Accurate Evaluation of Polynomials in Legendre Basis

Algorithm 7

The conversion algorithm from power basis to Legendre basis.

function 𝑏=𝐶𝑜𝑛𝑣𝑒𝑟𝑡(𝑎)

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

𝑏^(𝑛)=0,

𝑏₀^(𝑛)=𝑎_𝑛,

for 𝑗=𝑛−1∶−1∶0

 for 𝑘=0∶1∶𝑛−𝑗

  𝑏₀^(𝑗)=𝑎_𝑗−𝐶₁𝐴₁𝑏₁^(𝑗+1) (𝑘=0)

  𝑏_𝑘^(𝑗)=1𝐴_𝑘−1𝑏^(𝑗+1)_𝑘−1−𝐶_𝑘+1𝐴_𝑘+1𝑏^(𝑗+1)_𝑘+1  (𝑘=1,…,𝑛−𝑗−2)

  𝑏_𝑘^(𝑗)=1𝐴_𝑘−1𝑏^(𝑗+1)_𝑘−1  (𝑘≥𝑛−𝑗−1)

 end

end

𝑏=𝑏⁽⁰⁾