Rod Calculus - Polynomial Equation

Polynomial Equation

North Song dynasty mathematician Jia Xian invented Horner scheme for solving simple 4th order equation of the form

South Song dynasty mathematician Qin Jiushao improved Jia Xian's Horner method to solve polynomial equation up to 10th order. The following is algorithm for solving

in his Mathematical Treatise in Nine Sections vol 6 problem 2.

This equation was arranged bottom up with counting rods on counting board in tabular form

0 shang root
626250625 shi constant
0 fang coefficent of x
15245 shang lian positive coef of x^2
0 fu lian negative coef of x^2
0 xia lian coef of x^3
1 yi yu negative coef of X^4

Algorithm:

  1. Arrange the coefficents in tabular form, constant at shi, coeffienct of x at shang lian, the coeffiecnt of X^4 at yi yu;align the numbers at unit rank.
  2. Advance shang lian two ranks
  3. Advance yi yu three ranks
  4. Estimate shang=20
  5. let xia lian =shang * yi yu
  6. let fu lian=shang *yi yu
  7. merge fu lian with shang lian
  8. let fang=shang * shang lian
  9. subtract shang*fang from shi
  10. add shang * yi yu to xia lian
  11. retract xia lian 3 ranks,retract yi yu 4 ranks
  12. The second digit of shang is 0
  13. merge shang lian into fang
  14. merge yi yu into xia lian
  15. Add yi yu to fu lian, subtract the result from fang, let the result be denominator
  16. find the highest common factor =25 and simplies the fraction
  17. solution

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