NUMERICAL DIFFERENTIATION FORMULAE BY INTERPOLATING POLY- NOMIALS

Transcription

1 LECTURE 8 NUMERICAL DIFFERENTIATION FORMULAE BY INTERPOLATING POLY- NOMIALS Relationship Between Polynomials and Finite Difference Derivative Approimations We noted that N th degree accurate Finite Difference (FD) epressions for first derivatives have an associated error E h N dn + 1 f d N + 1 If f() is an N th degree polynomial then, d N + 1 f d N and the FD approimation to the first derivative is eact! Thus if we know that a FD approimation to a polynomial function is eact, we can derive the form of that polynomial by integrating the previous equation. f a 1 N + a 2 N a N + 1 p. 8.1

2 This implies that a distinct relationship eists between polynomials and FD epressions for derivatives (different relationships for higher order derivatives). We can in fact develop FD approimations from interpolating polynomials Developing Finite Difference Formulae by Differentiating Interpolating Polynomials Concept The approimation for the derivative of some function f can be found by p th taking the derivative of a polynomial approimation, g, of the function f. p th Procedure Establish a polynomial approimation g of degree N such that N p g is forced to be eactly equal to the functional value at N + 1 data points or nodes The derivative of the polynomial g is an approimation to the derivative of f p th p th p. 8.2

3 Approimations to First and Second Derivatives Using Quadratic Interpolation We will illustrate the use of interpolation to derive FD approimations to first and second derivatives using a 3 node quadratic interpolation function For first derivatives p1 and we must establish at least an interpolating polynomial of degree N1 with N+12 nodes For second derivatives p2 and we must establish at least an interpolating polynomial of degree N2 with N+13 nodes Thus a quadratic interpolating function will allow us to establish both first and second derivative approimations Apply a shifted coordinate system to simplify the derivation without affecting the generality of the derivation f 0 f 1 f 2 0 h 1 h 2 0 h h shifted ais p. 8.3

4 Develop a quadratic interpolating polynomial We apply the Power Series method to derive the appropriate interpolating polynomial Alternatively we could use either Lagrange basis functions or Newton forward or backward interpolation approaches in order to establish the interpolating polynomial The 3 node quadratic interpolating polynomial has the form g a o 2 + a 1 + a 2 The approimating Lagrange polynomial must match the functional values at all data points or nodes ( o 0, 1 h, 2 ) N + 1 g o f o a o a a 2 f o g 1 f 1 a o + a 1 h + a 2 f 1 g 2 f 2 a o 2 + a 1 + a 2 f 2 p. 8.4

5 Setting up the constraints as a system of simultaneous equations h 1 a o a 1 f o f a 2 f 2 Solve for a o, a 1, a 2 f 2 2f 1 + f o 4f a o, 1 f 2 3f o a , a 2 f o The interpolating polynomial and its derivative are equal to: g f 2 2f 1 + f o f 1 f 2 3f o f o g 1 f 2 2f 1 + f o f 1 f 2 3f o p. 8.5

6 Evaluating g (1) ( o ) to obtain a forward difference approimation to the first derivative Evaluating the derivative of the interpolating function at o 0 g 1 o g 1 0 g 1 o 3f o + 4f 1 f Since the function f is approimated by the interpolating function g f 1 g 1 o o Substituting in for the epression for g 1 o f 1 o 3f o + 4f 1 f p. 8.6

7 Generalize the node numbering for the approimation i i+1 i+2 generalized nodal numbering This results in the generic 3 node forward difference approimation to the first derivative at node i 1 f i 3f i + 4f i + 1 f i p. 8.7

8 Evaluating g (1) ( 1 ) to obtain a central difference approimation to the first derivative Evaluating the derivative of the interpolating function at 1 h g 1 1 g 1 h g 1 1 f 2 2f 1 + f o 4f 1 f 2 3f o h g 1 1 f 2 f o Again since the function f is approimated by the interpolating function g f 1 g Substituting in for the epression for g 1 1 f 2 f o f 1 p. 8.8

