Question

Adams Fourth-Order Predictor-Corrector Python ONLY!!

Adams Fourth-Order Predictor-Corrector To approximate the solution of the initial-value problem y' = f(t, y), ast<b, y(a) = a, at (N + 1) equally spaced numbers in the interval [a, b]: INPUT endpoints a, b; integer N; initial condition a. OUTPUT approximation w to y at the (N + 1) values of t. Step 1 Set h = (b − a)/N; to = a; Wo = a; OUTPUT (to, wo). Step 2 For i = 1, 2, 3, do Steps 3–5. (Compute starting values using Runge-Kutta method.) Step 3 Set K1 = hf (ti-1, W;-1); K2 = hf (ti-1 +h/2, Wi-1 + K1/2); K3 = hf (ti-1 +h/2, Wi-1 + K2/2); K4 = hf (ti-1 + h, Wi-1 + K3). Step 4 Set w; = Wi-1 + (K1 + 2K2 + 2K3 + K4)/6; ti = a + ih. Step 5 OUTPUT (ti, W;). Step 6 For i = 4, ... , N do Steps 7–10. Step 7 Set t = a +ih; w = wz + h[55f (t3, w3) – 59f (t2, w2) + 37f (t), wi) - 9f (to, wo)]/24; (Predict W;.) w = w3 + h[9f (t, w) + 19 f (tz, w3) – 5 f (t2, W2) + f(t1,w1)]/24. (Correct wi.) Step 8 OUTPUT (t, w). Step 9 For j = 0, 1, 2 set t; = t;+1; (Prepare for next iteration.) W;=Wj+1. Step 10 Set tz = t; W3 =W. Step 11 STOP.

Please translate this pseudocode into Python code, thanks!!

Answer 1

The python code for given pseudo code is given below. f(t, y) has been given a sample definition for testing purposes.

The definition of f(y,t) used here is

ft,y) = y-t +1

import numpy as np

def f(t, y):
#some function
return y-t**2+1;

def adamsFourthOrderPredictorCorrector():
#input
a = float(input("a:"))
b = float(input("b:"))
N = int(input("N:"))
alpha = float(input("alpha:"))
#step 1
h = (b-a)/N
t = np.zeros((N+1))
t[0] = a
w = np.zeros((N+1))
w[0] = alpha
print(t[0], w[0])
#step 2
for i in range(1,4):
#step 3
k1 = h*f(t[i-1], w[i-1])
k2 = h*f(t[i-1]+h/2, w[i-1]+k1/2)
k3 = h*f(t[i-1]+h/2, w[i-2]+k2/2)
k4 = h*f(t[i-1]+h, w[i-1]+k3)
#step 4
w[i] = w[i-1] + (k1+2*k2+2*k3+k4)
t[i] = a+i*h
#step 5
print(t[i], w[i])
#step 6
for i in range(4,N+1):
#step 7
T = a+i*h
W = w[3] + h*(55*f(t[3], w[3])-59*f(t[2], w[2])+37*f(t[1], w[1])-9*f(t[0], w[0]))/24
W = w[3] + h*(9*f(T, W)+19*f(t[3],w[3])-5*f(t[2],w[2])+f(t[1],w[1]))/24
#step 8
print(T,W)
#step 9
for j in range(0,3):
t[j] = t[j+1]
w[j] = w[j+1]
#step 10
t[3] = T
w[3] = W

adamsFourthOrderPredictorCorrector()

import numpy as np def f(t, y): #some function return y-t**2+1; def adams FourthorderPredictorCorrector(): #input a = int(input("a:")) b = int(input("b:")) N = int(input("N:")) alpha = float(input("alpha:")) #step 1 h = (b-a)/N t = np.zeros((N+1)) t[0] = a W = np.zeros((N+1)) W[0] = alpha print(t[@], W[0]) #step 2 for i in range(1,4): #step 3 kl = h f(t[i-1], w[i-1]) k2 = h f(t[i-1]+h/2, w[i-1]+k1/2) k3 = h f(t[i-1]+h/2, w[i-2]+k2/2) k4 = h f(t[i-1]+h, w[i-1]+k3) #step 4 W[i] = w[i-1] + (k1+2*k2+2*k3+k4) til = a+i h #step 5 print(t[i], w[i]) #step 6 for i in range (4,N+1): #step 7 T = a+ih W = W[3] + h (55*f(t[3], W[3]) -59*f(t[2], W[2])+37*f(t[1], w[1])-9 f(t[o], W[0]))/24, W = w[3] + h (9*f(T, W) +19 f(t[3],w[3])-5 f(t[2],W[2])+f(t[1],[1]))/24

