UleM | PORTAL UNI… Ulearn @ UTEM 9 Games Box Email – HAZIEQUE… Inbox HAZIEQUE BIN MUSA – 1313 ENGINEERING MATHEMATICS 1 Question 3 Not yet answered Marked out of 1.00 Generally, the higher the order of interpolation, the more accurate the estimated interpolated value. Secara amnya, semakin tinggi urutan interpolasi semakin tepat anggaran nilai interpolasi. Select one: True / Betul Flag question False / Salah Previous page Next page
The Correct Answer and Explanation is:
Correct Answer: False / Salah
Explanation:
While it might seem intuitive that higher-order interpolation would always produce more accurate estimates because it uses more data points, this is not always true. In fact, higher-order interpolation can often lead to inaccurate or unstable results due to a phenomenon known as Runge’s phenomenon.
What is Interpolation?
Interpolation is a method of estimating unknown values that fall between known data points. For example, if we know the values of a function at certain points, interpolation allows us to estimate the function’s value at intermediate points.
There are different types of interpolation:
- Linear interpolation (1st order)
- Quadratic interpolation (2nd order)
- Cubic interpolation (3rd order)
- Polynomial interpolation of higher orders (4th, 5th, etc.)
The Problem with High-Order Interpolation
When we use high-order polynomials to interpolate data over a wide range, the resulting curve may start to oscillate wildly, especially near the edges of the interval. This erratic behavior is called Runge’s phenomenon, and it can lead to poor approximations, even though more data points and higher-degree polynomials are being used.
Additionally:
- High-order interpolation is more sensitive to small errors in the data.
- It increases the risk of overfitting, where the interpolation curve fits the known data points perfectly but fails to generalize well.
- The computation becomes more complex and less stable.
Practical Approach
In engineering and numerical analysis, piecewise interpolation such as splines (e.g., cubic splines) is often preferred. Splines use low-degree polynomials over small intervals to avoid the instability of high-degree polynomial interpolation.
Conclusion
Although higher-order interpolation uses more data and might seem more accurate, it does not guarantee better results. For many practical applications, lower-order or piecewise interpolation provides more reliable and stable estimates. Therefore, the statement is False.