LECTURE 8 NUMERICAL DIFFERENTIATION FORMULAE BY ...• The approximation for the derivative of some function can be found by taking the derivative of a polynomial approximation, ,

CE 30125 - Lecture 8

p. 8.1

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 derivativeshave an associated error

• If f(x) is an Nth degree polynomial then,

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 canderive the form of that polynomial by integrating the previous equation.

E hN d

N 1+f

dxN 1+

----------------

dN 1+ f

dxN 1+

---------------- 0=

f x a1xN

a2xN 1– aN 1++ + +


p. 8.2

• This implies that a distinct relationship exists between polynomials and FD expressionsfor 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

• The approximation for the derivative of some function can be found by

taking the derivative of a polynomial approximation, , of the function.

Procedure

• Establish a polynomial approximation of degree such that

• is forced to be exactly equal to the functional value at data points or nodes

• The derivative of the polynomial is an approximation to the derivative of

pth

f x

pth g x f x

g x N N p

g x N 1+

pth

g x pth

f x


p. 8.3

Approximations to First and Second Derivatives Using Quadratic Interpolation

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

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

• For second derivatives p=2 and we must establish at least an interpolating polyno-mial of degree N=2 with N+1=3 nodes

• Thus a quadratic interpolating function will allow us to establish both first andsecond derivative approximations

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

hhx0 x1 x2

f0 f1 f2x

shifted x axish h0


p. 8.4

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 orbackward interpolation approaches in order to establish the interpolating polyno-mial

• The 3 node quadratic interpolating polynomial has the form

• The approximating Lagrange polynomial must match the functional values at all data points or nodes ( , , )

g x aox2

a1x a2+ +=

N 1+xo 0= x1 h= x2 2h=

g xo fo= ao02

a10 a2+ + fo=

g x1 f1= aoh2

a1h a2+ + f1=

g x2 f2= ao 2h 2

a1 2h a2+ + f2=


p. 8.5

• Setting up the constraints as a system of simultaneous equations

• Solve for , ,

, ,

• The interpolating polynomial and its derivative are equal to:

0 0 1

h2 h 1

4h2 2h 1

aoa1a2

fof1f2

=

ao a1 a2

aof2 2f1– fo+

2h2

----------------------------= a14f1 f2– 3fo–

2h-------------------------------= a2 fo=

g x f2 2f1– fo+

2h2----------------------------- x2

4f1 f2– 3fo–

2h-------------------------------- x fo+ +=

g1

x f2 2f1– fo+

h2

----------------------------- x4f1 f2– 3fo–

2h--------------------------------+=


p. 8.6

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

• Evaluating the derivative of the interpolating function at

• Since the function is approximated by the interpolating function

• Substituting in for the expression for

xo 0=

g1

xo g1

0 =

g1

xo 3fo– 4f1 f2–+

2h------------------------------------=

f x g x

f1

xog

1 xo

g1

xo

f1

xo

3fo– 4f1 f2–+

2h------------------------------------


p. 8.7

• Generalize the node numbering for the approximation

• This results in the generic 3 node forward difference approximation to the first deriva-tive at node i

i i+1 i+2

generalized nodal numbering

x0 x1 x2

fi1 3fi– 4fi 1+ fi 2+–+

2h------------------------------------------------


p. 8.8

Evaluating g(1)(x1) to obtain a central difference approximation to the first derivative


• Again since the function is approximated by the interpolating function


x1 h=

g1

x1 g1

h =

g1

x1 f2 2f1– fo+

h2

---------------------------------h4f1 f2 3fo––

2h-------------------------------+=

g1

x1 f2 fo–

2h--------------=

f x g x

f1

x1g

1 x1

g1

x1

f1

x1

f2 fo–

2h--------------


p. 8.9

• Generalize the node numbering

• This results in the generic expression for the three node central difference approxima-tion to the first derivative

x0

i-1

x1 x2

i i+1

fi1 fi 1+ fi 1––

2h--------------------------


p. 8.10

Evaluating g(1)(x2) to obtain a backward difference approximation to the first derivative


