Question

Using Wallis's method, interpolate the row \(p=3\) in his ratio table for \(n=1 / 2, n=3 / 2\), and \(n=5 / 2\).

Short answer

Answer: The formula for second-order interpolation in Wallis's method is: \(R_{n_2,2} = R_{n_1,1} + \frac{n_2-n_1}{n_3-n_1}(R_{n_3,1}-R_{n_1,1})\).

Step-by-step solution

Understand Wallis's method for interpolation

In Wallis's method, we interpolate using ratios of the given function values rather than the function values themselves. The formula for calculating the ratio of a function is: \(R_{n,p} = \frac{f(n+p)-f(n)}{f(n)-f(n-p)}\) where \(R_{n,p}\) is the ratio for the given value of n and p, and f(n) represents the unknown function.

Calculate the ratios for n=1/2, n=3/2, and n=5/2

First, we need to calculate the ratios for the given values of n. We have: \(n_1 = 1/2\), \(n_2 = 3/2\), and \(n_3 = 5/2\) Let's calculate the ratios \(R_{n_1,1}\), \(R_{n_2,1}\), and \(R_{n_3,1}\). To calculate these ratios, we need the values of the function f(n) at the given points, which will be given as below: \(f(1/2) = A\), \(f(3/2) = B\), and \(f(5/2) = C\) Now we will calculate the ratios: \(R_{1/2,1} = \frac{f(1/2+1)-f(1/2)}{f(1/2)-f(1/2-1)} = \frac{f(3/2)-A}{A-f(-1/2)}\) \(R_{3/2,1} = \frac{f(3/2+1)-f(3/2)}{f(3/2)-f(3/2-1)} = \frac{f(5/2)-B}{B-f(1/2)}\) \(R_{5/2,1} = \frac{f(5/2+1)-f(5/2)}{f(5/2)-f(5/2-1)} = \frac{f(7/2)-C}{C-f(3/2)}\)

Use interpolation to find the corresponding values for p=3

Now, we will use interpolation to find the corresponding values for p=3. Since we have three values of n, we will use second-order interpolation. The formula for second-order interpolation is: \(R_{n_2,2} = R_{n_1,1} + \frac{n_2-n_1}{n_3-n_1}(R_{n_3,1}-R_{n_1,1})\) Now, by substituting the values of the ratios and n values into the formula, we get: \(R_{3/2,3} = R_{1/2,1} + \frac{3/2-1/2}{5/2-1/2}(R_{5/2,1}-R_{1/2,1})\) This equation enables us to find the value of \(R_{3/2,3}\) in terms of the known ratios \(R_{1/2,1}\), \(R_{3/2,1}\), and \(R_{5/2,1}\).

Finalize the solution

In conclusion, we have used Wallis's method to interpolate the row p=3 in the ratio table for the given values of n (n=1/2, n=3/2, and n=5/2). We calculated the ratios for these values of n and then used second-order interpolation to find the value of \(R_{3/2,3}\) in terms of the known ratios. This method can be applied to any set of discrete data points to find intermediate values using Wallis's method.

Key concepts

Wallis's Method

Wallis's Method is a technique used in interpolation to estimate unknown values within a set of known discrete data points. Instead of using direct function values, Wallis's approach focuses on interpolating using ratios. This method is particularly useful for functions where only specific discrete values are known. It allows us to find intermediate values through a systematic ratio formula.

Here is the basic idea behind Wallis's Method:

• You calculate the ratio of the change in function values around the point of interest.
• For given values of a function, you use the formula:

\[ R_{n,p} = \frac{f(n+p)-f(n)}{f(n)-f(n-p)} \]
Where \( R_{n,p} \) represents the ratio for a given \( n \) and \( p \), while \( f(n) \) is the function value at \( n \).

Wallis's Method can be particularly advantageous in dealing with mathematical tables where the exact function is unknown, but the trend or pattern can provide insights through calculated ratios.

Ratio Table

A Ratio Table is a tool used to organize and calculate the ratios between different data points in Wallis's Method. It serves as a structured format to focus on the differences between these points rather than the absolute values. The table helps in applying Wallis's Method more effectively and systematically.

In practice, when creating a Ratio Table:

• Choose values for \( n \), which are the points at which the function is known. These might be termed \( n_1, n_2, n_3, ... \).
• Calculate the ratios using Wallis's formula for each segment between your chosen \( n \) values.
• Use these ratios to predict or interpolate unknown intermediate values through methods like second-order interpolation.

The Ratio Table helps illustrate the interaction between different intervals, highlighting similarities or variations in the underlying data trend, essential in estimating unknowns through interpolation.

Second-Order Interpolation

Second-Order Interpolation is a mathematical method used to estimate unknown values using a quadratic model derived from three known data points. In the context of interpolation using Wallis’s Method, second-order interpolation helps refine predictions for points within the scope of the original data by considering not just one or two points, but three points which allow for a parabola to fit the data more flexibly.

The formula for second-order interpolation used in this context is:\[ R_{n_2,2} = R_{n_1,1} + \frac{n_2-n_1}{n_3-n_1}(R_{n_3,1}-R_{n_1,1}) \]
This equation uses the ratios calculated from the known function values and the spacing between the \( n \) values:

• Choose three close data points and calculate their respective ratios.
• Estimate an intermediate value by using the formula to adjust \( R_{n_1,1} \) considering the proportional relationship between the points.

The advantage of second-order interpolation lies in its capacity to accommodate curvature in data, thereby providing more accurate interpolation results compared to linear methods, especially when the underlying data follows a trend that quadratic functions can better model. This method is versatile and can be applied to various practical applications requiring precise interpolation.