Accurate Evaluation of Polynomials in Legendre Basis

Algorithm 16

DDConvert algorithm for the polynomial from power basis to Legendre basis.

function 𝑏=𝐷𝐷𝐶𝑜𝑛𝑣𝑒𝑟𝑡(𝑎)

𝑎,𝑏 are the vectors of monomial polynomial and orthogonal polynomial’s coefficients, respectively

𝑏^(𝑛)=0,

𝑏ℎ₀^(𝑛)=𝑎_𝑛^(ℎ),

for 𝑗=𝑛−1∶−1∶0

  for 𝑘=0∶1∶𝑛−𝑗

    if 𝑘=0

   [ℎ₁,𝑙₁]=𝑝𝑟𝑜𝑑_𝑑𝑑_𝑑𝑑(𝐶₁^(ℎ)𝐶₁^(𝑙),𝑏ℎ₁^(𝑗+1),𝑏𝑙₁^(𝑗+1))

   [ℎ₂,𝑙₂]=𝑑𝑖𝑣_𝑑𝑑_𝑑𝑑(ℎ₁,𝑙₁,𝐴₁^(ℎ),𝐴₁^(𝑙))

   [𝑏ℎ₀^(𝑗),𝑏𝑙₀^(𝑗)]=𝑎𝑑𝑑_𝑑𝑑_𝑑𝑑(−ℎ₂,−𝑙₂,𝑎_𝑗^(ℎ),𝑎_𝑗^(𝑙))

    if 𝑘=1,2,…,𝑛−𝑗−2

   [ℎ₁,𝑙₁]=𝑑𝑖𝑣_𝑑𝑑_𝑑𝑑(𝑏ℎ^(𝑗+1)_𝑘−1,𝑏𝑙^(𝑗+1)_𝑘−1,𝐴^(ℎ)_𝑘−1,𝐴^(𝑙)_𝑘−1)

   [ℎ₂,𝑙₂]=𝑝𝑟𝑜𝑑_𝑑𝑑_𝑑𝑑(𝐶^(ℎ)_𝑘+1,𝐶^(𝑙)_𝑘+1,𝑏ℎ^(𝑗+1)_𝑘+1,𝑏𝑙^(𝑗+1)_𝑘+1)

   [ℎ₃,𝑙₃]=𝑑𝑖𝑣_𝑑𝑑_𝑑𝑑(ℎ₂,𝑙₂,𝐴^(ℎ)_𝑘+1,𝐴^(𝑙)_𝑘+1)

   [𝑏ℎ_𝑘^(𝑗),𝑏𝑙_𝑘^(𝑗)]=𝑎𝑑𝑑_𝑑𝑑_𝑑𝑑(ℎ₁,𝑙₁,−ℎ₃,−𝑙₃))

    if 𝑘≥𝑛−𝑗−1

   [𝑏ℎ_𝑘^(𝑗),𝑏𝑙_𝑘^(𝑗)]=𝑑𝑖𝑣_𝑑𝑑_𝑑𝑑(𝑏ℎ^(𝑗+1)_𝑘−1,𝑏𝑙^(𝑗+1)_𝑘−1,𝐴^(ℎ)_𝑘−1,𝐴^(𝑙)_𝑘−1)

  end

end

𝑏=[𝑏ℎ⁽⁰⁾,𝑏𝑙⁽⁰⁾]