Week01_Introduction to Numerical

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WEEK 1

Introduction to Numerical Methods Mathematical modeling

Approximation and round off errors

Truncation errors and Taylor Series 2

At the end of this topic, the students will be able:

To describe numerical techniques as

compared to analytical methods

To use Taylor series expansion to approximate

a function

To perform error analysis associated with

numerical methods

LESSON OUTCOMES

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Why Numerical Method?

Could handle large systems of equations, nonlinearity and complex geometries that is not

common

It provide approximate solutions to many of the engineering problems.

Powerful analysis tool in problem solving and understanding problem in mathematical language

Techniques by which mathematical problems are formulated, so that they can be solved with

arithmetic operations

The role of numerical method in solving engineering problem:

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What is Numerical Method

PROBLEM

FORMULATION

Fundamental laws are used to

develop mathematical

equations that can represent

the specific problem

SOLUTION

Suitable numerical methods

are then selected to solve

the mathematical equations

INTERPRETATION

The results obtained can

then be used to

predict/analyze/understand

the specific problem better

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Is the use of mathematics to

Describe real world phenomena

Investigate important questions about the observed world

Explain real world phenomena

Test ideas

Make predictions about the real world

The real world refers to Engineering Physics

Physiology Ecology

Wildlife management Chemistry

Economics Sports

Etc

Mathematical modeling

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A mathematical model is represented as a functional relationship of the form

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Dependent variable

Observed behaviour/state/phenomenon of a system

Characteristic that reflects behaviour or state of the system i.e. y, f(x), f(t)

Independent variable

Dimension that determine a system i.e. time, t , x

Parameter

Quantity that serves to relate to functions and variables Reflective of the systems properties or composition

Forcing functions

External influence that acts on system i.e. acceleration gravity, g

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Example

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Apply Newtons second law, F = ma

And also can write as

Fdt

xdm

or

Fdt

dvm

2

2

Eq. relates a linear position x (dependent variable) to the applied

force, F (forcing function) and the time, t (independent variable). The mass, m is the only parameter in the above model.

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Example of mathematical modeling

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Example

Assume that interested to predict the velocity of the falling parachutist with time

Use fundamental knowledge to find a mathematical equation correlates the velocity

to the various forces acting on the parachutist

Newtons 2nd law of Motion

the time rate change of momentum of a body is equal to the resulting force acting on

it.

The model is formulated as

F = ma

F=net force acting on the body (N)

m=mass of the object (kg)

a=its acceleration (m/s2)

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m

F

dt

dv

dt

dv

aon accelerati

resistanceair of force upward

gravity of force downward

UF

DF

UFDFF

cvUF

mgDF

vm

cg

m

cvmg

dt

dv

Model relates acceleration of falling

object to the forces acting on it,

(differential equation)

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Exact or analytical solution: it exactly satisfies the original equation

dependent variable

independent variable

forcing functions

parameter

t,s v,(m/s)

0 0.00

2 16.40

4 27.77

6 35.64

8 41.10

10 44.87

12 47.87

53.39

Analytical solution of the parachutist

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Unfortunately, there are many mathematical

models that cannot be solved exactly.

Numerical solution that approximates the exact solution

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)(1

)()1

(

1

)()1

(

it

ii

it

it

ii

it

it

vm

cg

tt

vv

tt

vv

t

v

dt

dv

vm

cg

m

cvmg

dt

dv

)( 1)()()( 1 iittt ttvm

cgvv

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The use of finite difference to approximate the

first derivative of v with respect to t

Approximate or

numerical solution

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t,s v,(m/s)

0 0.00

2 19.60

4 32.00

6 39.85

8 44.82

10 47.97

12 49.96

53.39

Numerical solution

)( 1)()()( 1 iittt ttvm

cgvv

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Comparison between the exact and

numerical solution

Approximation and Roundoff Errors

Significant figures

98

98.09

0.0098

Numbers to be used in confidence

2 significant figures



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Important of significance figures in numerical methods:

Numerical methods yield approximate results, therefore, need to develop criteria to specify the confident in

approximate result.

