Floating Point System
Floating Point System
Objective
 | Understand and handle error from floating point calculation |
| Role of stability in numerical calculation |
Vocabulary
Base, Sign bit, Exponent, Mantissa, underflow, overflow, rounding, chopping, roundoff error, absolute error, relative error, significant digits, truncation error, machine epsilon.
Concepts
| Distribution of floating point numbers |
| Addition of floating point numbers |
| Loss of associativity and distributivity |
| Catastrophic cancellation |
| Prevention of catastrophic cancellation |
| Forward error analysis |
| Backward error analysis |
Extra
: | IEEE standard |
| Subnormal numbers, guard bit |
Some External Links:
Floating Points, An excellent general introduction by Cleve Moler on the floating points and IEEE Standard.
Basic Issues in Floating Point Arithmetic and Error Analysis, Supplementary Lecture Notes of J. Demmel on Floating Point System, University of California at Berkeley, September 1995.
Miscalculating Area and Angles of a Needle-like Triangle, Notes of W. Kahan on common misconceptions of floating point calculations, University of California at Berkeley, September 1997.
IEEE Floating Point Arithmetic, More advanced Lecture Notes of W. Kahan on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic, University of California at Berkeley, September 1995.
Floating-Point Number Tutorial, Java applets are used to visualize the significance of mantissa size and exponent range and the meaning of underflow, overflow, and roundoff error.
References
Goldberg, David, What every computer scientist should know about floating-point arithmetic, ACM Computing Surveys, Vol.23, No. 1 (March 1991), pp. 5-48 (Electronic version not available.)
Correction to Misconceptions
| The machine epsilon is not the smallest machine representable number. |
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