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Chapter 3

Computational Errors

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Numerical "bugs"

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Numerical Errors

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Uncertainty: well- or ill-conditioned?

Errors in input data can cause uncertain results input

data can be experimental or rounded. leads to a certain

variation in the results

well-conditioned: numerical results are insensitive tosmall variations in the input

ill-conditioned: small variations lead to drastically

different numerical calculations (a.k.a. poorly

conditioned)

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Our Job

As numerical analysts, we need to

solve a problem so that the calculation is not susceptible to large

roundoff error

solve a problem so that the approximation has a tolerabletruncation error

How?

incorporate roundoff-truncation knowledge into

the mathematical model the method

the algorithm

the software design

awareness correct interpretation of results

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Intel

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Intel

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Intel

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Intel Timeline

June 1994: Intel engineers discover the division error. Managers

decide the error will not impact many people. Keep the issue

internal.

June 1994: Dr Nicely at Lynchburg College notices computation

problems

Oct 19, 1994: After months of testing, Nicely confirms that other

errors are not the cause. The problem is in the Intel Processor.

Oct 24, 1994: Nicely contacts Intel. Intel duplicates error.

Oct 30, 1994: After no action from Intel, Nicely sends an email

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Intel Timeline

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Intel Timeline

Nov 1, 1994: Software company Phar Lap Software receives Nicelys email.

Sends to collegues at Microsoft, Borland, Watcom, etc. decide the error will

not impact many people. Keep the issue internal.

Nov 2, 1994: Email with description goes global.

Nov 15, 1994: USC reverse-engineers the chip to expose the problem. Intellstill denies a problem. Stock falls.

Nov 22, 1994: CNN Moneyline interviews Intel. Says the problem is minor.

Nov 23, 1994: The MathWorks develops a fix.

Nov 24, 1994: New York Times story. Intel still sending out flawed chips. Will

replace chips only if it caused a problem in an important application. Dec 12, 1994: IBM halts shipment of Pentium based PCs

Dec 16, 1994: Intel stock falls again.

Dec 19, 1994: More reports in the NYT: lawsuits, etc.

Dec 20, 1994: Intel admits. Sets aside $420 million to fix.

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Pentium Bug

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Digital Representation

Can only use a finite number of digits to represent a number

+ reduces cost

+ reduces memory

- errors bit: binary (0 or 1)

byte: 8 bits

word: smallest addressable unit of memory for a certain

computer

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Integers

most computers: 16 or 32 bits for integers

leads to 216 1 = 65535 integers

spanned over -32768 to 32767

not many problems here

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Integer Algorithm

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Integers

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Floating Point

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Floating Point Algorithm

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Floating Point Representation

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Floating Point

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Overflow/Underflow

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Floating Point Number Line

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A problem with precision

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Finite Precision

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Finite Precision

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Solution

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Round-off

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On Symbolic versus Numerical

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So What?

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Floating point Arithmetic

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Where is the rounding error?

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Example: Quadratic Formula

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Example: Quadratic Formula

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A fix?

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Real (obscure) fix

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Catastrophic Cancellation

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Catastrophic Cancellation

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Summary

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How large is the roundoff error?

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To compute

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Practical Considerations

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How close is float x to float y?

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Why Relative?

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Relative vs. Absolute

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Truncation Error

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Taylor Series revisited

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Taylor Series revisited

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Big "O"

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Example

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Example

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Any Question??