Jan 16, 2002
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- class taught by Prof. Cal Ribbens

- Relative error vs. absolute error
	- application: testing for floating point equality

- Introduction to representations
	- self study (sec 2.1)
	- conversion between bases

- Floating point representation
	- Motivation: handle extremes of scale and provide
	  relatively uniform distribution of exactly representabl
	  numbers ... all in a fixed number of bits

- IEEE (32-bit) standard
	- fl(x) = mantissa * 2^(exponent), where
		- mantissa = 1.____________ (23 bits)
		- exponent = 8 bit "excess-127" representation
		- and one sign bit

- Related terminology
	- machine epsilon: 2^-23
	  (smallest positive numbr x s.t. fl(1+x) > 1)
	- overflow
	- underflow
	- double precision


Jan 18, 2002
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- Roundoff error and loss of significance
	- occurs when a number cannot be exactly represented
	  in available number of bits
	- error in representation
	- Implication: for 32-bit representation, no number has more
	  than 6 or 7 significant digits

- Propagation of roundoff error
	- multiplication
	- addition
	- example of "cancelation error"

- Avoiding error propagation
	- see hints in Sec 2.3
	- classic example: quadratic formula

Jan 21, 2002
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- Naren returns
	- asks questions about what happened

- Review number representation
	- finite decimal can correspond to infinite binary
	  representation
	- but finite binary always corresponds to finite decimal
          (why?)

- Moral of the story
	- Even before you started to compute anything, there is error!

- What happens to numbers on the computer?
	- chopping (Procrustes analogy)
		- easy to study
	- rounding (an example from House apportionment 
	  in the US Congress)
	  	- more difficult to study

- Review IEEE 32-bit standard
	- relative error for rounding <= 2^-24 (eps/2)
	- relative error for chopping <= 2^-23 (eps)
	- eps is smallest positive number > 1 - 1
	- eps: unit roundoff error, machine precision, or just eps

- Multiplication and division are benign
	- not so addition with opposite signs

- In addition and subtraction
	- Obs 1: coefficients can get arbitrarily large
	- Obs 2: If x & y are same sign, coefficients of relative error
	         are positive and bounded by 1 (addition is benign)
	- Obs 3: If x & y are different signs, coefficients of relative
	         error can get arbitrarily large (compared to x and y)
			=> cancelation erorr
			   (the Achilles Heel of numerical methods)

- Moral of the story:
	- Avoid subtraction or addition with different signs

- Revisit Quadratic Equation

- Review Problems

- Story of the Pentium Affair