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| A solution exists, is unique, and depends continuously on the problem data |
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| In order to find an accurate solution, a problem must be well-posed and well-conditioned, and an algorithm must be stable. |
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| The difference between exact function values due to error in input |
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| Difference between exact and approximate function values for the same input. Composed of truncation and rounding error. |
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| Difference between true result and that produced by a given algorithm using exact arithmetic. |
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| Difference between the result produced by a given algorithm using exact arithmetic and result produced by same algorithm using finite-precision. |
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Discrepency between computed and true values:
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| The quantity [image] where [image] |
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| Conditioning of a problem |
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Effects of propagated data error on the solution of a problem
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| Stability of an algorithm |
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| The effects of computational error on the result computed by an algorithm |
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Definition
| Expressed as a number of moderate size times a power of ten. Base (usually two), Precision(53), Lower exponent (-1022) and upper exponent (1023). |
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Definition
Round to zero (chop): fl(x) is truncated after the (p-1)st digit, where p is precision
Round to nearest: fl(x) is the nearest floating-point number to x |
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Bounds the relative error in representing any nonzero real number x within the normalized range of a floating-point system
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| Floating-point arithmetic |
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Definition
Adding/Subtracting: Exponents must match, so shifted first. Trailing digits of smaller number are lost.
Multiplication: exponents are summed and mantissas are multiplied. Two p-digit mantissas contains up to 2p digits. |
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| Subtraction between two p-digit numbers having the same sign and similar magnitudes yields a result with fewer than p significant digits; exactly representable; potential serious loss of information |
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| Describes limiting behavior of a function when the argument tends towards a particular value or infinity. |
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