04-Divided-Difference.ppt

Interpolation and Polynomial Approximation(p187)123Page 1894Page 190 Example 4.1 Reading5Pages 192&193 Examples 4.2&4.3 Home AssignmentPage 195 Exercises 1(b)&2(b)Reading6Polynomial Approximation (p200)78910Page 200 Example 4.4 Home AssignmentPage 205 Exercise 2 Reading11Different from previous method(using system of linear equation to calculate the polynomial coefficients),we could directly give the coefficients without derivation:-Lagrange Polynomial-Newton Polynomial12Lagrange ApproximationThe French mathematician Lagrange noticed which passes through(x0,y0)and(x1,y1).Lagrange linear polynomial13Lagrange quadratic interpolating polynomialPage 210which passes through(x0,y0),(x1,y1),and(x2,y2).14Lagrange cubic interpolating polynomialwhich passes through(x0,y0),(x1,y1),(x2,y2),and(x3,y3).15The general form of Lagrange interpolating polynomial(p208)16ReadingPage 207 Example 4.6 Page 210 Example 4.7 Home AssignmentPage 217 Exercises 1 and 2 17InterpolationNewtons Divided Difference Polynomial MethodWhat is Interpolation?Given(x0,y0),(x1,y1),(xn,yn),find the value of y at a value of x that is not given.19Interpolants Polynomials are the most common choice of interpolants because they are easy to:EvaluateDifferentiate,and Integrate.20Newtons Divided Difference MethodLinear interpolation:Given pass a linear interpolant through the datawhere 21Example The path of a rapid laser is given by these specifications.If the laser is traversing from x=2 to x=4.25 in a linear path,find the value of y at x=4 using the Newtons Divided difference polynomial method.22Linear Interpolation23Linear Interpolation(contd)24Quadratic Interpolation注意：与数值微分的关系！25Example The path of a rapid laser is given by these specifications.If the laser is traversing from x=2 to x=4.25 to x=5.25 using a quadratic path,find the value of y at x=4 using the Newtons Divided difference polynomial method.26Quadratic Interpolation(contd)27Quadratic Interpolation(contd)28Quadratic Interpolation(contd)29General Formwhere Rewriting 30General Form31General form32Example The path of a rapid laser is given by these specifications.Find the path traversed by the laser using the Newtons Divided Difference Polynomial method.33Example 34Example35Example36ReadingPage 225 Example 4.12 Home AssignmentPage 229 Exercise 7 37