Understanding Orthogonal Basis and Orthogonal Polynomials
Explore the theory of approximation through orthogonal basis and periodic functions, orthogonal polynomials like Tchebycheff and Legendre, least square solutions, normal equations, and the concept of orthonormal polynomials. Learn about orthogonality, symmetries, and recursion formulas involved in these mathematical concepts.
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Presentation Transcript
Theory of Approximation: Orthogonal Basis, Periodic Functions
Least Square Solution: Normal Equations ? ??? ,??? ?? = ? ? ,??? ; ? = 0,1,2, ? ?=0 If ?0,?1,?2, ?? is an orthogonal system: ?,?? ??,?? = ?? ? = 0,1,2, ? Orthogonal Polynomials: Tchebycheff Legendre Gram
Orthogonal Polynomials: Tchebycheff Define: ? = cos?,? 0,? and ??? = cos?? = cos ?cos 1? Recursion Formula: ??+1? = 2???? ?? 1? Example: ?0? = 1; ?1? = ?; ?2? = 2?2 1 Symmetry: ?? ? = 1???? Tn(x) constitutes an orthogonal family of polynomials in [-1, 1]!
Orthogonal Polynomials: Tchebycheff Orthogonality (continuous): ? ??? ,??? = cos??cos???? 0 1 1 = ??? ??? 1 ?2?? 1 0 ? 2 ? if ? = ? = 0 if ? ? = if ? = ? 0 For arbitrary f(x), a x b: ? =?+? 2+? ? 2? 1 ? 1
Orthogonal Polynomials: Legendre Solution of the Legendre s equation (for n non-negative: ? ?? 1 ?2?? + ? ? + 1 ? = 0 1 ? 1 ?? Solutions are an orthogonal set of polynomials given by: ?? ??? 1 ?2 1? ?0? = 1; ??? = 2??! Bonnet s recursive relation for n 2: ??? =2? 1 ??? 1? ? 1 ?? 2? ? ? Examples: ?0? = 1; ?1? = ?; ?2? =1 23?2 1 ; ?3? =1 25?3 3?
Orthogonal Polynomials: Legendre Symmetry: ?? ? = 1???? Orthogonality (continuous): 0 if ? ? 2 ??? ,??? = if ? = ? 2? + 1 ? ? = 1; ??? 1 For discrete data: equidistant data points.
Orthonormal Polynomials: Gram For Equidistant Discrete Data: For 1 ? 1, a net of (n + 1) equidistant points are given by: ??= 1 +2? for ? = 0,1,2, ? ? ? On this net, the orthonormal set of polynomials ??? are given by: ?=0 1 ?? ?? 1?? 1? ? 1? = 0; ?0? = ? + 1; ??+1? = ?????? 4 ? + 12 1 ? + 12 ? + 12 ? ??= ? = 0,1,2, ? 1 ? + 1 Example (for n = 5): 1 5? 70; ?2? =25 3 7?2 5 7 3 ?0? = 6; ?1? = 16 16
Orthonormal Polynomials: Gram Orthogonality: ? ???????? = 0 if ? ? ??? ,??? = 1 if ? = ? ?=0 When m << n1/2, Gm(x) are very similar to the Legendre polynomials When n << m1/2, Gm(x) have very large oscillations between the net points, large maximum norm in [-1, 1] When fitting a polynomial to equidistant data, one should never choose m larger than ~ 2n1/2
Least Square Solution: Example (Continuous) Approximate the function f(x) = 1/(1 + x2) for x in [0, 1] using a 2ndorder Legendre polynomial For Legendre polynomials, use x = (z + 1)/2 such that for x in [0, 1], z is in [-1, 1] The function is: f(z) = 4/(5 + 2z + z2) 2 ?? 23?2 1 ; ? ? = ?=0 The basis functions are: ?0= 1; ?2=1 ?1= ?; ?? 1 5 + 2? + ?2?? = 4 ?,?0 = 2= 1.5708 1 1 5 + 2? + ?2?? = 2ln2 4? ?,?1 = 2= 0.1845 1 141 5 + 2? + ?2?? = 12 6ln2 5 23?2 1 2= 0.1286 10 1 ?,?2 = 1 = 0.1286 10 1 =1.5708 = 0.1845 = 0.3216 10 1 ?0 = 0.7854;?1 = 0.2768;?2 2 5 2 2/3 2 ??? = 0.8015 0.2768? 0.4824 10 1?2 ? ? = ?? ?=0 If you now use, z = 2x 1 ? ? = 1.030 0.3605? 0.193?2 Least square polynomial is unique! It does not depend on the basis!
