Lecture 8 Numerical Differentiation Formulae By

CE 30125 - Lecture 8

LECTURE 8

NUMERICAL DIFFERENTIATION FORMULAE BY INTERPOLATING POLY- NOMIALS

Relationship Between Polynomials and Finite Difference Derivative Approximations • We noted that Nth degree accurate Finite Difference (FD) expressions for first derivatives have an associated error N + 1 N d f Eh ------dxN + 1 •If f(x) is an Nth degree polynomial then,

dN + 1f ------= 0 N + 1 dx and the FD approximation to the first derivative is exact!

• Thus if we know that a FD approximation to a polynomial function is exact, we can derive the form of that polynomial by integrating the previous equation.

N N – 1 fx a1x +++a2x  aN + 1

p. 8.1 CE 30125 - Lecture 8

• This implies that a distinct relationship exists between polynomials and FD expressions for derivatives (different relationships for higher order derivatives).

• We can in fact develop FD approximations from interpolating polynomials

Developing Finite Difference Formulae by Differentiating Interpolating Polynomials

Concept th • The approximation for the p derivative of some function fx can be found by th taking the p derivative of a polynomial approximation, gx , of the function fx.

Procedure

• Establish a polynomial approximation gx of degree N such that Np

•gx is forced to be exactly equal to the functional value at N + 1 data points or nodes

th th • The p derivative of the polynomial gx is an approximation to the p derivative of fx

p. 8.2 CE 30125 - Lecture 8

Approximations to First and Second Derivatives Using Quadratic Interpolation

• We will illustrate the use of interpolation to derive FD approximations to first and second derivatives using a 3 node quadratic interpolation function

• For first derivatives p=1 and we must establish at least an interpolating polynomial of degree N=1 with N+1=2 nodes

• For second derivatives p=2 and we must establish at least an interpolating polynomial of degree N=2 with N+1=3 nodes • Thus a quadratic interpolating function will allow us to establish both first and second derivative approximations

• Apply a shifted coordinate system to simplify the derivation without affecting the gener- ality of the derivation

f0 f1 f2 x x0 h x1 h x2 0 h h shifted x axis

p. 8.3 CE 30125 - Lecture 8

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

2 gx= aox ++a1xa2

• The approximating Lagrange polynomial must match the functional values at all N + 1

data points or nodes (xo = 0 , x1 = h , x2 = 2h )

2 gxo = fo  ao0 ++a10 a2 = fo

2 gx1 = f1 aoh ++a1ha2 = f1 

2 gx2 = f2  ao2h ++a12h a2 = f2

p. 8.4 CE 30125 - Lecture 8

• Setting up the constraints as a system of simultaneous equations

0 0 1 ao fo 2 h h 1 a1 = f1 2 4h 2h 1 a2 f2

• Solve for ao , a1 , a2

f – 2f + f 4f –3f – f a = ------2 1 o- , a = ------1 2 o- , a = f o 2 1 2 o 2h 2h

• The interpolating polynomial and its derivative are equal to:

f2 – 2f1 + fo 2 4f1 – f2 – 3fo gx= ------x ++------xfo 2h2 2h

1 f – 2f + f 4f – f – 3f gx = ------2 1 o- x + ------1 2 o-  2 h 2h

p. 8.5 CE 30125 - Lecture 8

(1) Evaluating g ( x o ) to obtain a forward difference approximation to the first derivative

• Evaluating the derivative of the interpolating function at xo = 0

1 1 g xo = g 0 

1 –43fo + f1 – f2 g x = ------o 2h

• Since the function fx is approximated by the interpolating function gx

1 1 f  g xo xo

1 • Substituting in for the expression for g xo

1 –43fo + f1 – f2 f  ------xo 2h

p. 8.6 CE 30125 - Lecture 8

• Generalize the node numbering for the approximation

x0 x1 x2 i i+1 i+2 generalized nodal numbering

• This results in the generic 3 node forward difference approximation to the first derivative at node i

1 – 3fi + 4fi + 1 – fi + 2 f  ------i 2h

p. 8.7 CE 30125 - Lecture 8

(1) Evaluating g ( x 1 ) to obtain a central difference approximation to the first derivative

• Evaluating the derivative of the interpolating function at x1 = h

1 1 g x1 = g h 

1 f2 – 2f1 + fo 4f1 – f2 – 3fo g x = ------h + ------ 1 2 h 2h

1 f2 – fo g x = ------1 2h

• Again since the function fx is approximated by the interpolating function gx

1 1 f  g x1 x1

1 • Substituting in for the expression for g x1

1 f2 – fo f  ------x1 2h

p. 8.8 CE 30125 - Lecture 8

• Generalize the node numbering

x0 x1 x2 i-1 i i+1

• This results in the generic expression for the three node central difference approximation to the first derivative

