Tiêu đề part2.pdf ✅

Numerical Methods and Computations : MTL107 Dr. Mani Mehra Department of Mathematics Indian Institute of Technology Delhi Dr. Mani Mehra (IIT Delhi) Numerical Methods and Computation: MTL107 1 / 50
Interpolation Interpolation methods are the basis for many other procedures that you will study in this course, such as numerical integration and di!erentiation. They are behind the ways that we use to solve ordinary and partial di!erential equations. They demonstrate important theory about polynomials and the accuracy of numerical methods. They are one of the more important ways that curves are drawn on your computer screen. There is a rich history behind interpolation. It really began with the early studies of astronomy when the motion of heavenly bodies was determined from periodic observations. An application of interpolation that you see every day is in weather forecasting. When you watch the weather forecasts on television, you may wonder where these (usually) correct projections come from. Dr. Mani Mehra (IIT Delhi) Numerical Methods and Computation: MTL107 2 / 50
Interpolating Polynomials Dr. Mani Mehra (IIT Delhi) Numerical Methods and Computation: MTL107 3 / 50
Interpolating Polynomials We will be most interested in techniques adapted to situations where the data are far from linear. The basic principle is to fit a polynomial curve to the data. The reason for using polynomials has already been stated-they are nice functions and their evaluation requires only those arithmetic operations that computers can do. Suppose that we have the following data pairs-x-values and f (x)-values where f (x) is some unknown function: x 3.2 2.7 1.0 4.8 5.6 f (x) 22.0 17.8 14.2 38.3 51.7 First, we need to select the points that determine our polynomial. The maximum degree of the polynomial is always one less than the number of points. Suppose we choose the first four points. If the cubic is ax3 + bx2 + cx + d, we can write four equations involving the unknown coe”cients a, b, c, and d. Dr. Mani Mehra (IIT Delhi) Numerical Methods and Computation: MTL107 4 / 50