• Again since the function is approximated by the interpolating function


x2 2h=

g1

x2 g1

2h =

g1

x2 f2 2f1– fo+

h2

---------------------------------2h4f1 f2 3fo––

2h-------------------------------+=

g1

x2 3f2 4f1– fo+

2h-------------------------------=

f x g x

f1

x2g

1 x2

g1

x2

f1

x2

3f2 4f1– fo+

2h-------------------------------


p. 8.11

• Generalizing the node numbering

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

x0

i-1

x1 x2

ii-2

fi1 3fi 4fi 1–– fi 2–+

2h-------------------------------------------


p. 8.12

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

• We note that in general can be computed as:

• Evaluating the second derivative of the interpolating function at :

• Again since the function is approximated by the interpolating function , thesecond derivative at node xo is approximated as:

g2

x

g 2 x f2 2f1– fo+

h2

-----------------------------=

xo 0=

g2

xo g2

0 =

g2

xo f2 2f1– fo+

h2

----------------------------=

f x g x

f2

xog

2 xo


p. 8.13


• Generalizing the node numbering

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

g2

xo

f2

xo

f2 2f1– fo+

h2

----------------------------

x2

i+2

x1

i+1

x0

i

fi2 fi 2+ 2fi 1+– fi+

h2

----------------------------------------


p. 8.14

Evaluating g(2)(x1) to obtain a central difference approximation to the second derivative

• Evaluating the second derivative of the interpolating function at :

• Again since the function is approximated by the interpolating function , thesecond derivative at node x1 is approximated as:


x1 h=

g2

x1 g2

h =

g2

x1 f2 2f1– fo+

h2

----------------------------=

f x g x

f2

x1g

2 x1

g2

x1

f2

x1

f2 2f1– fo+

h2

----------------------------


p. 8.15

• Generalizing node numbering

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

Notes on developing differentiation formulae by interpolating polynomials

• In general we can use any of the interpolation techniques to develop an interpolationfunction of degree . We can then simply differentiate the interpolating functionand evaluate it at any of the nodal points used for interpolation in order to derive anapproximation 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).

x0 x1 x2

i i+1i-1

fi2 fi 1+ 2fi– fi 1–+

h2

----------------------------------------

N p


p. 8.16

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 establishfinite difference formulae by simply differentiating the error estimate associated withthe 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 ( ) has 3 associated nodes ( ) orinterpolating points. We again assume that the nodes are evenly distributed as:

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

N 2= N 1+ 3=

x0 x1 x2

f0 f1 f2

h h

x


p. 8.17

• The approximating or interpolating function is defined using Newton forward interpola-tion as:

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

• These latter two forms which do not involve are more suitable for the necessary differ-entiation w.r.t. x since is functionally dependent on x, i.e.

f x g x e x +=

g x fo x xo– foh

--------12!----- x xo– x x1–

2foh

2----------+ +=

e x x xo– x x1– x x2–

3!--------------------------------------------------------f

3 = xo x2


3!--------------------------------------------------------fo

3


3!--------------------------------------------------------

3foh

3----------

x =


p. 8.18

• The forward difference operators are defined as:

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

• Evaluating the first derivative of the function at :

fo f1 fo–

2fo f2 2f1– fo+

x1

f1

x x1=g

1 x1 e

1 x1 +=

x0 x1 x2

central


p. 8.19

• Evaluating in the previous expression

g1

x

f1

x x1=

foh

--------12!----- x xo–

2foh

2----------

12!----- x x1–

2foh

2---------- e

1 x + ++

x x1==

f1

x x1=

foh

--------12!----- x1 xo–

2foh

2---------- e

1 x1 + +=

f1

x x1=

f1 fo–

h--------------

12---

h

h2

----- f2 2f1– fo+ e1

x1 + +=

f1

x x1=

f2 fo–

2h-------------- e

1 x1 +=


p. 8.20

• Now evaluate using a non dependent expression for the error term and evalu-ating this expression at

• Substituting for results in:

e1

x x1

e1

x1 13!----- f

3 xo x x1– x x2– x xo– x x2– x xo– x x1– + + x x1=

e1

x1 13!----- f

3 xo x1 xo– x1 x2–

e1

x1 h

2

3!----- f

3 xo –

e1

x1

f1

x x1=

f2 fo–

2h--------------

h2

3!----- f

3 xo –


p. 8.21

• Generalizing node numbering as:

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

x0 x1 x2

i i+1i-1

fi1 fi 1+ fi 1––

2h-------------------------- E+=

E h2

6----- f 3 xi 1– –


p. 8.22

• Notes

• The same discrete differentiation formulae can be derived using the Taylor seriesapproach.

