] of degree 3: – Irreducible polynomials in F [x] of degree 4+: – No simple characterization in general for high degree polynomials. (The conditions in Gallian are necess ...
The calculation of the degree of an approximate greatest common
... resultant matrix R(f, g). This computation is usually performed by placing a threshold on the small singular values of R(f, g), but this method suffers from disadvantages because the numerical rank of R(f, g) may not be defined, or the noise level imposed on the coefficients of f (y) and g(y) may no ...
... resultant matrix R(f, g). This computation is usually performed by placing a threshold on the small singular values of R(f, g), but this method suffers from disadvantages because the numerical rank of R(f, g) may not be defined, or the noise level imposed on the coefficients of f (y) and g(y) may no ...
2009-04-02 - Stony Brook Mathematics
... N is a (commutative) semigroup, which is a set of numbers that are commutative and associative, meaning that we have a set N and an operation (+) so that the following is true for any a, b, c € N: a + b € N (closure); a + b = b + a (commutativity); and a + (b + c) = (a + b) + c (associativity). Note ...
... N is a (commutative) semigroup, which is a set of numbers that are commutative and associative, meaning that we have a set N and an operation (+) so that the following is true for any a, b, c € N: a + b € N (closure); a + b = b + a (commutativity); and a + (b + c) = (a + b) + c (associativity). Note ...
H8
... (c) For any polynomial f (x) ∈ R[x] the numbers f (2), f (3), f ′ (2), and f ′ (3), determine the polynomial f (x) uniquely up to multiples of m(x) = (x − 2)2 (x − 3)2 , i.e., mod m(x). The remainder when dividing by m(x) is a polynomial of degree ≤ 3, and so can be written in the form c0 + c1 x + c ...
... (c) For any polynomial f (x) ∈ R[x] the numbers f (2), f (3), f ′ (2), and f ′ (3), determine the polynomial f (x) uniquely up to multiples of m(x) = (x − 2)2 (x − 3)2 , i.e., mod m(x). The remainder when dividing by m(x) is a polynomial of degree ≤ 3, and so can be written in the form c0 + c1 x + c ...
File
... Polynomial: One term or the sum/difference of two or more terms. To be a polynomial: • Variable Bases • Whole number exponents • Real number coefficients ...
... Polynomial: One term or the sum/difference of two or more terms. To be a polynomial: • Variable Bases • Whole number exponents • Real number coefficients ...