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Interpolation & Polynomial Approximation Divided Differences: A Brief Introduction Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c � 2011 Brooks/Cole, Cengage Learning

Introduction Notation Newton’s Polynomial Outline Introduction to Divided Differences 1 Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 2 / 16

Introduction Notation Newton’s Polynomial Outline Introduction to Divided Differences 1 The Divided Difference Notation 2 Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 2 / 16

Introduction Notation Newton’s Polynomial Outline Introduction to Divided Differences 1 The Divided Difference Notation 2 Newton’s Divided Difference Interpolating Polynomial 3 Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 2 / 16

Introduction Notation Newton’s Polynomial Outline Introduction to Divided Differences 1 The Divided Difference Notation 2 Newton’s Divided Difference Interpolating Polynomial 3 Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 3 / 16

Introduction Notation Newton’s Polynomial Introduction to Divided Differences A new algebraic representation for P n ( x ) Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 4 / 16

Introduction Notation Newton’s Polynomial Introduction to Divided Differences A new algebraic representation for P n ( x ) Suppose that P n ( x ) is the n th Lagrange polynomial that agrees with the function f at the distinct numbers x 0 , x 1 , . . . , x n . Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 4 / 16

Introduction Notation Newton’s Polynomial Introduction to Divided Differences A new algebraic representation for P n ( x ) Suppose that P n ( x ) is the n th Lagrange polynomial that agrees with the function f at the distinct numbers x 0 , x 1 , . . . , x n . Although this polynomial is unique, there are alternate algebraic representations that are useful in certain situations. Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 4 / 16

Introduction Notation Newton’s Polynomial Introduction to Divided Differences A new algebraic representation for P n ( x ) Suppose that P n ( x ) is the n th Lagrange polynomial that agrees with the function f at the distinct numbers x 0 , x 1 , . . . , x n . Although this polynomial is unique, there are alternate algebraic representations that are useful in certain situations. The divided differences of f with respect to x 0 , x 1 , . . . , x n are used to express P n ( x ) in the form P n ( x ) = a 0 + a 1 ( x − x 0 )+ a 2 ( x − x 0 )( x − x 1 )+ · · · + a n ( x − x 0 ) · · · ( x − x n − 1 ) for appropriate constants a 0 , a 1 , . . . , a n . Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 4 / 16

Introduction Notation Newton’s Polynomial Introduction to Divided Differences P n ( x ) = a 0 + a 1 ( x − x 0 )+ a 2 ( x − x 0 )( x − x 1 )+ · · · + a n ( x − x 0 ) · · · ( x − x n − 1 ) To determine the first of these constants, a 0 , note that if P n ( x ) is written in the form of the above equation, then evaluating P n ( x ) at x 0 leaves only the constant term a 0 ; that is, a 0 = P n ( x 0 ) = f ( x 0 ) Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 5 / 16

Introduction Notation Newton’s Polynomial Introduction to Divided Differences P n ( x ) = a 0 + a 1 ( x − x 0 )+ a 2 ( x − x 0 )( x − x 1 )+ · · · + a n ( x − x 0 ) · · · ( x − x n − 1 ) To determine the first of these constants, a 0 , note that if P n ( x ) is written in the form of the above equation, then evaluating P n ( x ) at x 0 leaves only the constant term a 0 ; that is, a 0 = P n ( x 0 ) = f ( x 0 ) Similarly, when P ( x ) is evaluated at x 1 , the only nonzero terms in the evaluation of P n ( x 1 ) are the constant and linear terms, f ( x 0 ) + a 1 ( x 1 − x 0 ) = P n ( x 1 ) = f ( x 1 ) Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 5 / 16