9 Generalize the node numbering 0 i i i+1 This results in the generic epression for the three node central difference approimation to the first derivative 1 f i + 1 f i 1 f i p. 8.9

10 Evaluating g (1) ( 2 ) to obtain a backward difference approimation to the first derivative Evaluating the derivative of the interpolating function at 2 g 1 2 g 1 g 1 2 f 2 2f 1 + f o 4f 1 f 2 3f o g 1 2 3f 2 4f 1 + f o Again since the function f is approimated by the interpolating function g f 1 g Substituting in for the epression for g 1 2 3f 2 4f 1 + f o f 1 p. 8.10

11 Generalizing the node numbering i-1 i-2 i This results in the generic epression for a three node backward difference approimation to the first derivative 3f i 4f i 1 + f i f i 1 p. 8.11

12 Evaluating g (2) ( o ) to obtain a forward difference approimation to the second derivative g 2 We note that in general can be computed as: g 2 f 2 2f 1 + f o Evaluating the second derivative of the interpolating function at o 0 : g 2 o g 2 0 g 2 o f 2 2f 1 + f o Again since the function f is approimated by the interpolating function g, the second derivative at node o is approimated as: f 2 g 2 o o p. 8.12

13 Substituting in for the epression for g 2 o f 2 2f 1 + f o o f 2 Generalizing the node numbering i 0 i+1 1 i+2 2 This results in the generic epression for a three node forward difference approimation to the second derivative + f i f i + 2 2f i + 1 f i p. 8.13

14 Evaluating g (2) ( 1 ) to obtain a central difference approimation to the second derivative Evaluating the second derivative of the interpolating function at h : g 2 1 g 2 h 1 g 2 1 f 2 2f 1 + f o Again since the function f is approimated by the interpolating function g, the second derivative at node 1 is approimated as: f 2 g Substituting in for the epression for g 2 1 f 2 2f 1 + f o f 2 p. 8.14

15 Generalizing node numbering i-1 i i This results in the generic epression for a three node central difference approimation to the second derivative + f i f i + 1 2f i f i Notes on developing differentiation formulae by interpolating polynomials In general we can use any of the interpolation techniques to develop an interpolation function of degree N p. We can then simply differentiate the interpolating function and evaluate it at any of the nodal points used for interpolation in order to derive an approimation for the p th derivative. Orders of accuracy may vary due to the accuracy of the interpolating function varying. Eact accuracy can be obtained by substituting in Taylor series epansions or by considering the accuracy of the approimating polynomial g(). p. 8.15

16 Approimations and Associated Error Estimates to First and Second Derivatives Using Quadratic Interpolation We can derive an error estimate when using interpolating polynomials to establish finite difference formulae by simply differentiating the error estimate associated with the interpolating function. We will illustrate the use of a 3 node Newton forward interpolation formula to derive: A central approimation to the first derivative with its associated error estimate A forward approimation to the second derivative with its associated error estimate Developing a 3 node interpolating function using Newton forward interpolation A quadratic interpolating polynomial ( N 2) has 3 associated nodes ( N + 1 3) or interpolating points. We again assume that the nodes are evenly distributed as: f 0 f 1 f h h With a quadratic interpolating polynomial, we can derive differentiation formulae for both the first and second derivatives but no higher p. 8.16

17 The approimating or interpolating function is defined using Newton forward interpolation as: f g + e g f o o f o ! o 1 2 f o e o f 3 3! o 2 The error can be approimately epressed in either of the following forms: e o f 3! o e o f o ! h 3 These latter two forms which do not involve are more suitable for the necessary differentiation w.r.t. since is functionally dependent on, i.e. p. 8.17

18 The forward difference operators are defined as: f o f 1 f o 2 f o f 2 2f 1 + f o Deriving a central approimation to the first derivative and the associated error estimate Evaluating the first derivative of the function at : 1 f 1 1 g e central p. 8.18

