Interpolation Of A Twice Continuously Differentiable Function By A Linear Polynomial
In the realm of numerical analysis, interpolation stands as a cornerstone technique for approximating functions. When dealing with functions that are twice continuously differentiable, understanding the error bounds associated with polynomial interpolation becomes crucial. This article delves into the intricacies of interpolating a function f, which is twice continuously differentiable on the interval I = [-1, 1], using a linear polynomial. We will explore the theoretical underpinnings, practical implications, and error analysis involved in this process. Specifically, we aim to interpolate the function f by a linear polynomial through the support points (xi, f(xi)), i = 0, 1, with x0, x1 in I. This exploration will provide a comprehensive understanding of how well a linear approximation can represent a twice continuously differentiable function within a given interval.
Problem Statement
Consider a function f which is twice continuously differentiable on the interval I = [-1, 1]. This implies that the function f and its first two derivatives, f'(x) and f''(x), are continuous on I. We aim to interpolate this function by a linear polynomial P(x) using two support points (x0, f(x0)) and (x1, f(x1)), where x0 and x1 belong to the interval I. The linear polynomial P(x) can be expressed in the form:
P(x) = a + bx
where a and b are constants to be determined. The interpolation conditions require that the polynomial P(x) coincides with the function f(x) at the support points, i.e.,
P(x0) = f(x0) P(x1) = f(x1)
These two conditions allow us to solve for the coefficients a and b, thus defining the linear interpolating polynomial. This problem sets the stage for a deeper analysis into the accuracy of this approximation and the factors influencing its performance. The use of a linear polynomial provides a simple yet fundamental approach to understanding the broader concepts of polynomial interpolation and its error behavior.
Constructing the Linear Interpolating Polynomial
To construct the linear interpolating polynomial P(x), we need to determine the coefficients a and b in the equation P(x) = a + bx. Using the interpolation conditions, we have the following system of equations:
a + bx0 = f(x0) a + bx1 = f(x1)
Subtracting the first equation from the second, we get:
b(x1 - x0) = f(x1) - f(x0)
If x0 ≠x1, we can solve for b:
b = (f(x1) - f(x0)) / (x1 - x0)
This expression for b represents the slope of the line passing through the points (x0, f(x0)) and (x1, f(x1)), which aligns with the geometric interpretation of linear interpolation. Substituting this value of b back into either of the original equations, we can solve for a. Let's use the first equation:
a = f(x0) - bx0 a = f(x0) - [(f(x1) - f(x0)) / (x1 - x0)] * x0
Thus, we have determined the coefficients a and b in terms of the function values f(x0) and f(x1) and the support points x0 and x1. With a and b now defined, the linear interpolating polynomial P(x) is completely specified. This explicit construction of P(x) is a critical step in analyzing the interpolation error, as it allows us to compare P(x) directly with the original function f(x).
Lagrange Form of the Linear Interpolating Polynomial
An alternative and often more insightful way to express the linear interpolating polynomial is using the Lagrange form. This form highlights the contribution of each support point to the overall polynomial and simplifies the analysis of interpolation error. The Lagrange form of the linear interpolating polynomial P(x) is given by:
P(x) = f(x0) L0(x) + f(x1) L1(x)
where L0(x) and L1(x) are the Lagrange basis polynomials, defined as:
L0(x) = (x - x1) / (x0 - x1) L1(x) = (x - x0) / (x1 - x0)
It's easy to verify that L0(x0) = 1, L0(x1) = 0, L1(x0) = 0, and L1(x1) = 1. These properties ensure that the interpolation conditions P(x0) = f(x0) and P(x1) = f(x1) are satisfied. The Lagrange form provides a clear and structured representation of the interpolating polynomial, making it particularly useful for theoretical analysis and understanding the influence of each data point on the approximation. This form also facilitates the generalization to higher-degree polynomial interpolation, where the Lagrange basis polynomials become more complex but retain the same fundamental properties.
Error Analysis
The cornerstone of interpolation theory lies in understanding and quantifying the error between the function f(x) and its interpolating polynomial P(x). The error, denoted by E(x), is defined as:
E(x) = f(x) - P(x)
For a twice continuously differentiable function f interpolated by a linear polynomial P(x), the error can be expressed using the following theorem:
Error Bound Theorem: If f is twice continuously differentiable on the interval I = [-1, 1], and P(x) is the linear polynomial interpolating f at x0 and x1 in I, then for any x in I, there exists a point ξ in the interval (x0, x1*) such that:
E(x) = *f''(ξ) / 2! * (x - x0) (x - x1)
This theorem is a fundamental result in interpolation theory, providing a precise expression for the error in terms of the second derivative of the function f and the support points x0 and x1. The presence of f''(ξ) indicates that the error is directly influenced by the curvature of the function; functions with larger second derivatives will generally exhibit larger interpolation errors. The term (x - x0) (x - x1) reflects the distance of the point x from the support points, showing that the error tends to be smaller near the support points and larger in the middle of the interval. This error bound provides a theoretical foundation for understanding the limitations of linear interpolation and for developing strategies to minimize the error, such as choosing appropriate support points or using higher-degree polynomials.
Bounding the Error
To obtain a practical estimate of the error, we often seek to bound the error term E(x). Let M be an upper bound for the absolute value of the second derivative of f on the interval I = [-1, 1], i.e.,
|f''(x)| ≤ M for all x in I
Then, we can bound the absolute value of the error as follows:
|E(x)| = |*f''(ξ) / 2! * (x - x0) (x - x1)| ≤ (M / 2) |(x - x0) (x - x1)|
The term |(x - x0) (x - x1)| represents the product of the distances from x to the support points x0 and x1. To further bound this term, we can analyze its maximum value on the interval I. Let h(x) = (x - x0) (x - x1). Assuming x0 < x1, the maximum value of |h(x)| on the interval [x0, x1*] occurs at the midpoint x = (x0 + x1) / 2. At this point,
|h(x)| = |((x0 + x1) / 2 - x0) ((x0 + x1) / 2 - x1)| = |(x1 - x0)2 / 4|
Therefore, the error bound becomes:
|E(x)| ≤ (M / 2) |(x1 - x0)2 / 4| = M (x1 - x0)2 / 8
This final error bound provides a clear indication of how the error depends on the second derivative of the function and the spacing between the support points. Specifically, the error is proportional to the square of the distance between the support points, highlighting the importance of choosing support points that are close together to minimize the error. This analysis underscores the practical considerations in applying linear interpolation and provides a basis for comparing it with higher-degree interpolation methods, which can offer improved accuracy at the cost of increased complexity.
Conclusion
In conclusion, this article has provided a comprehensive exploration of interpolating a twice continuously differentiable function f using a linear polynomial P(x). We have meticulously examined the construction of the linear interpolating polynomial, both in its standard form and the Lagrange form, and delved into the theoretical error analysis. The error bound theorem provides a fundamental understanding of the factors influencing the interpolation error, with the error being proportional to the second derivative of the function and the square of the distance between the support points. The derived error bound serves as a practical tool for estimating the accuracy of linear interpolation and for guiding the selection of appropriate support points. While linear interpolation offers simplicity and ease of implementation, its accuracy is limited, particularly for functions with significant curvature or when the support points are widely spaced. For applications requiring higher accuracy, higher-degree polynomial interpolation methods or other approximation techniques may be more suitable. This exploration lays a solid foundation for further study in numerical analysis and approximation theory, highlighting the importance of understanding both the strengths and limitations of different interpolation methods.
