Birge Vieta Method

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Iteration scheme is simple to implement.
Iterations always converge for any continuous function f(x).

Disadvantages

Convergence process is slow.
Always one has to find the interval [a,b] which brackets the zero of f(x)= 0 more importantly f(a) * f(b) must be less than zero.

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1. Find the root of x⁴ - 3x³ + 3x²- 3x + 2 = 0

In this problem the coefficients are a₀ = 2, a₁ = -3, a₂ = +3, a₃ = -3, a₄ = +1

Let the initial approximation to p be p₀ = 0.5

a₀ = 2.0              a₁= -3.0             a₂ = 3.0       a₃ = -3.0       a₄ = 1.0
b₀ = 1.0              b₁ = -2.5           b₂ = 1.75     b₃ = -2.125    b₄ = 0.9375
c₀ = 1.0               c₁ = -2.0           c₂ = 0.75     c₃ = -1.75

p₁= p₀- (b₄/ c₃) = 1.0357143

a₀ = 2.0            a₁ = -3.0            a₂ = 3.0            a₃ = -3.0            a₄ = 1.0
b₀ = 1.0            b₁ = -1.964       b₂ = 0.965        b₃ = -1.999        b₄ = -0.0714
c₀ = 1.0            c₁ = -0.928       c₂ = 0.0038       c₃ = -1.996

p₂ = 0.9999518

a₀= 2.0              a₁ = -3.0            a₂ = 3.0           a₃ = -3.0           a₄ = 1.0
b₀ = 1.0             b₁ = -2.00          b₂ = 1.00        b₃ = -2.0           b₄ = 9.64E-5
c₀ = 1.0             c₁ = -1.00           c₂ = 2.38E-7   c₃ = -1.999

p₃ = 1.0

So one of the root of the give equation is 1.0

2. Find the root of x⁴ - x - 10 = 0

In this problem the coefficients are a₀ = 2, a₁ = -3, a₂ = +3, a₃ = -3, a₄ = +1

Let the initial approximation to p be p₀ = 1.5

a₀ = -10.0     a₁ = -1.0      a₂ = 0.0     a₃ = 0.0        a₄ = 1.0
b₀ = 1.0        b₁ = 1.5       b₂ = 2.25   b₃= 2.375    b₄ = -6.4375
c₀ = 1.0        c₁ = 3.0        c₂ = 6.75   c₃ = 12.5

p₁ = 2.0149999

a₀ = -10.0        a₁ = -1.0      a₂ = 0.0     a₃= 0.0     a4 = 1.0
b₀ = 1.0           b₁ = 2.01     b₂ = 4.06   b₃ = 7.18   b₄ = 4.47
c₀ = 1.0           c₁ = 4.029   c₂ = 12.18 c₃ = 31.72

p₂ = 1.87409

a₀ = -10.0           a₁ = -1.0           a₂ = 0.0               a₃ = 0.0            a₄ = 1.0
b₀ = 1.0              b₁ = 1.87          b₂ = 3.51             b₃ = 5.58          b₄ = 0.46
c₀ = 1.0             c₁ = 3.74          c₂ = 10.54            c₃ = 25.32

p₃ = 1.8558675

a₀ = -10.0     a₁ = -1.0         a₂ = 0.0            a₃ = 0.0       a₄ = 1.0
b₀ = 1.0        b₁ = 1.85        b₂ = 3.44          b₃ = 5.39     b₄ = 0.0069
c₀ = 1.0        c₁ = 3.71         c₂ = 10.33        c₃ = 24.56

p₄ = 1.8555846

a₀ = -10.0         a₁ = -1.0            a₂ = 0.0              a₃ = 0.0         a₄ = 1.0
b₀ = 1.0            b₁ = 1.855          b₂ = 3.44           b₃ = 5.38       b₄ = 2.8E-6
c₀ = 1.0            c₁ = 3.71            c₂ = 10.329        c₃ = 24.556

p₅ = 1.8555845

so one of the root of the given equation is 1.8555845.

3. Find the root of x³- 6x² + 11x - 6 = 0

In this problem the coefficients are a₀ = 2, a₁ = -3, a₂ = +3, a₃ = -3, a₄ = +1
Let the initial approximation to p be p₀ = 0.5

a₀ = -6.0           a₁ = 11.0            a₂ = -6.0             a₃ = 1.0
b₀ = 1.0            b₁ = -5.5            b₂ = 8.25             b₃ = -1.875
c₀ = 1.0            c₁ = -5.0             c₂ = 5.75
p₁ = 0.826087

a₀ = -6.0       a₁ = 11.0             a₂ = -6.0             a₃ = 1.0
b₀ = 1.0        b₁ = -5.17           b₂ = 6.72             b₃ = -0.444
c₀ = 1.0        c₁ = -4.35           c₂ = 3.134
p₂ = 0.96769285

a₀ = -6.0       a₁ = 11.0        a₂ = -6.0       a₃ = 1.0
b₀ = 1.0        b₁ = -5.03       b₂ = 6.130    b₃ = -0.0677
c₀ = 1.0        c₁ = -4.06       c₂ = 2.196
p₃ = 0.998544

a₀ = -6.0           a₁ = 11.0           a₂ = -6.0        a₃ = 1.0
b₀ = 1.0            b₁ = -5.001       b₂ = 6.005    b₃ = -0.002
c₀ = 1.0            c₁ = -4.002       c₂ = 2.0087

p₄ = 0.99999696

a₀ = -6.0       a₁ = 11.0        a₂ = -6.0       a₃ = 1.0
b₀ = 1.0        b₁ = -5.00      b₂ = 6.000    b₃ = -5.7E-6
c₀ = 1.0        c₁ = -4.00       c₂ = 2.00

p₅ = 0.9999998

so one of the root of the given equation is 0.9999998.