Introduction Notation Newton’s Polynomial Introduction to Divided Differences P n ( x ) = a 0 + a 1 ( x − x 0 )+ a 2 ( x − x 0 )( x − x 1 )+ · · · + a n ( x − x 0 ) · · · ( x − x n − 1 ) To determine the first of these constants, a 0 , note that if P n ( x ) is written in the form of the above equation, then evaluating P n ( x ) at x 0 leaves only the constant term a 0 ; that is, a 0 = P n ( x 0 ) = f ( x 0 ) Similarly, when P ( x ) is evaluated at x 1 , the only nonzero terms in the evaluation of P n ( x 1 ) are the constant and linear terms, f ( x 0 ) + a 1 ( x 1 − x 0 ) = P n ( x 1 ) = f ( x 1 ) f ( x 1 ) − f ( x 0 ) ⇒ a 1 = x 1 − x 0 Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 5 / 16

Introduction Notation Newton’s Polynomial Outline Introduction to Divided Differences 1 The Divided Difference Notation 2 Newton’s Divided Difference Interpolating Polynomial 3 Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 6 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation We now introduce the divided-difference notation, which is related to Aitken’s ∆ 2 notation ∆ Definition Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 7 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation We now introduce the divided-difference notation, which is related to Aitken’s ∆ 2 notation ∆ Definition The zeroth divided difference of the function f with respect to x i , denoted f [ x i ] , is simply the value of f at x i : f [ x i ] = f ( x i ) Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 7 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation We now introduce the divided-difference notation, which is related to Aitken’s ∆ 2 notation ∆ Definition The zeroth divided difference of the function f with respect to x i , denoted f [ x i ] , is simply the value of f at x i : f [ x i ] = f ( x i ) The remaining divided differences are defined recursively. Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 7 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation The first divided difference of f with respect to x i and x i + 1 is denoted f [ x i , x i + 1 ] and defined as f [ x i , x i + 1 ] = f [ x i + 1 ] − f [ x i ] x i + 1 − x i Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 8 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation The first divided difference of f with respect to x i and x i + 1 is denoted f [ x i , x i + 1 ] and defined as f [ x i , x i + 1 ] = f [ x i + 1 ] − f [ x i ] x i + 1 − x i The second divided difference, f [ x i , x i + 1 , x i + 2 ] , is defined as f [ x i , x i + 1 , x i + 2 ] = f [ x i + 1 , x i + 2 ] − f [ x i , x i + 1 ] x i + 2 − x i Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 8 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation Similarly, after the ( k − 1 ) st divided differences, f [ x i , x i + 1 , x i + 2 , . . . , x i + k − 1 ] and f [ x i + 1 , x i + 2 , . . . , x i + k − 1 , x i + k ] have been determined, Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 9 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation Similarly, after the ( k − 1 ) st divided differences, f [ x i , x i + 1 , x i + 2 , . . . , x i + k − 1 ] and f [ x i + 1 , x i + 2 , . . . , x i + k − 1 , x i + k ] have been determined, the k th divided difference relative to x i , x i + 1 , x i + 2 , . . . , x i + k is f [ x i , x i + 1 , . . . , x i + k − 1 , x i + k ] = f [ x i + 1 , x i + 2 , . . . , x i + k ] − f [ x i , x i + 1 , . . . , x i + k − 1 ] x i + k − x i Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 9 / 16

Introduction Notation Newton’s Polynomial The Divided Difference Notation Similarly, after the ( k − 1 ) st divided differences, f [ x i , x i + 1 , x i + 2 , . . . , x i + k − 1 ] and f [ x i + 1 , x i + 2 , . . . , x i + k − 1 , x i + k ] have been determined, the k th divided difference relative to x i , x i + 1 , x i + 2 , . . . , x i + k is f [ x i , x i + 1 , . . . , x i + k − 1 , x i + k ] = f [ x i + 1 , x i + 2 , . . . , x i + k ] − f [ x i , x i + 1 , . . . , x i + k − 1 ] x i + k − x i The process ends with the single n th divided difference, f [ x 0 , x 1 , . . . , x n ] = f [ x 1 , x 2 , . . . , x n ] − f [ x 0 , x 1 , . . . , x n − 1 ] x n − x 0 Numerical Analysis (Chapter 3) Divided Differences: A Brief Introduction R L Burden & J D Faires 9 / 16