Lecture 7 - Numerical Errors

Question 1

Absolute error

Answer 1

Δx = x - x’
where:
x - accurate value
x’ - approximated one

Question 2

Relative error

Answer 2

𝛿x = (Δx)/x

often expressed in percentage => 𝛿x * 100%

Question 3

Floating points

Answer 3

Real numbers are stored in most modern digital machines in the so-called form floating point.
They form a subset of the R file, denoted as F
They are defined as:
x = M * N^W
M - mantissa
W - exponent
usually N = 2

The real number is thus represented by two groups of bits

Two important restrictions:

1. Represented numbers cannot be arbitrarily large/small
2. There is a space between numbers representation

Question 4

IEEE 754 Standard

Answer 4

SINGLE PRECISION

• > 32 bits
• > 23 mantissa
• > 8 exponent

DOUBLE PRECISION

• > 64 bits
• > 52 mantissa
• > 11 exponent
• > biggest number 1.79 * 10e308
• > smallest number 2.23 * 10e-308
• > the resolution of a subset F is represented by the machine epsilon Em =~ 2 ^ (-52)

Question 5

Arithmetic / Wilkinson lemma

Answer 5

All mathematical operation performed on a computer can be represented by elementary operation + - * /

However in floating-point arithmetic we use the F space in which analogous operations exists e.g.:
x +’ y = fl(x+y)

Wilkinson Lemma:
every elementary operation  in F: +' -' *' /' introduced given error
x +' y = x(1+e) + y(1+e)
x -' y = x(1+e) - y(1+e)
x *' y = xy(1+e)
x /' y = (x/y)(1+e) 
where e < Em 
Conclusion:
Each arithmetic operation in the space F is performed with an accuracy of relative error less or equal them Em -> This fact causes an ROUND-OFF ERRORS

Question 6

Input data errors

Answer 6

Data error- an error that is caused by the input data of numeric tasks.
Origins:
-> measurement errors
-> errors from the previous calculation phase
-> replacing the values of constant that are non-measurable numbers approximations

Even SMALL errors in the input data can result in LARGE relative error of the result

Question 7

Truncation errors

Answer 7

Difference between the true derivative of a function and its derivative obtained by numerical approximation

Examples:

• > approximation of the infinite sum of series with the sum of the finite number of its elements
• > approximation of the derivative with the finite differences
• > approximation of the integral with a finite sum