Using Strassen's algorithm to accelerate the solution of linear systems
The Journal of Supercomputing
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Practical integer division with Karatsuba complexity
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Modern Computer Algebra
A computational introduction to number theory and algebra
A computational introduction to number theory and algebra
Approximate polynomial gcd: Small degree and small height perturbations
Journal of Symbolic Computation
Counting decomposable multivariate polynomials
Applicable Algebra in Engineering, Communication and Computing
Interval Partitions and Polynomial Factorization
Algorithmica
A Modified Split-Radix FFT With Fewer Arithmetic Operations
IEEE Transactions on Signal Processing
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The classical division algorithm for polynomials requires O(n2) operations for inputs of size n. Using reversal technique and Newton iteration, it can be improved to O(M(n)), where M is a multiplication time. But the method requires that the degree of the modulo, xl, should be the power of 2. If l is not a power of 2 and f(0)=1, Gathen and Gerhard suggest to compute the inverse, f−1, modulo $x^{\lceil l/2^r\rceil}, x^{\lceil l/2^{r-1}\rceil}, \cdots, x^{\lceil l/2\rceil}, x^ l$ , separately. But they did not specify the iterative step. In this paper, we show that the original Newton iteration formula can be directly used to compute f−1 mod xl without any additional cost, when l is not a power of 2.