Accurate Evaluation of Polynomials in Legendre Basis

Algorithm 5

The Clenshaw algorithm for finite Legendre series.

function 𝑝(𝑥)=𝐶𝑙𝑒𝑛𝑠ℎ𝑎𝑤(𝑝,𝑥)

𝑏_𝑛+2=𝑏_𝑛+1=0

for 𝑗=𝑛∶−1∶0

 𝑏_𝑗=2𝑗+1𝑗+1𝑥𝑏_𝑗+1−𝑗+1𝑏𝑗+2^𝑗+2+𝑎_𝑗

end

𝑝(𝑥)=𝑏₀