We will be considering Algorithm 2.3.3.A from Knuth's 'Fundamental Algorithms' Knuth (1968) (p. 357) which is an algorithm for the addition of polynomials represented using four-directional ...
This example and others inducing Froissart doublets have led to the construction of an even more stable representation of rational functions that has been used to develop the so-called AAA algorithm [38]. In AAA, a rational function is represented as a ratio of rationals: \begin{aligned} r(...
Comprehensive univariate polynomial class. All arithmetic performed symbolically. Some advanced features include: Arithmetic of polynomial rings over a finite field, the Tonelli-Shanks algorithm, GCD, exponentiation by squaring, irreducibility checking,
A Divide-and-conquer algorithm can be found here: split two polynomials in half, then recursively do 44 polynomial multiplications, and finally combine them together (polynomial addition is O(n)O(n) anyway) P.s As A[0]=A[0](x2)A[0]=A[0](x2) and A[1]=A[1](x2)A[1]=A[1...
Matrix addition: The sum B + C of two matrices B and C having the same order is obtained by adding the corresponding elements in B and C. That is, B+C=[bij]+[cij]=[bij+cij] So, for example, if B=(53−127−5)and C=(32810−1−3)then B+C=(857126−8) Matri...
The polynomial a must be strictly positive on \mathcal {K} because P is positive definite on that set, so we can apply one step of the Cholesky factorization algorithm to write \begin{aligned} L(x) P(x) L(x)^{{\mathsf T}}= \begin{bmatrix} a(x) &{}\quad 0 &{}\quad 0 \...
This paper develops a new grey prediction model with quadratic polynomial term. Analytical expressions of the time response function and the restored values of the new model are derived by using grey model technique and mathematical tools. With observati
The full PC2 is constructed using Algorithm 1 (KKT) and the sparse adaptive PC2 is constructed using Conclusions & further work A novel methodology for constructing physics-informed non-intrusive regression-based PCE, referred to as PC2 was proposed in this paper. Physical constraints in the ...
, use the division algorithm to get \(n=mg+i\) . we hope to show that \(y(n)=y(n-p)\) . suppose that \(i\in \{\gamma ,\ldots ,g-1\}\) . then by condition (ii), $$\begin{aligned} y(n)=y_i(m)={\left\{ \begin{array}{ll} y_{i-r}(m-x)& i\ge r\\ y...
In the 1920s, Alexander gave an algorithm for computing a polynomial invariant ΔK(t) (a Laurent polynomial in t) of a knot K, called the Alexander polynomial, by using its projection on a plane. He also gave its topological interpretation as an annihilator of a certain cohomology module ...