4. Find the root of x³- 4x²+ 5x - 2 = 0

In this problem the coefficients are a₀ = 2, a₁ = -3, a₂ = +3, a₃ = -3, a₄ = +1

Let the initial approximation to p be p₀ = 1.9

a₀ = -2.0        a₁ = 5.0             a₂ = -4.0            a₃ = 1.0
b₀ = 1.0         b₁ = -2.1            b₂ = 1.01           b₃ = -0.081
c₀ = 1.0         c₁ = -0.19          c₂ = 0.63
p₁ = 2.0285707

a₀ = -2.0        a₁ = 5.0           a₂ = -4.0           a₃ = 1.0
b₀ = 1.0         b₁ = -1.97       b₂ = 1.01           b₃ = 0.030
c₀ = 1.0          c₁ = 0.057      c₂ = 1.116
p₂ = 2.0015035

a₀ = -2.0           a₁ = 5.0           a₂ = -4.0           a₃ = 1.0
b₀ = 1.0            b₁ = -1.99       b₂ = 1.00          b₃ = 0.0015
c₀ = 1.0            c₁ = 0.003       c₂ = 1.0060205
p₃ = 2.0000048

a₀ = -2.0               a₁ = 5.0           a₂ = -4.0            a₃ = 1.0
b₀ = 1.0                b₁ = -1.99        b₂ = 1.0             b₃ = 4.7E-6
c₀ = 1.0                c₁= 9.5E-6      c₂ = 1.0000191

root = 2.0

so one of the root of the given equation is 2.0.

5. Find the root of x⁴- x³+ 3x²+ x - 4 = 0

In this problem the coefficients are a₀ = 2, a₁ = -3, a₂ = +3, a₃ = -3, a₄ = +1

Let the initial approximation to p be p₀ = 1.5
a₀ = -4.0     a₁ = 1.0    a₂ = 3.0     a₃ = -1.0       a₄ = 1.0
b₀ = 1.0       b₁ = 0.5   b₂ = 3.75   b₃ = 6.625    b₄ = 5.9375
c₀ = 1.0       c₁ = 2.0    c₂ = 6.75   c₃ = 16.75

p₁ = 1.1455224

a₀ = -4.0      a₁ = 1.0         a₂ = 3.0     a₃ = -1.0     a₄ = 1.0
b₀ = 1.0       b₁ = 0.145     b₂ = 3.16   b₃ = 4.63    b₄ = 1.3009324
c₀ = 1.0       c₁ = 1.29       c₂ = 4.64    c₃ = 9.95

p₂ = 1.0147647

a₀ = -4.0          a₁ = 1.0           a₂ = 3.0       a₃ = -1.0              a₄ = 1.0
b₀ = 1.0           b₁ = 0.015       b₂ = 3.015   b₃ = 4.06             b₄ = 0.12
c₀ = 1.0           c₁ = 1.029       c₂ = 4.059    c₃ = 8.179151

p₃ = 1.0001624

a₀ = -4.0         a₁ = 1.0                 a₂ = 3.0      a₃ = -1.0          a₄ = 1.0
b₀ = 1.0          b₁ = 1.62E-4         b₂ = 3.00    b₃ = 4.001       b₄ = 0.0013
c₀ = 1.0          c₁ = 1.0003           c₂ = 4.001   c₃ = 8.001948
p₄ = 1.0

a₀ = -4.0          a₁ = 1.0         a₂ = 3.0           a₃ = -1.0            a₄ = 1.0
b₀ = 1.0           b₁ = 0.0         b₂ = 3.0          b₃ = 4.0             b₄ = 0.0
c₀ = 1.0           c₁ = 1.0          c₂ = 4.0          c₃ = 8.0

p₅ = 1.0

so one of the root of the given equation is 1.0.

6. Find the root of x³- x - 4 = 0

In this problem the coefficients are a₀ = 2, a₁ = -3, a₂ = +3, a₃ = -3, a₄ = +1

Let the initial approximation to p be p₀ = 1.5
a₀ = -4.0     a₁ = -1.0       a₂ = -0.0              a₃ = 1.0
b₀ = 1.0      b₁ = 1.5         b₂ = 1.25            b₃ = -2.125
c₀ = 1.0       c₁ = 3.0       c₂ = 5.75
p₁ = 1.8695652

a₀ = -4.0       a₁ = -1.0        a₂ = -0.0        a₃ = 1.0
b₀ = 1.0        b₁ = 1.86       b₂ = 2.49        b₃ = 0.66
c₀ = 1.0        c₁ = 3.73        c₂ = 9.48
p₂ = 1.7994524

a₀ = -4.0       a₁ = -1.0         a₂ = -0.0              a₃ = 1.0
b₀ = 1.0        b₁ = 1.79        b₂ = 2.24             b₃ = 0.027
c₀ = 1.0        c₁ = 3.59        c₂ = 8.71
p₃ = 1.796328

a₀ = -4.0      a₁ = -1.0      a₂ = -0.0     a₃ = 1.0
b₀ = 1.0       b₁ = 1.79      b₂ = 2.22   b₃ = 5.27E-5
c₀ = 1.0       c₁ = 3.59       c₂ = 8.68

p₄ = 1.7963219

so one of the root of the given equation is 1.796.

Problems to Work-Out:

7. Find the root of x⁴- x - 4 = 0

8. Find the root of 2x³ - 3x²+ 2x - 3 = 0

9. Find the root of x³- 5x²+ 4x - 3 = 0

10. Find the root of x³- x²- x + 1 = 0

11. Find the root of 9x⁴+ 30x³+ 34x²+ 30x + 25 = 0

12. Find the root of x⁵- 2x4+4x3-x2-7x+5= 0