Mastering Algorithms with C

Questions and Answers

Q: In the discussion of polynomial interpolation, we stated that we need to choose enough points to give an accurate impression of the function we are interpolating. What happens if we do not use enough points?

A: Interpolating a function with not enough points, or poorly placed points, leads to an interpolating polynomial that does not accurately reflect the function we think we are interpolating. A simple example is interpolating a quadratic polynomial (a parabola when plotted) with only two points. Interpolation with two points results in a line, which is far from a parabola!

Q: Using the guidelines presented in this chapter, how many interpolation points should we use to interpolate the function f (x) = x ⁵ + 2.8x ³ - 3.3x ² - x + 4.1?

A: When interpolating a function that we know is a polynomial itself, we can get a good impression of the function by using n + 1 well-placed points, where n is the degree of the polynomial. In this example, the polynomial has a degree of 5, so we should use six well-placed interpolation points. This results in an interpolating polynomial that has the same degree as f (x).

Q: Recall that to approximate a root of an equation using Newton’s method, we select an interval [a, b] on which the root exists and iterate closer and closer to it. What if we choose this interval much larger than needed, but in such a way that both rules mentioned in this chapter are still satisfied?

A: The discussion of Newton’s method mentioned two ...