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**1 - 6**of**6**### Algorithms, Certification, and CryptographyTable of contents

"... 6.2.1. Mixed-precision fused multiply-and-add 11 6.2.2. Multiplication by rational constants versus division by a constant 11 6.2.3. Floating-point exponentiation on FPGA 11 6.2.4. Arithmetic around the bit heap 11 6.2.5. Improving computing architectures 11 ..."

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6.2.1. Mixed-precision fused multiply-and-add 11 6.2.2. Multiplication by rational constants versus division by a constant 11 6.2.3. Floating-point exponentiation on FPGA 11 6.2.4. Arithmetic around the bit heap 11 6.2.5. Improving computing architectures 11

### On Polynomial Multiplication in Chebyshev Basis

, 2013

"... In a recent paper, Lima, Panario and Wang have provided a new method to multiply polynomials expressed in Chebyshev basis which reduces the total number of multiplication for small degree polynomials. Although their method uses Karatsuba’s multiplication, a quadratic number of operations is still ne ..."

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In a recent paper, Lima, Panario and Wang have provided a new method to multiply polynomials expressed in Chebyshev basis which reduces the total number of multiplication for small degree polynomials. Although their method uses Karatsuba’s multiplication, a quadratic number of operations is still needed. In this paper, we extend their result by providing a complete reduction to polynomial multiplication in monomial basis, which therefore offers many subquadratic methods. Our reduction scheme does not rely on basis conversions and we demonstrate that it is efficient in practice. Finally, we show a linear time equivalence between the polynomial multiplication problem under monomial basis and under Chebyshev basis.

### RIGOROUS UNIFORM APPROXIMATION OF D-FINITE FUNCTIONS USING CHEBYSHEV EXPANSIONS

, 2013

"... A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we ne ..."

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A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-known that the order-n truncation of

### Distributed under a Creative Commons CC0- Public Domain Dedication 4.0 International License RIGOROUS UNIFORM APPROXIMATION OF D-FINITE FUNCTIONS USING CHEBYSHEV EXPANSIONS

, 2014

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