l1 - error analysis

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ESO 208A/ 218Computational Methods in Engineering

Summer Semester 2014-15 I

L1 – Error Analysis

Dr. Anubha Goel

[email protected]; x 7027



Mathematical model

A mathematical model can be broadly defined

as a formulation or equation that expresses

the essential features of a physical system or

process in mathematical terms.

5/20/2015 2L1 - Error Analysis



Numerical Methods

• Numerical methods are scientific in the sense that they

represent systematic techniques for solving mathematical

problems.

IMPORTANT – They are approximations and hence involve errors

• Out of several methods available choice of numerical

methods for solving a problem reduces to one of cost andaccuracy.

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Accuracy and Precision

Definitions

• Accuracy refers to how

closely a computed or

measured value agrees with

the true value.

• Precision refers to how

closely individual computed

or measured values agreewith each other.


An example from marksmanship illustrating the

concepts of accuracy and precision. (a) Inaccurate

and imprecise; (b) accurate and imprecise; (c)

inaccurate and precise; (d) accurate and precise.



Lecture contents

1. Numerical Methods and Numerical AnalysisWhen they are methods of choice, objective

3. Approximation4. Concept of convergence

Cauchy sequence

4. Error Type- Model and Data Error

- Round Off and Truncation Error – Taylor, Mc Laurin series5. Error Analysis

– Forward, Backward

– Error propagation in a function

– Condition number of a problem – Cp

– Norms of a vector

– Round off error, Machine precision unit (u)

– Condition number of an algorithm – Ca

– Floating Point error

– Total Relative error




Significant Figures

Basic rules for significant digits:-

1) All nonzero digits are significant.

2) All zeroes between significantdigits are significant.

3) All zeroes which are both(a) to the right of the decimal point

and,

(b) to the right of all non-zero

significant digits are themselvessignificant.


Example:

Number Significant

Digit

1.234 4

0.001 1

0.100 3

0.00100 3100 1

100.0 4

1.23 3

1.230 4



1. Concept of Numerical Methods and

Numerical Analysis

• Numerical method is an APPROXIMATION

→ Error exists

•

Define Numerical Method – The branch of mathematics that deals with transformation

of the mathematical problems to numerical calculations

• Need for Computation/ Numerical Methods in Engineering – Handling large datasets, repeated calculations,

– Examine result of variation in parameters




The solutions often quick and is easily adapted for parametric

sensitivity studies.

• Objective

To obtain a reasonably accurate solution with optimum use of

resources




Numerical Analysis

• Mathematical analysis conducted to determine whether thecomputational scheme converges or not is called numericalanalysis.

• Involves

– convergence analysis and – Error Analysis


• Problem - among the options available to solve a

problem, what is a numerical method?

• Answer - The computational scheme that converges to

the true solution can be termed as a numerical

method for computing the square root of a number.



Situations when numerical methods would be

the method of choice

1. integration of a function for which either the integral cannot

be expressed as analytical expressions or it is too

cumbersome and time consuming to evaluate,

2. solution of differential equations for complicated geometry,

and/or boundary conditions,

3. Large systems of equations, repeated solution of the same

system under changing conditions, etc.


Need to know how the numerical methods work on

computers to trouble shoot or design new algorithms



Concept of convergence – Cauchy sequence

• Function CONVERGES → Limit exists

• Convergence

– monotonically decreasing sequence,

–

lower bound, – Cauchy sequence




Error

Error = True value – approximation

3 basic kinds1. Data and Model Error

2. Round off/ Chop off

3. Truncation




• Model and Data Error

Description Related to

Model error due to assumptions made

in arriving at an algorithm

describing a process

Precision

refers to how closely individual

computed or measured values

agree with each other.

Data error due to approximation of

values such as value of g

as 9.81 (may actually be

9.8088)

accuracy

refers to how closely a

computed or measured value

agrees with the true value.


Ex. Data error : in case of function y = 5 ekx

Different values of k can lead to error in values of y.



• Truncation and Round off error

Error = True value – approximation

Two types – truncation and round off

Error Type Description

Truncation

Result when approximations are used to

represent exact mathematical procedures

Round-off Result when numbers having limited significant

figures are used to represent exact numbers.

E.g. value of ‘g’ is used as 9.81 (may actuallybe 9.8088)


Question: Chopping vs. Round off ?



Note on value of ‘g’

• Although the precise strength of Earth's gravity variesdepending on location, the nominal "average" value atthe Earth's surface, known as standard gravity is, bydefinition, 9.80665 m/s2

• Parameters affecting the apparent or actual strength ofEarth's gravity include latitude, altitude, and the localtopography and geology.

• Gravity decreases with altitude, since greater altitudemeans greater distance from the Earth's centre




What causes Truncation error (TE)?

This error is due to approximation of an infinite series by afinite number of terms or in other words, due to

truncation of the series after some points. Error

encountered in this way is termed as truncation error.

