Chapter 1 Scientific Computing 1.Computer Arithmetic (1.3) 2.Approximation in Scientific Computing (1.2) January 7 Floating-Point Number System (FPNS) Mantissa Exponent Fraction Examples 54 In base-10 system as 54 = (5 + 4/10) x 10 1 = 5.4 x 10 1 mantissa = 5.4, fraction = 0.4, exponent = 1 In base-2 system 54 = ( 0 + 1x x x x x2 5 ) = ( ) = ( 1 + 1/ (2 1 ) + 0/(2 2 ) + 1/(2 3 ) + 1/(2 4 ) + 0/(2 5 )) x 2 5 mantissa = , fraction = , exponent=5 Normalization 54 = ( 1 + 1/ (2 1 ) + 0/(2 2 ) + 1/(2 3 ) + 1/(2 4 ) + 0/(2 5 )) x 2 5 = x 32 (normalized) = ( 0 + 1/(2 1 ) + 1/(2 2 )+ 0/(2 3 ) + 1/(2 4 ) + 1/(2 5 )+0/(2 6 ) ) x 2 6 (not normalized) There are = 254 possible exponent values How to represent zero? OFL = when all d 0, , d p-1 = beta - 1 Underflow level Overflow level What are the 25 numbers? Online Demo at Absolute and Relative Errors Example Approximate with 4.3x10 has absolute error = relative error =approx= Floating-Point Arithmetics 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= , b=-98.78, c=5.015 The correct roots (to ten significant digits) , b 2 -4ac = 9756, its square-root is The computed roots using standard formula 1972, Using the second formula 1003,