• The results for the error estimates are the same regardless of whether you:

• Apply differentiation to , which represents the error estimate for .

• Apply a Taylor series analysis to the differentiation formula you derived.

• The error estimate for the interpolating function with is most precise in aformal sense (i.e. it includes all H.O.T. as well!). However there is a weak depen-dence of on and therefore some inaccuracies may be incurred when differenti-ating to obtain an error estimate for the corresponding finite differenceapproximation.

• Practically, for estimating the error of the differentiating formula derived by esti-

mating , we can apply the procedure used and examine an estimate of theerror which does not depend on . This is equivalent to examining the leading orderterm in the truncated series.

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

e x g x

g x

xe x

e1

x

xi


p. 8.23

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

• Evaluating the second derivative of the function at :

• Evaluate and substitute

xo

f2

x xo=g

2 xo e

2 xo +=

x0 x1 x2

forward

g2

x

f2

x xo=12!-----2foh

2----------

12!-----2foh

2---------- e

2 x + +

x xo==

f2

x xo=

2foh

2---------- e

2 xo +=

f2

x xo=

f2 2f1– fo+

h2

---------------------------- e2

xo +=


p. 8.24

• Now evaluate using a non dependent expression for the error term and evalu-ating this expression at

• Substituting for results in:

e2

x xo

e2

xo 13!----- f

3 xo x x1– x x2– x xo– x x2– x xo– x x1– + + + + + x xo=

e2

xo 13!----- f

3 xo xo x1– xo x2– xo xo– xo x2– xo xo– xo x1– + + + + +

e2

xo 13!----- f

3 xo h– 2h– 0 2h– 0 h–+ +

e2

xo h f3

xo –

e2

xo

f2

x xo=

f2 2f1– fo+

h2

---------------------------- hf3

xo –


p. 8.25

• Generalizing the node numbering:

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

x0 x1 x2

i i+2i+1

f 2 fi 2+ 2fi 1+– fi–

h2--------------------------------------- E+=

E hf3

xi –


p. 8.26

SUMMARY OF LECTURE 6, 7 AND 8

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

• In general, to develop a difference formula for you need nodes for accu-racy and nodes for O(h)N accuracy. Central approximations for even order deriva-tives require fewer nodes due to a fortunate cancelation of error terms.

• The generic form to evaluate a difference formula

• when nodes are used

• Substitute in Taylor series expansions for , etc. about node , re-arrange equa-tions such that coefficients multiply equal order derivatives at node and generate

algebraic equations by setting the coefficient of equal to 1 and the other coefficients equal to zero.

• Solve for , , ...

fip

p 1+ O h p N+

fip E–

a f a f a f+ + +

hp---------------------------------------------------------------=

E O h N= p N+

f f i

i

fip

p N 1–+

a a


p. 8.27

• 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 estab-lished. However, general formulae for any order accuracy are difficult to establish. Alsoyou still need Taylor series analysis to derive the accuracy of the approximations.

• Numerical differentiation formulae can be established by defining an interpolating poly-

nomial for at least nodes (to evaluate the derivative). Any interpolating tech-nique formula can be used. The numerical differencing formula is simply thedifferentiated interpolating polynomial evaluated at one of the nodes used for interpola-tion. The node at which the formula is evaluated establishes whether the approximationis forward, backward, central, etc.

p 1+ pth

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LECTURE 8 NUMERICAL DIFFERENTIATION FORMULAE BY ...• The approximation for the derivative of some function can be found by taking the derivative of a polynomial approximation, ,