Graeffe's Root Squaring 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 x²- 10x + 23 = 0

a[] 1.0 -10.0 23.0

b[] 1.0 54.0 529.0

roots = 7.35 3.129

b[] 1.0 1858.0 279841.0

roots = 6.565 3.503

b[] 1.0 2892482.0 7.831E10

roots = 6.4218 3.585

b[] 1.0 8.2098E12 6.1326E21

roots = 6.414 3.585

6.414

3.585

Thus the absolute values of the roots are 6.414 and 3.585.

Since f(6.414) = 0 and f(3.585) = 0, the signs of the roots 6.414 and 3.585 are all positive.

So one of the root of the give equation is 1.0

2. Find the root of x³ + 3x² - 4 = 0

a[] 1.0 3.0 -0.0 -4.0

b[] 1.0 9.0 24.0 16.0

roots = 3.0 1.633 0.816

b[] 1.0 33.0 288.0 256.0

roots = 2.39 1.718 0.971

b[] 1.0 513.0 66048.0 65536.0

roots = 2.18 1.835 0.999

b[] 1.0 131073.0 4.295E9 4.2949E9

roots = 2.088 1.915 0.9999

Thus the absolute values of the roots are 2.088, 1.915 and 0.9999.

Since f(-2.088) = 0, f(-1.915) = 0 and f(0.9999) = 0, the roots are -2.088, -1.915 and 0.9999.

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

a[] 1.0 -4.0 5.0 -2.0

b[] 1.0 6.0 9.0 4.0

roots = 2.449 1.2247 0.666

b[] 1.0 18.0 33.0 16.0

roots = 2.059 1.1636 0.834

b[] 1.0 258.0 513.0 256.0

roots = 2.002 1.0897 0.9167

b[] 1.0 65538.0 131073.0 65536.0

roots = 2.000 1.0443 0.9576

b[] 1.0 4.295E9 8.5899E9 4.295E9

roots = 2.00 1.0218 0.978

Thus the absolute values of the roots are 2.00, 1.0218 and 0.978.

Since f(2.00) = 0, f(1.0218) = 0 and f(0.978) = 0, the signs of the roots 2.00, 1.0128 and 0.978 are all positive.

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

a[] 1.0 -6.0 11.0 -6.0

b[] 1.0 14.0 49.0 36.0

roots = 3.74 1.87 0.857

b[] 1.0 98.0 1393.0 1296.0

roots = 3.146 1.942 0.982

b[] 1.0 6818.0 1686433.0 1679616.0

roots = 3.014 1.99 0.999

b[] 1.0 4.31E7 2.82E12 2.82E12

roots = 3.0002 1.999 0.9999

b[] 1.0 1.853E15 7.958E24 7.958E24

roots = 3.00 1.999 0.9999

Thus the absolute values of the roots are 3.00, 1.999 and 0.9999.

Since f(3.00) = 0, f(1.999) = 0 and f(0.9999) = 0, the signs of the roots 3.00, 1.999 and 0.9999 are all positive.

5. Find the root of x³- x²- x + 1 = 0

a[] 1.0 -1.0 -1.0 1.0

b[] 1.0 3.0 3.0 1.0

roots = 1.73 1.0 0.577

b[] 1.0 3.0 3.0 1.0

roots = 1.316 1.0 0.759

b[] 1.0 3.0 3.0 1.0

roots = 1.147 1.0 0.8717

b[] 1.0 3.0 3.0 1.0

roots = 1.0711 1.0 0.9336

b[] 1.0 3.0 3.0 1.0

roots = 1.0349 1.0 0.9663

Thus the absolute values of the roots are 1.0349, 1.0 and 0.9663.

Since f(1.0349) = 0, f(-1.0) = 0 and f(0.9663) = 0, the roots are 1.0349, -1.0 and 0.9663.

6. Find the root of x²+ 6x - 3 = 0

a[] 1.0 6.0 -3.0

b[] 1.0 42.0 9.0

roots = 6.481 0.463

b[] 1.0 1746.0 81.0

roots = 6.464 0.4641

b[] 1.0 3048354.0 6561.0

roots = 6.464 0.464

Thus the absolute values of the roots are 6.464 and 0.464.

Since f(-6.464) = 0 and f(0.464) = 0, the roots are -6.464 and 0.464.

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 x⁵- 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