Journal of Applied Mathematics

Research Article

A Family of Three-Point Methods of Ostrowski's Type for Solving Nonlinear Equations

Algorithm 1

Program (written in Mathematica).
fx=f1a*(e+c2 e ̂ ‌ 2+c3 e ̂ ‌ 3+c4 e ̂ ‌ 4); f1x=D[fx,e];
ey=e-Series[fx/f1x, { e,0,8 } ];
fy=f1a*(ey+c2 ey ̂ ‌ 2+c3 ey ̂ ‌ 3+c4 ey ̂ ‌ 4);
t=Series[fy/fx, { e,0,8 } ];
ez=ey-Series[1/(1−2t)*fy/f1x, { e,0,8 } ];
fz=f1a*(ez+c2 ez ̂ ‌ 2+c3 ez ̂ ‌ 3);
s=Series[fz/fy, { e,0,8 } ];
v=Series[fz/fx, { e,0,8 } ];
gt=t0+t10*t+t20*t ̂ ‌ 2/2+t30*t ̂ ‌ 3/6;
gs=s0+s10*s+s20*s ̂ ‌ 2/2;
gv=v0+v10*v+v20*v ̂ ‌ 2/2;
f1z=f1x*gt*gs*gv;
e1=ez-Series [fz/f1z, { e,0,8 } ]//Simplify
C1: “Out[a4] = 𝑐 2 ( 𝑐 2 2 − 𝑐 3 ) (-1+t0 s0 v0)/(t0 s0 v0)”
t0=1; s0=1; v0=1; a5=Coefficient[e1,e ̂ ‌ 5] //Simplify
C2: “Out [ a5 ] = 𝑐 2 2 ( 𝑐 2 2 − 𝑐 3 ) (2 + t10)”
t10=−2;  a6=Coefficient[e1,e ̂ ‌ 6]//Simplify
C3: “Out [ a6 1 ] = 2 𝑐 2 ( 𝑐 2 2 − 𝑐 3 ) ( − 2 𝑐 3 (1+s10) + 𝑐 2 2 (4 + t20 + 2v10) ) ”
s10=−1;  t20=-2;  a7=Coefficient[e1,e ̂ ‌ 7]//Simplify
C4: “Out [ a7 1 ] = 6 𝑐 2 2 ( 𝑐 2 2 − 𝑐 3 ) ( − 6 𝑐 3 ( 2 +  v10 ) + 𝑐 2 2 ( t30 + 6 ( 2 + v10 ) ) ) ”
v10=−2;  t30=0;  a8=Coefficient[e1,e ̂ ‌ 8]//Simplify
C5: “Out [ a8 1 ] = 2 𝑐 2 ( 𝑐 2 2 − 𝑐 3 ) ( 2 𝑐 2 𝑐 4 + 𝑐 3 2 𝑣 2 0 + 𝑐 2 4 ( 4 + 𝑠 2 0 ) − 2 𝑐 2 2 𝑐 3 ( 4 + 𝑠 2 0 ) ) 𝑒 8 + 𝑂 [ 𝑒 9 ] ”