1 / 24

Exact Computation

Exact Computation. Theory and Techniques. Serge Adamowsky & Lina Ourima Seminar Algorithmen WS 03 / 04. Contents. Evaluating One Determinant Expression Error-bounds Versus Precision-bounds Error-Bounds Precision-bounds Theory of Exact Computation. Evaluating One Determinant Expression.

vian
Download Presentation

Exact Computation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Exact Computation Theory and Techniques Serge Adamowsky & Lina Ourima Seminar Algorithmen WS 03 / 04

  2. Contents • Evaluating One Determinant Expression • Error-bounds Versus Precision-bounds • Error-Bounds • Precision-bounds • Theory of Exact Computation

  3. Evaluating One Determinant Expression • View Expression as labled dag (=directed acyclic graph) • Each sink node is a number • Each internal node is an operator

  4. Example

  5. Error-bounds Bottom-up Deterministic Depends on error of input Precision-bounds Top down Not deterministic User-specified quantity Error-bounds Versus Precision-bounds

  6. Example: Error-bounds

  7. Example: Precision-bounds

  8. Error Bounds • Number Representation (Notation) • Normalizing Error Digits • Examples for Normalization • Algorithms for Maintaining bigFloat Ranges • Addition • