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Chapter 1 Scientific Computing Computer Arithmetic (1.3)

Chapter 1 Scientific Computing Computer Arithmetic (1.3) 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

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Chapter 1 Scientific Computing Computer Arithmetic (1.3)

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  1. Chapter 1 • Scientific Computing • Computer Arithmetic (1.3) • Approximation in Scientific Computing (1.2) • January 7

  2. Floating-Point Number System (FPNS) Mantissa Exponent Fraction

  3. Examples 54 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

  4. 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)

  5. There are 126+127+1 = 254 possible exponent values How to represent zero?

  6. Underflow level Overflow level OFL = when all d0, …, dp-1 = beta - 1

  7. What are the 25 numbers?

  8. Online Demo at http://www.cse.illinois.edu/iem/floating_point/rounding_rules/

  9. Absolute and Relative Errors Example Approximate 43.552 with 4.3x10 has absolute error = 0.552 relative error =approx= 0.01267

  10. Floating-Point Arithmetics

  11. Cancellation

  12. Cancellation

  13. 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?

  14. 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

  15. 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

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