Accurate Evaluation of Polynomials in Legendre Basis

Algorithm 15

DDClenshaw algorithm of evaluating an Legendre series in double-double arithmetic.

function 𝑟𝑒𝑠=𝐷𝐷𝐶𝑙𝑒𝑛𝑠ℎ𝑎𝑤(𝑝,𝑥)

𝑏_𝑛+2=𝑏_𝑛+1=0

for 𝑗=𝑛∶−1∶0

  [ℎ1,𝑙1]=𝑝𝑟𝑜𝑑_𝑑𝑑_𝑑(𝑏^(ℎ)_𝑗+1,𝑏^(𝑙)_𝑗+1,𝑥)

  [ℎ2,𝑙2]=𝑝𝑟𝑜𝑑_𝑑𝑑_𝑑𝑑(ℎ1,𝑙1,𝐴_𝑗^(ℎ),𝐴_𝑗^(𝑙))

  [ℎ3,𝑙3]=𝑝𝑟𝑜𝑑_𝑑𝑑_𝑑𝑑(𝑏^(ℎ)_𝑗+2,𝑏^(𝑙)_𝑗+2,𝐶^(ℎ)_𝑗+1,𝐶^(𝑙)_𝑗+1)

  [ℎ4,𝑙4]=𝑎𝑑𝑑_𝑑𝑑_𝑑𝑑(ℎ2,𝑙2,ℎ3,𝑙3)

  [𝑏_𝑗^(ℎ),𝑏_𝑗^(𝑙)]=𝑎𝑑𝑑_𝑑𝑑_𝑑𝑑(ℎ4,𝑙4,𝑎_𝑗^(ℎ),𝑎_𝑗^(𝑙))

end

𝑟𝑒𝑠=[𝑏₀^(ℎ),𝑏₀^(𝑙)]