An Efficient Iteration Method for Toeplitz-Plus-Band Triangular Systems Generated from Fractional Ordinary Differential Equation

Algorithm 2

The efficient iteration method for FODE.

 input: 𝑀,𝜂,𝐹,𝐴

 output:  𝑈

(1) 𝜖←10⁻⁶,𝛿=1.0

(2) while   𝛿>𝜖   do

(3)     set 𝑣𝑏_{1→𝑀−𝜂} with 𝑢𝑎_{𝑀−𝜂→1}

(4)    𝑉1←FFT(𝑣𝑏)

(5)    𝑉2←FFT(𝑣𝑎)

(6)    𝑣←𝑉1⨀𝑉2

(7)    𝑢_1→𝑀←𝑓_1→𝑀

(8)    for   𝑖=1   to   𝑀−𝜂  by 1   do

(9)     𝑢_𝜂+𝑖←𝑢_𝜂+𝑖+𝑣_{2(𝑀−𝜂)−𝑖}

(10)   for   𝑗=1   to   𝑀  by 1   do

(11)   𝑢_𝑗←𝑢_𝑗𝑑_𝑗

(12)   for   𝑖=𝑗+1   to   min𝑗+𝜂−1,𝑀  by 1   do

(13)      𝑢_𝑖←𝑢_𝑖−𝑏_{𝑖−𝑗+1}𝑢_𝑗

(14)   𝛿←max𝑢_1→𝑀−𝑢𝑎_1→𝑀

(15)   𝑢𝑎_1→𝑀←𝑢𝑎_1→𝑀