Accurate Evaluation of Polynomials in Legendre Basis

Algorithm 6

Compensated Clenshaw algorithm.

function 𝑟𝑒𝑠=𝐶𝑜𝑚𝑝𝐶𝑙𝑒𝑛𝑠ℎ𝑎𝑤(𝑝,𝑥)

𝑏̂_𝑛+2=𝑏̂_𝑛+1=0

𝜖𝑏̂_𝑛+2=𝜖𝑏̂_𝑛+1=0

for 𝑗=𝑛∶−1∶0

 [𝐴_𝑗^(ℎ),𝐴_𝑗^(𝑙)]=𝑑𝑖𝑣_𝑑_𝑑(2𝑗+1,𝑗+1)

 [𝐶_𝑗^(ℎ),𝐶_𝑗^(𝑙)]=𝑑𝑖𝑣_𝑑_𝑑(−𝑗−1,𝑗+2)

 [𝑟,𝛼_𝑗]=𝑇ℎ𝑟𝑒𝑒𝑃𝑟𝑜𝑑(𝐴_𝑗^(ℎ),𝑏̂_𝑗+1,𝑥)

 [𝑠,𝛽_𝑗]=𝑇𝑤𝑜𝑃𝑟𝑜𝑑(𝐶^(ℎ)_𝑗+1,𝑏̂_𝑗+2)

 [𝑡,𝜂_𝑗]=𝑇𝑤𝑜𝑆𝑢𝑚(𝑟,𝑠)

 [𝑏̂_𝑗,𝜉_𝑗]=𝑇𝑤𝑜𝑆𝑢𝑚(𝑡,𝑎_𝑗)

 𝑠_𝑗̂=𝛼_𝑗⊕𝛽_𝑗⊕𝜂_𝑗⊕𝜉_𝑗⊕𝐴_𝑗^(𝑙)⊗𝑥⊗𝑏̂_𝑗+1⊕𝐶^(𝑙)_𝑗+1⊗𝑏̂_𝑗+2⊕𝑎_𝑗^(𝑙)

 𝜖𝑏̂_𝑗=𝐴_𝑗^(ℎ)⊗𝑥⊗𝜖𝑏̂_𝑗+1⊕𝐶^(ℎ)_𝑗+1⊗𝜖𝑏̂_𝑗+2⊕𝑠_𝑗̂

end

𝑟𝑒𝑠=𝑦⊕𝜖𝑏̂₀