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Roots by Recursion

Roots by Recursion. Olivia Sutherland. Brief History. Leonhard Euler found a way to extract roots without use of the Quadratic formula or Cardano’s formula but with recursion equations. This method is not as effective as it only finds an approximation and only one root. . Quadratic Formula.

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Roots by Recursion

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  1. Roots by Recursion Olivia Sutherland

  2. Brief History • Leonhard Euler found a way to extract roots without use of the Quadratic formula or Cardano’s formula but with recursion equations. This method is not as effective as it only finds an approximation and only one root.

  3. Quadratic Formula

  4. Rewrite equation

  5. Recursion Equations • Let A = 1 and B = 2 (They can be any numbers) • α=4 β=1 • C = 4B +A = 4(2) +1 = 9 • D = 4C +B = 4(9) + 2 = 38 • E = 4D +C = 4(38) +9 = 161 • F = 4E +D = 4(161) +38 = 682 • G = 4F +E = 4(682) + 161 = 2889 • And this continues but I’m going to stop there.

  6. Compare From the Quadratic formula: These values are only off by the 5th decimal point. Now take the next term H: The value is now only off by the 7th decimal point.

  7. Generalize Can break up fraction by partial fraction decomposition

  8. Geometric Series Each fraction on the right hand side can be put in a geometric series:

  9. Matching Terms Now matching constant, linear, and quadratic terms: Now for more general and larger exponents:

  10. The Root Remember that p is the largest term The root is so just take the reciprocal and you now have an approximation for the root.

  11. The End Thank you for listening!

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