1 fi + 1 – fi – 1 f  ------i 2h

p. 8.9 CE 30125 - Lecture 8

(1) Evaluating g ( x 2 ) to obtain a backward difference approximation to the first derivative

• Evaluating the derivative of the interpolating function at x2 = 2h

1 1 g x2 = g 2h 

1 f – 2f + f 4f – f – 3f  gx = ------2 1 o-2h + ------1 2 o- 2 2 h 2h

1 3f2 – 4f1 + fo g x = ------2 2h

• Again since the function fx is approximated by the interpolating function gx

1 1 f  g x2 x2

1 • Substituting in for the expression for g x2

1 3f2 – 4f1 + fo f  ------x2 2h

p. 8.10 CE 30125 - Lecture 8

• Generalizing the node numbering

x0 x1 x2 i-2 i-1 i

• This results in the generic expression for a three node backward difference approximation to the first derivative

1 3fi – 4fi – 1 + fi – 2 f  ------i 2h

p. 8.11 CE 30125 - Lecture 8

(2) Evaluating g ( x o ) to obtain a forward difference approximation to the second derivative

2 • We note that in general g x can be computed as:

f – 2f + f g2 x = ------2 1 o-  2 h

• Evaluating the second derivative of the interpolating function at xo = 0 :

2 2 g xo = g 0 

2 f2 – 2f1 + fo g xo = ------h2

• Again since the function fx is approximated by the interpolating function gx , the

second derivative at node xo is approximated as:

2 2 f  g xo xo

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2 • Substituting in for the expression for g xo

f – 2f + f f2 ------2 1 o- x  2 o h

• Generalizing the node numbering

i i+1 i+2

x0 x1 x2

• This results in the generic expression for a three node forward difference approximation to the second derivative

f – 2f + f f2 ------i + 2 i + 1 -i i  2 h

p. 8.13 CE 30125 - Lecture 8

(2) Evaluating g ( x 1 ) to obtain a central difference approximation to the second derivative

• Evaluating the second derivative of the interpolating function at x1 = h :

2 2 g x1 = g h 

2 f2 – 2f1 + fo g x1 = ------h2

• Again since the function fx is approximated by the interpolating function gx , the

second derivative at node x1 is approximated as:

2 2 f  g x1 x1

2 • Substituting in for the expression for g x1

f – 2f + f f2 ------2 1 o- x  1 h2

p. 8.14 CE 30125 - Lecture 8

• Generalizing node numbering

i-1 i i+1

x0 x1 x2

• This results in the generic expression for a three node central difference approximation to the second derivative

2 f – 2f + f f ------i + 1 i i – 1- i  2 h

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 Np . 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 approximation for the pth derivative. • Orders of accuracy may vary due to the accuracy of the interpolating function varying. • Exact accuracy can be obtained by substituting in Taylor series expansions or by consid- ering the accuracy of the approximating polynomial g(x).

p. 8.15 CE 30125 - Lecture 8

Approximations and Associated Error Estimates to First and Second Derivatives Us- ing 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 approximation to the first derivative with its associated error estimate • A forward approximation 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 +31 = ) or interpolating points. We again assume that the nodes are evenly distributed as:

f0 f1 f2 x x0 x1 x2 hh

• With a quadratic interpolating polynomial, we can derive differentiation formulae for both the first and second derivatives but no higher

p. 8.16 CE 30125 - Lecture 8

• The approximating or interpolating function is defined using Newton forward interpolation as:

fx= gx+ ex

2 f 1  f gx = f ++xx– ------o------xx– xx– ------o  o o o 1 2 h 2! h

xx– o xx– 1 xx– 2 3 ex= ------f  x  x 3! o 2

• The error can be approximately expressed in either of the following forms:

xx– o xx– 1 xx– 2 3 ex ------f 3! o

3 xx– xx– xx–  f ex ------o 1 2 ------o  3 3! h

• These latter two forms which do not involve  are more suitable for the necessary differentiation w.r.t. x since  is functionally dependent on x, i.e. = x

p. 8.17 CE 30125 - Lecture 8

• The forward difference operators are defined as:

fo  f1 – fo

2  fo  f2 – 2f1 + fo

Deriving a central approximation to the first derivative and the associated error estimate

• Evaluating the first derivative of the function at x1 :

1 1 1 f = g x1 + e x1 xx= 1

x0 x1 x2

central

p. 8.18 CE 30125 - Lecture 8

1 • Evaluating g x in the previous expression

2 2 1 fo 1  fo 1  fo 1 f = ------+ -----xx– o ------++-----xx– 1 ------e x xx= 1 h 2! 2 2! 2 h h xx= 1

