1. direct calculation method: this is the most basic solution and the most direct method. First of all, we need to make a cubic difference of the original polynomial to get a new polynomial. Then, we divide this new polynomial by the third derivative of the original polynomial to get the third difference quotient. The advantage of this method is intuitive and easy to understand, but it requires a lot of calculation.
2. Newton interpolation method: Newton interpolation method is a commonly used numerical analysis method, which can be used to solve the approximation of high-order polynomials. We can use Newton interpolation to solve the third-order difference quotient. First, we need to construct a cubic polynomial about X, so that this polynomial is equal to the original polynomial at a given value of X. Then, we make a cubic difference on this cubic polynomial and get a new polynomial. Finally, we divide this new polynomial by the third derivative of the original polynomial to get the third difference quotient. The advantage of this method is that the amount of calculation is small, but it needs to construct a cubic polynomial, which may increase the complexity of calculation.
3. Using Taylor series expansion: Taylor series is a method to express a function as an infinite series, which can be used to solve the approximate value of the function. We can use Taylor series to solve the third-order difference quotient. First of all, we need to expand the original polynomial by Taylor series and get an infinite series. Then, we make the third difference of this infinite series and get a new infinite series. Finally, we divide this new infinite series by the third derivative of the original polynomial and get the third difference quotient. The advantage of this method is that it can use the known mathematical formula, but it needs to have a certain understanding of Taylor series.
The above is the common solution of the third-order difference quotient. Different solutions have their own advantages and disadvantages, so it is necessary to choose the appropriate solution according to specific problems and conditions.