Although quantities such as , e, or 7 represent specific quantities, they cannot be expressed exactly by a limited

number of digits. Computers retain only a finite number of

significant figures.

= 3.141592653589793238462643

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Accuracy and Precision

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Inaccurate & imprecise

accurate & precise Inaccurate & precise

accurate & imprecise

Increasing accuracy In

cre

as

ing

pre

cis

ion

How closely a computed

or measured value agree

with the true value

How closely individual

computed or measured

values agree with each

other

Error definitions

True value

Approximation

value

Error

True value = Error + Approximation value

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Truncation errors ~ result

when approximations are

used to represent exact

mathematical procedures

Round-off errors ~ result

when numbers having a

limited significant figures

are used to represent

exact numbers

t designates true percent relative error

True value = error + approximation value

True error (Et)= true value approximation

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(1)

(2)

(3)

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Calculation of errors

True value of length of a bridge is 10,000 cm.

When you measure, the length recorded is

9,999 cm. Compute the true error and true

percentage relative error of the bridge.

Answer: 1 cm and 0.01%

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However, in actual situation, true value is rarely available. Therefore, need to estimate the true value approximation

In numerical method, iterative approach is used to compute answer, in which error is estimated as the

difference between previous and current approximations.

The signs of error can be negative or positive,

Absolute value of error, |a| need to be lower than prespecified percent tolerance, s

n is significant figures. 23

(4)

(5)

Error estimates for iterative method

Suppose that we have exponential function as,

Starting with the simplest version, ex=1, add terms to estimate e0.5. Compute true (t) and approximate error (a) after each term is added until |a| falls below s , conforming to 3 significant figures. Note that true value of e0.5

is 1.648721.

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Answer

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Round-off errors

ln 2 = 0.693 147 180 559 945 309 41...

A device only shows 8

significant numbers, so

round-off error is discrepancy

introduced by omission of

significant figures.

Round-off error for this case is

0.00000000055994530941

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Results when numbers having limited significant

figures are used to represent exact numbers

Other example of roundoff error

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Truncation errors and Taylor series

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Truncation errors

Truncation error is the discrepancy introduced by the fact that numerical methods may employ approximations to represent exact

mathematical operations and quantities.

Truncation error are errors resulted from using an approximation in place of an exact mathematical procedure.

The difference between the calculated value using exact mathematical equation and approximation mathematical equation.

Zero order

First order

Second

order

nth order

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Provides a means to predict a function value at one point in terms of the

function value and its derivative at another point

Taylor series

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(6)

(7) 1

)1(

)!1(

)(

n

n

n hn

fR

Taylor series by defining a step size h = xi+1 - xi

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Let say we truncated the Taylor series expansion after zero-order term to yield

Remainder term, Rn for zero order version

Let truncate the remainder itself,

This result is still inexact because neglected second and higher order terms.

)()( 1 ii xxff

...!3!2

''' 3

)(3

2)(

)(0 hf

hf

hfR iii

xx

x

hfRix )(0

'

Remainder term, Rn, accounts for all terms from (n+1) to infinity.

It also usually expressed as:

)( 1 nn hOR

1)1(

)!1(

)(

n

n

n hn

fR

Remainder for the Taylor series Expansion

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Alternative simplification that tranforms the approximation into an equivalence based on graphical insight

derivative mean-value theorem states that if a function f(x) and its derivative are continous over interval from xi to xi+1, there exist at least one point on the function that has a slope, designated by f(), parallel to line joining f (xi) and f(xi+1)

Thus,

So,

Zero order version

First order version

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Numerical differentiation

Forward finite divided difference

Backward finite divided difference

Centered finite divided difference

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Forward finite divided difference approximation of first derivative

Backward finite divided difference approximation of first derivative

Centered finite divided difference approximation of first derivative

Where, (14)

(15)

(16)

(17)

O(h)h

f)f'(x

h

R

h

)f(x)f(x)f'(x

ii

iii

111 ii xxhWhere,

11 iiii xxxxhWhere,

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Use forward and backward difference approximations of O(h) and a centered

difference approximation of O(h2) to estimate the first derivative of

2.125.05.015.01.0f 234 xxxx)(xi

at x = 0.5 using a step size h = 0.5. Repeat using h = 0.25. Also calculate the

true percent relative error for each approximation.

