Q6Advanced Engineering Mathematics

In numerical analysis, differentiating tabulated data is a frequent requirement. When the independent variable (often denoted as $x$ or time) is measured at strictly uniform, equally spaced intervals, standard techniques like Newton's Forward or Backward difference formulas are employed. However, real-world data collection is often imperfect, resulting in observations taken at irregular, unequal intervals. In such complex scenarios, those standard formulas fail completely, and we must utilize Newton's Divided Difference Interpolation formula to construct an interpolating polynomial, which we then analytically differentiate.

1. Fundamentals of Divided Differences

The core concept of this method is the "divided difference," which is essentially a finite approximation of a derivative. For a given set of data points $(x_0,y_0),(x_1,y_1),\dots ,(x_n,y_n)$, the zeroth-order divided difference is simply the function value itself: $f[x_i]=y_i$.

The first-order divided difference between two adjacent points is the slope of the secant line connecting them:

Higher-order divided differences are built recursively from lower-order ones. The $k$-th order divided difference is defined as:

2. Constructing the Divided Difference Table

Given the unequal intervals $x=\{0,2,3,4,6,7,9\}$ and corresponding function values $f(x)=\{4,26,58,112,466,922,3544\}$, we systematically construct a comprehensive divided difference table to calculate all necessary coefficients for the interpolating polynomial.

First Divided Differences ($\Delta y/\Delta x$):

• $f[x_0,x_1]=(26-4)/(2-0)=22/2=11$
• $f[x_1,x_2]=(58-26)/(3-2)=32/1=32$
• $f[x_2,x_3]=(112-58)/(4-3)=54/1=54$
• $f[x_3,x_4]=(466-112)/(6-4)=354/2=177$
• $f[x_4,x_5]=(922-466)/(7-6)=456/1=456$
• $f[x_5,x_6]=(3544-922)/(9-7)=2622/2=1311$

Second Divided Differences:

• $f[x_0,x_1,x_2]=(32-11)/(3-0)=21/3=7$
• $f[x_1,x_2,x_3]=(54-32)/(4-2)=22/2=11$
• $f[x_2,x_3,x_4]=(177-54)/(6-3)=123/3=41$
• $f[x_3,x_4,x_5]=(456-177)/(7-4)=279/3=93$
• $f[x_4,x_5,x_6]=(1311-456)/(9-6)=855/3=285$

Third Divided Differences:

• $f[x_0,\dots ,x_3]=(11-7)/(4-0)=4/4=1$
• $f[x_1,\dots ,x_4]=(41-11)/(6-2)=30/4=7.5$
• $f[x_2,\dots ,x_5]=(93-41)/(7-3)=52/4=13$
• $f[x_3,\dots ,x_6]=(285-93)/(9-4)=192/5=38.4$

Fourth Divided Differences:

• $f[x_0,\dots ,x_4]=(7.5-1)/(6-0)=6.5/6=1.083$
• $f[x_1,\dots ,x_5]=(13-7.5)/(7-2)=5.5/5=1.1$
• $f[x_2,\dots ,x_6]=(38.4-13)/(9-3)=25.4/6=4.233$

Normally, higher-order differences approach zero if the underlying function is a low-degree polynomial. The non-zero fourth differences indicate a complex curve.

3. Constructing and Differentiating the Polynomial

Newton's General Interpolation Formula is formed using the top row of the divided difference table (the leading differences):

To find the numerical derivative $f^{\prime }(x)$ at a specific point, we differentiate this polynomial equation analytically with respect to $x$.

By substituting $x=6$ into this derived formula and plugging in the carefully calculated divided differences from the table, we obtain the precise numerical estimate for the derivative at that specific point. This process demonstrates the immense power of numerical analysis to extract continuous derivative information from sparse, irregularly sampled discrete data sets.