# Shared Flashcard Set

## Details

Numerical Computation
General Error Analysis, Finite-Precision Computation, Operation Counts
16
Computer Science
03/02/2014

Term
 Well-Posed Problems
Definition
 A solution exists, is unique, and depends continuously on the problem data
Term
 Problems vs. Algorithms
Definition
 In order to find an accurate solution, a problem must be well-posed and well-conditioned, and an algorithm must be stable.
Term
 Data Error
Definition
 The difference between exact function values due to error in input
Term
 Computational Error
Definition
 Difference between exact and approximate function values for the same input.  Composed of truncation and rounding error.
Term
 Truncation Error
Definition
 Difference between true result and that produced by a given algorithm using exact arithmetic.
Term
 Rounding error
Definition
 Difference between the result produced by a given algorithm using exact arithmetic and result produced by same algorithm using finite-precision.
Term
 Forward Error
Definition
 Discrepency between computed and true values:  [image]
Term
 Backward Error
Definition
 The quantity [image] where [image]
Term
 Conditioning of a problem
Definition
 Effects of propagated data error on the solution of a problem [image]
Term
 Stability of an algorithm
Definition
 The effects of computational error on the result computed by an algorithm
Term
 Floating-point numbers
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).
Term
 Rounding Rules
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
Term
 Machine Precision
Definition
 Bounds the relative error in representing any nonzero real number x within the normalized range of a floating-point system [image]
Term
 Floating-point arithmetic
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.
Term
 Cancellation
Definition
 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
Term
 Big O notation
Definition
 Describes limiting behavior of a function when the argument tends towards a particular value or infinity.
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