Biger Vieta polynomial Algorithm

https://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/polynomial%20methods/bv%20examples.html#exp1

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

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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.

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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.

 

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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.

 
 

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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.

 

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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.

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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