NUMERICAL METHODS II . Csaba J. Hegedüs ELTE, Faculty of Informatics Budapest, 2016 January "Financed from the financial support ELTE won from the Higher Education Restructuring Fund of the Hungarian Government" Referee: Dr. Levente Lócsy, ELTE Hegedüs: Numerical Methods II. 2 Contents 11. Lagrange interpolation and its error .................................................................................................. 4 11.1. Interpolating function with linear parameters ............................................................................ 4 11.2. Polynomial interpolation ............................................................................................................ 4 11.3. Interpolation with Lagrange base polynomials .......................................................................... 5 11.4. Example...................................................................................................................................... 6 11.5. The barycentric Lagrange interpolation ..................................................................................... 6 11.6. Problems..................................................................................................................................... 8 12. Some properties of polynomial interpolation.................................................................................... 9 12.1. Theorem on uniform convergence.............................................................................................. 9 12.2. Lemma on upper bound for ωn (x ) .......................................................................................... 9 12.3. Another theorem on error bound.............................................................................................. 10 12.4. Skilful choice of support abscissas, Chebyshev polynomials .................................................. 10 12.5. Theorem on the best zero approximating monic polynomial in ∞ -norm................................ 10 12.6. Problems................................................................................................................................... 11 13. Iterated interpolations (Neville, Aitken, Newton)........................................................................... 13 13.1. Neville and Aitken interpolations............................................................................................. 13 13.2. Divided differences .................................................................................................................. 14 13.3. Recursive Newton interpolation............................................................................................... 16 13.4. Problems................................................................................................................................... 16 14. Newton and Hermite interpolations ................................................................................................ 18 14.1. Theorem, interpolation error with divided differences............................................................. 18 14.2. Hermite’s interpolation............................................................................................................. 18 14.3. Base polynomials for Hermite’s interpolation ......................................................................... 20 14.4. The Heaviside „cover up” method for partial fraction expansion and interpolation................ 21 14.5. Inverse interpolation................................................................................................................. 23 14.6. Problems................................................................................................................................... 23 15. Splines ............................................................................................................................................. 25 15.1. Spline functions........................................................................................................................ 25 15.2. Splines of first degree: s() x∈ S 1 ( Θ n ) ................................................................................... 26 15.3. Splines of second degree: s() x∈ S 2 ( Θ n ) .............................................................................. 26 15.4. Splines of third degree: s() x∈ S 3 ( Θ n ) .................................................................................. 27 15.5. Example.................................................................................................................................... 29 15.6. Problems................................................................................................................................... 29 3 16. Solution of nonlinear equations I. ................................................................................................... 31 16.1. The interval of the root............................................................................................................. 31 16.2. Fixed-point iteration................................................................................................................. 31 16.3. Speed of convergence............................................................................................................... 33 16.4. Newton iteration (Newton-Raphson method) and the secant method...................................... 34 16.5. Examples .................................................................................................................................. 37 16.6. Problems................................................................................................................................... 38 17. Solution of nonlinear equations II. .................................................................................................. 40 17.1. The method of bisection........................................................................................................... 40 17.2. The method of false position (regula falsi).............................................................................. 40 17.3. Newton iteration for functions of many variables.................................................................... 41 17.4. Roots of polynomials................................................................................................................ 42 17.5. Problems................................................................................................................................... 43 18. Numerical quadrature I.................................................................................................................... 44 18.1. Closed and open Newton-Cotes quadrature formulas.............................................................. 44 18.2. Some simple quadrature formulas............................................................................................ 45 18.3. Examples .................................................................................................................................. 48 18.4. Problems................................................................................................................................... 49 19. Gaussian quadratures....................................................................................................................... 50 19.1. Theorem on the roots of orthogonal polynomials .................................................................... 50 19.2. Theorem on the order of the Gaussian quadrature ................................................................... 50 By applying the mean value theorem of integrals, we get the statement from...................................... 51 19.3. Examples .................................................................................................................................. 52 19.4. Problems................................................................................................................................... 53 Hegedüs: Numerical Methods II. 4 11. Lagrange interpolation and its error Interpolation is a simple way of approximating functions by demanding that the interpolant function assumes the values of the approximated function at specified places. Collect the support points of the interpolation into the set Ωn = {,x0 x 1 ,… , x n } , where xi -s are not necessarily ordered. We shall consider interpolation in the interval [a , b ] . The relation [ab , ]= [min xi , max x i ] holds in many cases, but all support points may also be inner points of i i [a , b ] . 11.1. Interpolating function with linear parameters Let n be a natural number, n ∈ ℕ and assume the values of the function f( x ) are known at the points xk ∈ ℝ, k= 0,1,..., n . In the case of a linear interpolation problem, we choose the interpolant as n Φ()x = ∑ aiϕ i () x , (11.1) i=0 where ϕi (x ) -s are base functions, and the unknown coefficients ai are determined from the conditions fx(i )= Φ ( xi i ), = 0,..., n . (11.2) i The functions ϕi (x ) in (11.1) may be powers of x : ϕi (x ) = x , which will lead to a polynomial, but it is also possible to choose other functions: ϕi (x )= sin( i ω x ), ϕi()x= cos( ix ωϕ ), i () x = exp( ix ω ). In the case of n = 2 the interpolation problem (11.2) leads to the linear system of equations ϕ00()()x ϕ 10 x ϕ 20 () xa 0 fx () 0 ϕ01()x ϕ 11 () x ϕ 21 () xa 1= fx (). 1 (11.3) ϕ02()()x ϕ 12 x ϕ 22 () xa 2 fx () 2 The obtained system can be solved uniquely