h = (b-a)/N t = np.zeros((N+1)) t[0] = a W = np.zeros((N+1)) W[0] = alpha print(t[o], W[O]) #step 2 for i in range(1,4): #step 3 ki = h f(t[i-1], w[i-1]) k2 = h f(t[i-1]+h/2, w[i-1]+k1/2) k3 = h f(t[i-1]+h/2, w[i-2]+k2/2) k4 = h f(t[i-1]+h, w[i-1]+k3) #step 4 W[i] = w[i-1] + (k1+2k2+2*k3+k4) t[i] = a+ih #step 5 print(t[i], w[i]) #step 6 for i in range (4,N+1): #step 7 T = a+ih W = w[3] + h*(55*f(t[3], W[3])-59*f(t[2], W[2])+37*f(t[1], w[1])-9*f(t[0], W[0]))/24 W = w[3] + h (9*f(T, W) +19*f(t[3],[3])-5 f(t[2],W[2] ) +f(t[1],[1]))/24 #step 8 print(TW) #step 9 for j in range(0,3): t[j] = t[j+1] W[j] = w[j+1] #step 10 t[3] = T W[3] = W 49 adams FourthörderpredictorCorrector();

Sample Input and Output:

a:20 b:30 N:5 alpha:23.4 20.0 23.4 22.0 -14673.4 24.0 -443813.4 26.0 - 13009855.800000003 28.0 -86303798.5 30.0 -532012009.48125

If you need more information, please reach out to me in comment section.

ft,y) = y-t +1

import numpy as np def f(t, y): #some function return y-t**2+1; def adams FourthorderPredictorCorrector(): #input a = int(input("a:")) b = int(input("b:")) N = int(input("N:")) alpha = float(input("alpha:")) #step 1 h = (b-a)/N t = np.zeros((N+1)) t[0] = a W = np.zeros((N+1)) W[0] = alpha print(t[@], W[0]) #step 2 for i in range(1,4): #step 3 kl = h f(t[i-1], w[i-1]) k2 = h f(t[i-1]+h/2, w[i-1]+k1/2) k3 = h f(t[i-1]+h/2, w[i-2]+k2/2) k4 = h f(t[i-1]+h, w[i-1]+k3) #step 4 W[i] = w[i-1] + (k1+2*k2+2*k3+k4) til = a+i h #step 5 print(t[i], w[i]) #step 6 for i in range (4,N+1): #step 7 T = a+ih W = W[3] + h (55*f(t[3], W[3]) -59*f(t[2], W[2])+37*f(t[1], w[1])-9 f(t[o], W[0]))/24, W = w[3] + h (9*f(T, W) +19 f(t[3],w[3])-5 f(t[2],W[2])+f(t[1],[1]))/24

h = (b-a)/N t = np.zeros((N+1)) t[0] = a W = np.zeros((N+1)) W[0] = alpha print(t[o], W[O]) #step 2 for i in range(1,4): #step 3 ki = h f(t[i-1], w[i-1]) k2 = h f(t[i-1]+h/2, w[i-1]+k1/2) k3 = h f(t[i-1]+h/2, w[i-2]+k2/2) k4 = h f(t[i-1]+h, w[i-1]+k3) #step 4 W[i] = w[i-1] + (k1+2k2+2*k3+k4) t[i] = a+ih #step 5 print(t[i], w[i]) #step 6 for i in range (4,N+1): #step 7 T = a+ih W = w[3] + h*(55*f(t[3], W[3])-59*f(t[2], W[2])+37*f(t[1], w[1])-9*f(t[0], W[0]))/24 W = w[3] + h (9*f(T, W) +19*f(t[3],[3])-5 f(t[2],W[2] ) +f(t[1],[1]))/24 #step 8 print(TW) #step 9 for j in range(0,3): t[j] = t[j+1] W[j] = w[j+1] #step 10 t[3] = T W[3] = W 49 adams FourthörderpredictorCorrector();

a:20 b:30 N:5 alpha:23.4 20.0 23.4 22.0 -14673.4 24.0 -443813.4 26.0 - 13009855.800000003 28.0 -86303798.5 30.0 -532012009.48125