Periodic Functions A function of period p: f(x + p) = f(x) for all x We shall study functions of period 2 0 ? 2? or ? ? ? For any function f(x) with a period p, transform t = 2 x/p Allow functions to have complex values. Definition: Inner product of two complex-valued functions f and g of period 2 2? ?,? = ? ? ? ? ?? (??????????) 0 ? 2?? ? + 1 = ? ?? ? ?? ; ??= (????????) ?=0
Periodic Orthogonal Basis Functions ??? = ????; 0 ? 2? ? = 0, 1, 2, or ? ? ? Continuous Case: ? ????? ????? = 0 if ? ? ??,?? = 2? if ? = ? ? Discrete Case: ??=2?? ?+1 ? ? 2?? ? + 1 ?????? ????= ??,?? = exp ? ? ? ?=0 ?=0 ? ? ? + 1 = ? + 1 if is an integer 0 Otherwise
Periodic Orthogonal Basis Functions ??? = cos??; 0 ? 2? or ? ? ? ? 0 ? 2? if ? = ? = 0 if ? ? if ? = ? 0 ??,?? = cos??cos?? ?? = ? ??? = sin??; 0 ? 2? or ? ? ? ? sin??sin?? ?? = 0 if ? ? if ? = ? ??,?? = ? ? Wave Number: ? = 0,1,2,
Least Square Approximation of Periodic Functions Consider a periodic function f(x) of period 2 We shall find the least square approximation of this function, f*(x), using orthogonal basis functions as follows: ? ? = ????? + ????? ?=0 ?=0 where, ??? = cos?? and ??? = sin?? 0 ? 2? if ? = ? = 0 if ? ? and ??,?? = 0 if ? ? if ? = ? We have: ??,?? = if ? = ? 0 ? ??,?? = 0 ?,?
Least Square Approximation of Periodic Functions If f*(x) is the least square approximation of f(x), normal equations must be satisfied: ? ? ,?? = ? ? ,?? = 0 ? ?,?? = ? ,?? ??? ?,?? = ? ,?? ? As ? ? = ?=0 ????? + ?=0 ????? ?,?? ??,?? ?,?? = ????,?? + ????,?? ??= ?=0 ?=0 ?,?? ??,?? ?,?? = ????,?? + ????,?? ??= ?=0 ?=0
Least Square Approximation of Periodic Functions ? ? ?,?? ??,?? 1 1 ??= = ? ? ??? ?? = ? ? cos???? ??,?? ??,?? ? ? ? ? 1 ??=1 ?0= 2? ? ? ??; ? ? ? ? cos????; ? = 1,2, ? ? ? ?,?? ??,?? 1 1 ??= = ? ? ??? ?? = ? ? sin???? ??,?? ??,?? ? ? ? ??=1 ?0= 0; ? ? ? ? sin????; ? = 1,2,
Least Square Approximation of Periodic Functions ? ? = ????? + ????? ?=0 ?=0 where, ??? = cos?? and ??? = sin?? ? ? = ??cos?? + ??sin?? ?=0 ?=0 = ?0+ ??cos?? + ??sin?? ?=1 ?=1 where, ? ? ? 1 ??=1 ??=1 ?0= 2? ? ? ??; ? ? ? ? cos????; ? ? ? ? sin???? ? Fourier Series is the Least Square Approximation of a periodic function!
Fourier Series Approximation: Alternate Basis Functions Consider a periodic function f(x) of period 2 We shall find the least square approximation of this function f*(x) using orthogonal basis functions as follows: ? ? = ??? = ???? ????? ; ?= We have: ??,?? = 0 if ? ? ? if ? = ?
Least Square Approximation of Periodic Functions If f*(x) is the least square approximation of f(x), normal equations must be satisfied: ? ? ,?? = 0 ? ? ? = ?= ?????? As, ?,?? ??,?? ?,?? = ????,?? ??= ?= ? ?,?? ??,?? 1 ? ? ? ????? 2? ??= = ?
Least Square Approximation of Periodic Functions 1 ? ? = ??????= ?0+ ??????+ ?????? ?= ?= ?=1 ? ?? ???+ ?????? = ?0+ ?=1 ?=1 = ?0+ ??+ ? ? cos?? + ? ?? ? ? sin?? ?=1 ?=1 = ?0+ ??cos?? + ??sin?? ?=1 ?=1 where, ?0= ?0;??= ??+ ? ?;??= ? ?? ? ?