• e.g. Taylor Series

• Mc Laurin series (Taylor series with a = 0)




Truncation Error in Series

• Taylor Series

• Mc Laurin series (Taylor series with a = 0)

For more details on these series – revise MTH 101, Thomas and Finney




Round Off error, Machine precision unit u

• Concept of significant figures and link to round off error

• The concept of significant figures has two important

implications for our study of numerical methods:

1. numerical methods yield approximate results.

• We must, therefore, develop criteria to specify how confident

we are in our approximate result.

•

One way to do this is in terms of significant figures.




For example, we might decide that our approximation is

acceptable if it is correct to four significant figures.

2. All computing machines have finite maximum length for

storing a number, they are always rounded-off after a few

significant digits following decimal.

• This brings us to Floating Point representation. Lets revise.




Floating Point representation of a number

• Fractional quantities are typically represented in computers

using floating-point form.

• In this approach, the number is expressed as a fractional part,

called a mantissa or significant. and an integer part, called an

exponent or characteristic, as in

m . be

• where m = the mantissa, b = the base of the number system

being used, and e = the exponent.

• b=2 – binary, 10 – decimal, 16 - hexadecimal




• For instance, the number 156.78 could be represented as

0.15678 x 103 in a floating point base-I0 system.

Mantissa normalization

• Mantissa is usually normalized if it has leading zero digits.• The consequence of normalization is that the absolute value

of m is limited. That is,

I/b < = m < 1

• where b = the base.




Example

• for a base-l0 system, m would range between 0. I and I,

• for a base-2 system, between 0.5 and I.

•

Important: Normalization introduces a source of error – why? because the mantissa holds only a finite number of

significant figures.

– Thus, a round-off error is introduced.

Self Study:• Derivation of machine precision unit (u)

• Calculation of Backward error




Error Analysis

CONTENT

• Forward, Backward Error Analysis

• Error propagation in a function

•

Condition number of a problem – Cp• Condition number of an algorithm – Ca

• Total Error




Forward, Backward Error Analysis

Given error at the beginning (∆x), determine how much error

will be caused in final result (∆y).

– More relevant Data Error

Backward error analysis:

• Given an error incurred, determine what it is equivalent to

in parameter values

• OR process of estimating perturbation (∆) required in the

input variable (x) to explain the final result




Backward Error Analysis

• In short: you know – final error (∆y)

• You predict – what change/adjustment is needed to

cause/ remove the final error

• Advantage of Backward error analysis over Forward Error

we do not need to know the true value of the result, which

was needed in a forward error analysis




Error propagation in a function

Question: How the error in input data is translated to output

(even when no other error type such as truncation/ round off)

is encountered?

Answer:

• We know that Error in y is a function of data itself.

• Hence, objective is estimating the error introduced in the final

computation of the functional value as a result of these errors

in the variables (due to errors discussed above – data, model,

truncation, round off)

26

Error Bounds:In general, error is estimated using - Linear forward analysis

• Retain only the first order approximation term

Question: Why only first order?

Answer - Generally value ∆x is small and higher order analysis makes iteven smaller



True, and Relative Error• True error (e) = True value - Approximate value

• True Relative error

Why do we calculate Relative error?

Answer:

• To give a better idea of size of error vs. value concerned.

• Sometimes the error may be large, but only say 1% of True value.

Example:

TV = 1000, Calculated 950.

• Absolute error = 50, Relative error = 50/1000 = 0.05)

• Note: It is always better to use Relative Error than True Error

When TV is not known and needs to be approximated by a sequence:




Error Estimates for Iterative Methods

Approximate Error

Approximate error (e) = Current approximation - Previous

approximation

• Approximate relative error


Tutorial 1 problem: Ex. 3.2 Chapra Ed. 6



Condition number of a problem (Cp)

• Definition: ratio of the relative change in the function f(x), to

the relative change in the variable x for a small perturbation

dx,

• Ill conditioned vs. well conditioned

• derive values of Cp for each

• How can each type of problem be handled p

– USE higher precision

•

– i.e. reduce truncation error (include higher order terms)• Use different algorithm




Condition number of an algorithm (Ca)

• Definition: A measure of the change required in the input data toexplain the errors in the output result is termed the conditionnumber of the algorithm (C4).

How can this be obtained?

• Answer: This is done through a backward error analysis, whichdetermines the change necessary in the input variable (or data) toexplain the final error in the output (or result) obtained througherrors introduced at various stages of calculation.

• Importance/ Usefulness: among several algorithms available for aproblem, use the one with lowest value of Ca.

• Actual error in a computation –

• combines Cp and Ca




Total Numerical Error


A graphical depiction of the trade-off between round-off and truncation error that

sometimes comes into play in the course of a numerical method. The point of diminishing

returns is shown, where round-off error begins to negate the benefits of step-size

reduction.




Error Analysis of Numerical Differentiation

l1 - error analysis

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