Chapter 1Scientific Computing

1. Computer Arithmetic (1.3)2. Approximation in Scientific Computing (1.2)

January 7

Floating-Point Number System (FPNS)

Mantissa Exponent

Fraction

Examples54

In base-10 system as

54 = (5 + 4/10) x 101 = 5.4 x 101

mantissa = 5.4, fraction = 0.4, exponent = 1

In base-2 system

54 = ( 0 + 1x21 + 1x 22 + 0x23 + 1x24 + 1x25)

= (0 + 2 + 4 + 0 + 16 + 32 )

= ( 1 + 1/ (21) + 0/(22) + 1/(23) + 1/(24) + 0/(25)) x 25

mantissa = 1.6875, fraction = 0.6875, exponent=5

Normalization

54 = ( 1 + 1/ (21) + 0/(22) + 1/(23) + 1/(24) + 0/(25)) x 25

= 1.6875 x 32 (normalized) = ( 0 + 1/(21) + 1/(22)+ 0/(23) + 1/(24) + 1/(25)+0/(26) ) x 26

(not normalized)

There are 126+127+1 = 254 possible exponent values

How to represent zero?

OFL = when all d0, …, dp-1 = beta - 1

Underflow level

Overflow level

What are the 25 numbers?

Online Demo at

http://www.cse.illinois.edu/iem/floating_point/rounding_rules/

Absolute and Relative Errors

Example

Approximate 43.552 with 4.3x10 has

absolute error = 0.552

relative error =approx= 0.01267

Floating-Point Arithmetics

Cancellation

Cancellation

For example: with base = 10, p =3.

Take x= 23115, y = 23090, there difference of 25 is comparatively much smaller than either x or y

(using chopping) what is the difference x-y in this FPNS?

Quadratic Formula

If the coefficients are too large or too small, overflow and underflow could occur.

Overflow can be avoided by scaling the coefficients.

Cancellation between –b and square root can be avoid by using

Example (pages 26-27)

Take ( base = 10, p=4)

a=0.05010, b=-98.78, c=5.015

The correct roots (to ten significant digits)

1971.605916, 0.05077069387

b2-4ac = 9756, its square-root is 98.77

The computed roots using standard formula

1972, 0.09980

Using the second formula

1003, 0.05077