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Introduction To Numerical Analysis For Engineers: - Fundamentals of Digital Computing

This document provides an introduction to numerical analysis concepts for engineers, including: - Digital computer models and number representation - Error analysis techniques such as error propagation, estimation, and cancellation - Numerical stability and condition numbers - Examples of applying these concepts to solving differential equations and propagating errors in numerical solutions.

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AryanFrogh
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109 views12 pages

Introduction To Numerical Analysis For Engineers: - Fundamentals of Digital Computing

This document provides an introduction to numerical analysis concepts for engineers, including: - Digital computer models and number representation - Error analysis techniques such as error propagation, estimation, and cancellation - Numerical stability and condition numbers - Examples of applying these concepts to solving differential equations and propagating errors in numerical solutions.

Uploaded by

AryanFrogh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Introduction to Numerical Analysis for

Engineers
Fundamentals of Digital Computing

Digital Computer Models


Convergence, accuracy and stability
Number representation
Arithmetic operations
Recursion algorithms

Error propagation numerical stability


Error estimation
Error cancellation
Condition numbers

Error Analysis

13.002

Numerical Methods for Engineers

Lecture 2

Floating Number Representation

Examples

m
b
e

Mantissa
Base
Exponent

Decimal
Binary
Convention
Decimal
Binary

Max mantissa
Min mantissa
Max exponent

General
13.002

Min exponent
Numerical Methods for Engineers

Lecture 2

Error Analysis
Number Representation

Addition and Subtraction

Absolute Error
Shift mantissa of largest number
Relative Error

Result has exponent of largest number


Absolute Error
Relative Error
Unbounded

Multiplication and Division

Relative Error
13.002

Numerical Methods for Engineers

Bounded
Lecture 2

Error Propagation
Spherical Bessel Functions
Differential Equation

Solutions

13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation
Spherical Bessel Functions
Forward Recurrence

Forward Recurrence
Unstable

Backward Recurrence

Millers algorithm
Stable

N ~ x+20
13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation
Differential Equation

Eulers Method

Example

Discretization
Finite Difference (forward)

Recurrence
euler.m

Central Finite Difference

13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation

Absolute Errors
'y~f (x)'x

Function of one variable

'x = x - x

General Error Propagation Formula


x

13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation
Example
Error Propagation Formula

Multiplication

=>
=>

=>

=>

Relative Errors Add for Multiplication

13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation
Expectation of Errors
Addition
Standard Error

Truncation

Error Expectation

Rounding

Standard Error better measure


of expected errors
13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation
Error Cancellation
Function of one variable

Max. error

Error cancellation

Stand. error

13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation
Condition Number
y = f(x)

x = x(1 + D)

y = y(1 + E)

Problem Condition Number


Problem ill-conditioned
Error cancellation example

Well-conditioned problem
13.002

Numerical Methods for Engineers

Lecture 2

Error Propagation
Condition Number
Problem Condition Number

K A is algorith condition number, which


may be much larger than the K P due
to limited number representation.
Solution
Higher precision
Rewrite algorithm

4 Significant Digits

Well-conditioned Algorithm

Algorithm Condition Number


13.002

Numerical Methods for Engineers

Lecture 2

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