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
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- in his Mathematical Treatise in Nine Sections vol 6 problem 2.
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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:
- 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.
- Advance shang lian two ranks
- Advance yi yu three ranks
- Estimate shang=20
- let xia lian =shang * yi yu
- let fu lian=shang *yi yu
- merge fu lian with shang lian
- let fang=shang * shang lian
- subtract shang*fang from shi
- add shang * yi yu to xia lian
- retract xia lian 3 ranks,retract yi yu 4 ranks
- The second digit of shang is 0
- merge shang lian into fang
- merge yi yu into xia lian
- Add yi yu to fu lian, subtract the result from fang, let the result be denominator
- find the highest common factor =25 and simplies the fraction
- solution
Read more about this topic: Rod Calculus
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