2 1 f 1  f 1 f = ------o- ++----- x – x ------o ex xx= 1 o 1 1 h 2! h2

1 f – f 1 h 1 f = ------1 o- ++------f – 2f + f ex xx= 22 1 o 1 1 h 2h

1 f2 – fo 1 f = ------+ e x1 xx= 1 2h

p. 8.19 CE 30125 - Lecture 8

1 • Now evaluate e x using a non  dependent expression for the error term and evaluating this expression at x1

1 1 3 e x1  ----- f xo xx– 1 xx– 2 ++xx– o xx– 2 xx– o xx– 1 3! xx= 1

1 1 3 e x  ----- f x x – x x – x 1 3! o 1 o 1 2

2 1 h 3 e x  – ----- f x 1 3! o

1 • Substituting for e x1 results in:

2 1 f2 – fo h 3 f  ------– ----- f xo xx= 1 2h 3!

p. 8.20 CE 30125 - Lecture 8

• Generalizing node numbering as:

x0 x1 x2

i-1 i i+1

• This results in the generic expression for a three node central difference approximation to the first derivative with an appropriate error estimate

f – f f1 = ------i + 1 i – 1 + E i 2h

2 h 3 E  – ----- f x 6 i – 1

p. 8.21 CE 30125 - Lecture 8

•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 ex , which represents the error estimate for gx . • Apply a Taylor series analysis to the differentiation formula you derived.

• The error estimate for the interpolating function gx with  is most precise in a formal sense (i.e. it includes all H.O.T. as well!). However there is a weak depen- dence of  on x and therefore some inaccuracies may be incurred when differentiating ex to obtain an error estimate for the corresponding finite difference approximation.

• Practically, for estimating the error of the differentiating formula derived by esti- 1 mating e x , we can apply the procedure used and examine an estimate of the error which does not depend on  . This is equivalent to examining the leading order term in the truncated series.

• Note that the derivative in the error formula on the previous page may also be esti-

mated at xi

p. 8.22 CE 30125 - Lecture 8

Deriving a forward difference approximation to the second derivative and the associated error estimate

• Evaluating the second derivative of the function at xo :

2 2 2 f = g xo + e xo xx= o

x0 x1 x2

forward

2 • Evaluate g x and substitute

2 2 2 1  fo 1  fo 2 f = ------++------e x xx= o 2! 2 2! 2 h h xx= o

2 2  f 2 f = ------o + ex xx= 2 o o h

2 f – 2f + f 2 f = ------2 1 o- + ex xx= 2 o o h

p. 8.23 CE 30125 - Lecture 8

2 • Now evaluate e x using a non  dependent expression for the error term and evaluating this expression at xo

2 1 3 e xo  ----- f xo xx– 1 +++++xx– 2 xx– o xx– 2 xx– o xx– 1 3! xx= o

2 1 3 e x  ----- f x x – x +++++x – x x – x x – x x – x x – x o 3! o o 1 o 2 o o o 2 o o o 1

2 1 3 e x  ----- f x –2h –02h ++–0h – h o 3! o

2 3 e xo  –h f xo

2 • Substituting for e xo results in:

f – 2f + f f2 ------2 1 o- – hf3 x xx=  o o h2

p. 8.24 CE 30125 - Lecture 8

• Generalizing the node numbering:

x0 x1 x2 i i+1 i+2

• This results in the generic expression for a three node forward difference approximation to the second derivative with an appropriate error estimate

f – 2f – f f2 = ------i + 2 i + 1 i + E h2

3 Ehf – xi

p. 8.25 CE 30125 - Lecture 8

SUMMARY OF LECTURE 6, 7 AND 8

• Difference formulae can be developed such that linear combinations of functional values at various nodes approximate a derivative at a node.

p • In general, to develop a difference formula for fi you need p + 1 nodes for Oh accuracy and pN+ nodes for O(h)N accuracy. Central approximations for even order derivatives require fewer nodes due to a fortunate cancelation of error terms. • The generic form to evaluate a difference formula

p a f +++a f  a f fi – E = ------hp

N • EOh=  when pN+ nodes are used

• Substitute in Taylor series expansions 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 fi equal to 1 and the pN+ – 1 other coefficients equal to zero.

• Solve for a , a , ...

p. 8.26 CE 30125 - Lecture 8

• Forward, central and backward difference operators can be manipulated in ways analo- gous to differentiation to develop higher order differences

• Formulae to approximate 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 approximations.

• Numerical differentiation formulae can be established by defining an interpolating polynomial for at least p + 1 nodes (to evaluate the pth derivative). Any interpolating tech- nique 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 approximation is forward, backward, central, etc.

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