Example

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Exercise

Use forward and backward and a centered difference to estimate the first

derivative of the function

7.08.01.05.0f 23 xxx)(xi

at x = 0.5 using a step size h=0.5. Repeat using h = 0.25. Also calculate the

true percent relative error for each approximation.

Ans:

h=0.5

FDM:1.525 41.9%

BDM:0.875 18.60%

CDM:1.200 11.63%

h=0.25

FDM:1.26875 18.02%

BDM:0.94375 12.21%

CDM:1.10625 2.91%

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Second forward finite difference approximation of higher derivatives

Second backward finite difference approximation of higher derivatives

Second centred finite difference approximation of higher derivatives

Higher derivatives

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Error propagation This section is to study how errors in numbers can

propagate through mathematical functions. If we multiply

two numbers that have errors, we would like to estimate

the error in the product.

If a function f is dependent on

(a) a single independent variable x : f(x)

(b) two independent variables x and y : f(x, y)

(c) several independent variables x1, x2, x3, ... ,xn : f(x1 , x2 ,..., xn ).

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Function of a single variable.

Let x be the true value and

x* be an approximate value of x

Then, TSE for f(x) computed near f(x*) is given by

...*)(

2

*)(''*)(* 2 xx

xfxx)f'(xf(x*)f(x)

Truncating after the first derivative term and rearranging the remaining terms

to give

where

**)'

*)*)('

x(xff(x*)

xx(xff(x*)f(x)

xxxx

xfxf(xf

t variableindependen oferror theof estimatean is **

function theoferror theof estimatean is *)()(*)'

Eq.(21) provides 2 capabilities:

1. to approximate the error in f(x) knowing its derivative.

2. to approximate the error in the independent variable x.

(21)

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Example

Given a value of x* = 2.5 with an error of x* = 0.01, estimate the

resulting error in the function, f(x)=x3.

Solution

187506251552

atpredict thcan it ,625.15)5.2( Because

1875.0)01.0()5.2(3

So,

**)'

2

. .).f(

f

f(x*)

x(xff(x*)

Or the true value lies between 15.4375 and 15.8125. In fact, if x ~2.49,

f(x) could be 15.4382 and if x ~ 2.51, it would be 15.8132.

The first order error analysis provides a fairly close estimate of the true error.

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Exercise

Knowing a value of x* = 2.0 with an error of x* = 0.01, estimate the resulting error in the function

f(x) = 0.5x3-0.1x2+0.8x-0.7

Ans: f(2.0)=4.5 0.064

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Function of a more than One variable.

(22)

(23)

Refer section 4.2.2

for examples

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Relative error

Condition number

Refer section 4.2.3

for example

=

=

Condition no equals 1 indicates that functions relative error is identical to the relative error in

x

Condition no greater than 1 indicates relative error is amplified.

Condition no less than 1 indicates relative error is attenuated.

Function with very large values are said to be ill-conditioned.

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Total numerical errors = truncation error + round off error

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Roundoff error by increase no. of significant figures or

reduce no. of computation in analysis

Truncation error by decreasing step size (h) or increase

no. of computation in analysis

Total Numerical Error

Control numerical error

avoid subtract 2 nearly equal numbers to avoid loss of significance

Use Taylor series for truncation and roundoff error analysis

Perform numerical experiments

- repeat computation with different step size or method and compare results

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