Fast algorithms for Taylor shifts and certain difference equations

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

“…, that is as a univariate polynomial in x2 with coefficients in x1; we write it as [32]; hence the cost of all of them is OB (d 3 + d 2 τ ). We also have to compute (expand) the d+1 polynomials (x2 +a2) i .…”

The basic polynomial algebra subprograms

“…Another example of Level 2 functionality is parallel Taylor shift computation for which four different algorithms are available: the two plain algorithms presented in [3], Algorithm (E) of [7] and an optimized version of Algorithm (F) of [7].…”

“…To this end, we perform the Taylor Shift operation, that is, the map f (x) −→ f (x + 1), by means of Algorithm (E) in [7], which reduces calculations to integer polynomial multiplication in large degrees and to using algorithm of [3] in small degrees. In Tables 3, we call BPAS this adaptive algorithm combining FFT-based arithmetic (via Algorithm (E)) and plain arithmetic (via [3]).…”

GPU-Accelerated Generation of Correctly Rounded Elementary Functions

“…The shifts of the form R t (x + iN ) are called Taylor shifts [22]. If we denote ǫ shif t the error propagated by the shift such that |P t+i (x) − R t (x + iN )| < ǫ shif t , then we set ǫ approx to ǫ approx = ǫ ′ approx + ǫ shif t (see Sect.…”

“…• and the straightforward shift which computes the P t+i 's from R t in parallel but requires multi-precision multiplications and additions [22]. Finally we propose an hybrid CPU-GPU Taylor shift algorithm which efficiently combines these two shifts with the hierarchical method, and which requires only fixed size multi-precision addition on GPU.…”