O metodě konečných prvkůLect_6.ppt

M. Okrouhlík

Ústav termomechaniky, AV ČR, PrahaLiberec, 2010

Pár slov o Matlabu ao zobrazení čísla na počítači

Recommended reading

Stejskal, V., Okrouhlík, M.: Kmitání s Matlabem, Vydavatelství ČVUT, Praha 2002, ISBN 80-01-02435-0

E:\edu_mkp_liberec_2\pdf_jpg_my_old_texts\KmiMat_240901_final\vibrace_1.pdf

downloaded from www.mathworks.com/moler

By C. Moler

Fortran programs to [1] can be downloaded from www.pdas.com/programs/fmm.f90

References to Moler’s book

E:\edu_mkp_liberec_2\pdf_jpg_my_old_texts\skripta_jaderna\aplik_mechanika_kontinua_1989.pdf

www.it.cas.cz/cs/elektronicka-kniha-numerical-methods-computation-mechanics

All computers designed from 1985 use so called IEEE floating point arithmetics which means that there is a machine independent standard of the of floating point number treatment. This means that the floating point numbers are expressed in the form, where is normalized integer mantisa represented by 52 bits and e is another integer within the interval

related to the number bits reserved for exponents representation. It is the finiteness of exponent which limits the interval of real numbers that can be represented by floating point numbers. The smallest floating-point number is

is the underflow limit and can be viewed as the computational threshold.

The maximum floating point number, pointing to the overflow limit, is

These two limits should be distinguished from another important quantity associated with representation of floating point numbers, namely a unity round-off error, also called machine epsilon, corresponding to the distance from 1.0 to the next larger floating point number. Its value is

and it is closely associated with the build up of roundoff errors. The number of decimal digits corresponding to 52 binary digits is approximately 16. It can be determined from , which gives .

efx 21 f

10231022 e

3081022 1085072012.225073852

3081023 10 48623161.797693132

-1652 10 92503132.220446042

x10252 745270215.6535597(10)log(2)/log*52 x

unit_roundoff = u, where 1 + u is different from 1

machine-epsilon = a – 1; where a is smallest representable number greater than 1

machine_epsilon = 2*u

http://www.physics.ohio-state.edu/~dws/grouplinks/floating_point_math.pdf

http://www.cs.berkeley.edu/~wkahan/Mindless.pdf

In Matlab: c = a*b;

Příklad

Užitečné procedurypro programování MKP na koleně,

a to pomocí Matlabu