19 g 1 Evaluating in the previous epression f 1 1 f o h ! o 2 f o ! 1 2 f o e 1 1 f 1 1 f o h ! 1 o 2 f o e 1 1 f 1 1 f 1 f o h f 2 2f 1 + f o + e 1 1 f 1 1 f 2 f o e 1 1 p. 8.19

20 e 1 Now evaluate using a non dependent epression for the error term and evaluating this epression at 1 e f 3 3! o o 2 + o 1 1 e f 3 3! o 1 o 1 2 e h2 f 3 3! o e 1 1 Substituting for results in: f 1 1 f 2 f o f 3 3! o p. 8.20

21 Generalizing node numbering as: i-1 i i+1 This results in the generic epression for a three node central difference approimation to the first derivative with an appropriate error estimate E 1 f i + 1 f i 1 f i E h f 3 6 i 1 p. 8.21

22 Notes The same discrete differentiation formulae can be derived using the Taylor series approach. The results for the error estimates are the same regardless of whether you: Apply differentiation to e, which represents the error estimate for g. Apply a Taylor series analysis to the differentiation formula you derived. The error estimate for the interpolating function g with is most precise in a formal sense (i.e. it includes all H.O.T. as well!). However there is a weak dependence of on and therefore some inaccuracies may be incurred when differentiating e to obtain an error estimate for the corresponding finite difference approimation. Practically, for estimating the error of the differentiating formula derived by estimating e 1, we can apply the procedure used and eamine an estimate of the error which does not depend on. This is equivalent to eamining the leading order term in the truncated series. Note that the derivative in the error formula on the previous page may also be estimated at i p. 8.22

23 Deriving a forward difference approimation to the second derivative and the associated error estimate Evaluating the second derivative of the function at : o f 2 o g 2 o + e 2 o forward g 2 Evaluate and substitute f 2 o f o ! f o e 2 2! o f 2 o 2 f o e 2 o f 2 o f 2 2f 1 + f o e 2 o p. 8.23

24 e 2 Now evaluate using a non dependent epression for the error term and evaluating this epression at o e 2 o f 3 3! o o o + 1 o e 2 o f 3 3! o o 1 + o 2 + o o + o 2 + o o + o 1 e 2 o f 3 3! o h h e 2 o h f 3 o e 2 o Substituting for results in: f 2 o f 2 2f 1 + f o hf 3 o p. 8.24

25 Generalizing the node numbering: i i+1 i+2 This results in the generic epression for a three node forward difference approimation to the second derivative with an appropriate error estimate f i E f 2 f i + 2 2f i + 1 E hf 3 i p. 8.25

26 SUMMARY OF LECTURE 6, 7 AND 8 Difference formulae can be developed such that linear combinations of functional values at various nodes approimate a derivative at a node. In general, to develop a difference formula for you need p + 1 nodes for Oh accuracy and p + N nodes for O(h) N accuracy. Central approimations for even order derivatives require fewer nodes due to a fortunate cancelation of error terms. p f i The generic form to evaluate a difference formula p f i E a f + a f + + a f h p E Oh N when p + N nodes are used Substitute in Taylor series epansions for f, f etc. about node i, re-arrange equations such that coefficients multiply equal order derivatives at node i and generate p algebraic equations by setting the coefficient of f i equal to 1 and the p + N 1 other coefficients equal to zero. Solve for,,... a a p. 8.26

27 Forward, central and backward difference operators can be manipulated in ways analogous to differentiation to develop higher order differences Formulae to approimate differentiation using the difference operators can be established. However, general formulae for any order accuracy are difficult to establish. Also you still need Taylor series analysis to derive the accuracy of the approimations. Numerical differentiation formulae can be established by defining an interpolating polynomial for at least p + 1 nodes (to evaluate the p th derivative). Any interpolating technique formula can be used. The numerical differencing formula is simply the differentiated interpolating polynomial evaluated at one of the nodes used for interpolation. The node at which the formula is evaluated establishes whether the approimation is forward, backward, central, etc. p. 8.27