Fourier Series Approximation: Alternate Basis Functions ? 1 ? ? ? ????? ??= 2? ? ? 1 ?0= ?0= 2? ? ? ?? ? ? ? 1 ? ???+ ?????? =1 ??= ??+ ? ? = 2? ? ? ? ? ? ? cos???? ? ? ? 1 ? ? ??? ?????? =1 ??= ? ?? ? ? = 2? ? ? ? ? ? ? sin???? ? Both are Fourier Series, one is the exponential form and the other is the sine- cosine form!
The Fourier Series For a periodic function f(x) with a period of 2 , i.e., (- , ) or (0, 2 ) ? ? =?0 2+ ?=1 ??cos?? + ??sin?? ?=1 ? ? ? ?0=1 ? ? ??; ??=1 ? ? cos?? ??; ??=1 ? ? ? ? ? ? ? ? sin?? ?? Alternatively: ? 1 ?????? ? ? ? ????? ? ? = ??= 2? ?= ? ??=1 ? ?=1 2?? ???; 2??+ ???
The Fourier Series For a periodic function of any period 2L i.e., (-L, L) or (0, 2L), simply scale the variable x. That is, replace x by x/L ? ? =?0 ??cos??? ??sin??? 2+ + ? ? ?=1 ?=1 ? ? ??=1 ? ? cos??? ??; ??=1 ? ? sin??? ? ? ? ? ?? ? ? ? 1 2? ??????? ? ? ? ???? ? ? = ; ??= ? ?? ? ?= ?
The Discrete and Finite Fourier Series ? 1 ?????? ? ? ? ????? ? ? = ??= 2? ?= ? Finite (only finite frequencies or wave numbers): ? ?????? ? ? = ?=? Discrete points in x: ? 2?? ? + 1; ?,?? ??,?? 1 ? ??? ???? ??= ??= = ? + 1 ?=0
The Fourier Series Conditions for convergence: f(x) is continuous or piecewise continuous in (- , ) or (-L, L) finite number of finite discontinuities, i.e., finite number of maxima, minima f(x) is periodic on the entire x-axis with period 2 or 2L Sufficient Condition (Dirichlet Condition for Fourier Series): ? ?? ? 2= 2? ? ?? < ? ? ? ? ? ?? ?
Theory of Approximation: Interpolation Abhas Singh Department of Civil Engineering IIT Kanpur Acknowledgements: Profs. Saumyen Guha and Shivam Tripathi (CE)
Approximation of Discrete data or tab (f): Regression Approximation of Continuous Function f(x) Complicated Analytical Function, Analog Signal from a measuring device Discrete measurements of continuous experiments or phenomena Approximation of Discrete data or tab (f): Regression Missing Data, Derivative, Integration for or tab (f): Interpolation
Discrete Data (n + 1) observations or data pairs [(x0, y0), (x1, y1), (x2, y2) (xn, yn)] (m + 1) basis functions: ?0,?1,?2, ?? Approximating polynomial: ? ? = ?=0 ?0?0?0 + ?1?1?0 + ?2?2?0 + + ?????0 = ?0 ?0?0?1 + ?1?1?1 + ?2?2?1 + + ?????1 = ?1 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ?0?0?? + ?1?1?? + ?2?2?? + + ?????? = ?? n equations, m unknowns: m < n: over-determined system, least square regression m = n: unique solution, interpolation m > n: under-determined system ? ?????
Interpolation Polynomials Newton s Divided Difference Lagrange Polynomials Gram s polynomials (introduced earlier) Spline Interpolation: piecewise continuous, smoothing
Newtons Divided Difference Triangular set of basis polynomials ?0? = 1 ?1? = ? ?0 ?2? = ? ?0 ? ?1 ?3? = ? ?0 ? ?1 ? ?2 ??? = ? ?0 ? ?1 ? ?? 1 ??? = ? ?0 ? ?1 ? ?? 1
Newtons Divided Difference Consider a net of points ?0,?1,?2, ?? and the corresponding function values as ?0,?1,?2, ?? Newton s polynomial is: ? ? = ?0+ ?1? ?0 + ?2? ?0 ? ?1 +?3? ?0 ? ?1 ? ?2 +??? ?0 ? ?1 ? ?? 1 True function: ? ? = ? ? +??+1? ? + 1 !? ?0 ? ?1 ? ?? for some? int ?,?0,?1 ??
