Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

Algorithm 1

The Maple code for finding the error equation.

fxn:= c1*(e+c2*e∧2+c3*e∧3+c4*e∧4+c5*e∧5+c6*e∧6+c7*e∧7+c8*e∧8+c9*e∧9+c10*e∧10+

    c11*e∧11+c13*e∧12+c13*e∧13+c14*e∧14+c15*e∧15+c16*e∧16+c17*e∧17);

dfxn:= diff (fxn, e);

ye:= simplify (taylor (e−fxn/dfxn, e = 0, 17));

fyn:= c1*(ye+c2*ye∧2+c3*ye∧3+c4*ye∧4+c5*ye∧5+c6*ye∧6+c7*ye∧7+c8*ye∧8+c9*ye∧9+

    c10*ye∧10+c11*ye∧11+c12*ye∧12+c13*ye∧13+c14*ye∧14+c15*ye∧15+c16*ye∧16+c17*ye∧17);

fyn:= simplify (taylor (fyn, e = 0, 17));

t1:= simplify (taylor (fyn/fxn, e = 0, 17));

fydfx:= simplify (taylor (fyn/dfxn, e = 0, 17));

ze:= factor (simplify (taylor (ye−(1+2*t1−t1∧2)*fydfx/(1−6*t1∧2), e = 0, 17)));

factor (simplify (taylor (ye−(1+2*t1−t1∧2)*fydfx/(1−6*t1∧2), e = 0, 7)));

  −c2*c3*e∧4+(−2*c3∧2+2*c3*c2∧2+2*c2∧4−2*c2*c4)*e∧5+(−3*c2*c5−7*c3*c4+6*c2*c3∧2+

      12*c3*c2∧3+3*c4*c2∧2−14*c2∧5)*e∧6+O(e∧7)

fzn:= c1*(ze+c2*ze∧2+c3*ze∧3+c4*ze∧4+c5*ze∧5+c6*ze∧6+c7*ze∧7+c8*ze∧8+c9*ze∧9+

    c10*ze∧10+c11*ze∧11+c12*ze∧12+c13*ze∧13+c14*ze∧14+c15*ze∧15+c16*ze∧16+c17*ze∧17);

fzn:= simplify (taylor (fzn, e = 0, 17));

t2:= simplify (taylor (fzn/fyn, e = 0, 17));

t3:= simplify (taylor (fzn/fxn, e = 0, 17));

fzdfx:= simplify (taylor (fzn/dfxn, e = 0, 17));

we:= simplify (taylor (ze−(1−t1+t3)*fzdfx/(1−3*t1+2*t3−t2), e = 0, 17));

simplify (taylor (ze−(1−t1+t3)*fzdfx/(1−3*t1+2*t3−t2), e = 0, 10));

    −c4*c3*c2∧2*e∧8+(−2*c5*c3*c2∧2−4*c2*c4*c3∧2+4*c3*c4*c2∧3−2*c4∧2*c2∧2+2*c4*c2∧5

    −3*c3∧3*c2∧2+4*c3∧2*c2∧4+4*c3*c2∧6)*e∧9+O(e∧10)

fwn:= c1*(we+c2*we∧2+c3*we∧3+c4*we∧4+c5*we∧5+c6*we∧6+c7*we∧7+c8*we∧8+c9*we∧9+c10*we∧10+

     c12*we∧12+c13*we∧13+c14*we∧14+c15*we∧15+c16*we∧16+c17*we∧17);

fwn:= simplify (taylor (fwn, e = 0, 17));

fwdfx:= simplify (taylor (fwn/dfxn, e = 0, 17));

t4:= simplify (taylor (fwn/fxn, e = 0, 17));

t5:= simplify (taylor (fwn/fyn, e = 0, 17));

t6:= simplify (taylor (fwn/fzn, e = 0, 17));

q1:= simplify (taylor (1/(1−2*(t1+t1∧2+t1∧3+t1∧4+t1∧5+t1∧6+t1∧7)), e = 0, 17));

q2:= simplify (taylor (4*t3/(1−(31/4)*t3), e = 0, 17));

q3:= simplify (taylor (t2/(1−t2−20*t2∧3), e = 0, 17));

q4:= simplify (taylor (8*t4/(1−t4)+2*t5/(1−t5)+t6/(1−t6), e = 0, 17));

q5:= simplify (taylor (15*t1*t3/(1−(131/15)*t3), e = 0, 17));

q6:= simplify (taylor (54*t1∧2*t3/(1−t1∧2*t3), e = 0, 17));

q7:= simplify (taylor (7*t2*t3+2*t1*t6+6*t6*t1∧2+188*t3*t1∧3+18*t6*t1∧3+

              9*t2∧2*t3+648*t1∧4*t3, e = 0, 17));

x[n+1]:= simplify (taylor (we−fwdfx*(q1+q2+q3+q4+q5+q6+q7), e = 0, 17));

for i to 16 do p:= factor (simplify (coeff (x[n+1], e, i))) end do;

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 −c4*c3*c2∧2*(c5*c3*c2∧2+2*c2*c4*c3∧2−20*c3∧4−51*c3∧3*c2∧2+522*c3∧2*c2∧4−2199*c3*c2∧6

 +2*c2∧8−30*c3*c4*c2∧3+54*c4*c2∧5)