For n given points there exists a unique polynomial of degree n − 1 which passes through these points. For example, suppose you are given the three points (x0, y0), 2 (x1, y1) and (x2, y2). Then there is polynomial of degree 2, p2(x) = a0 +a1x+a2x , such that p2(x0) = y0, p2(x1) = y1, and p2(x2) = y2. This is the quadratic interpolating polynomial through the three given points. Example: Find the quadratic function passing through (0, 1), (1, 4) and (2, 8). 2 Solution: We want a second degree polynomial p2(x) = a0 + a1x + a2x such that the following is true. 2 p2(0) = a0 + a1(0) + a2(0) = 1 =⇒ a0 = 1 2 p2(0) = a0 + a1(1) + a2(1) = 4 =⇒ a0 + a1 + a2 = 4 2 p2(0) = a0 + a1(2) + a2(2) = 8 =⇒ a0 + 2a1 + 4a2 = 8
This produces a linear system of three equations with three unknowns, a0, a1, a2. This can be solved by Gaussian Elimination.
Exercise 1: Solve the above for a0, a1, a2 to find the quadratic interpolating polynomial. Check your answer. Exercise 2: Write down the linear system of equations generated when finding 2 the coefficients for the interpolating cubic polynomial p3(x) = a0 + a1x + a2x + 3 a3x passing through the points (x0, y0), (x1, y1), (x2, y2), and (x3, y3). DO NOT SOLVE. This idea can be generalized to finding an nth degree polynomial which interpolates n + 1 given points. However, we will see later that this will lead to instabilities such as high oscillations. Thus, in interpolation, another method known as cubic spline interpolation is much more preferable. Exercise 3: To see how higher degree polynomial interpolation is not a good thing, visit the following web site: http://www-hm.ma.tum.de/ws0304/in1/links/Interpol/Lagrange.html This is a Java applet for interpolating polynomials. First click on the button for ”Polynomial” . Try to put about eight points in a straight line by clicking them in with your mouse . Then move one of the points in the middle up and down. Click on ”Move points” to do this. You will see that the interpolating polynomial will change drastically even far away from the perturbed node. Next, click on the button for ”cubic spline”. For the cubic spline, there will not be any high oscillations. 1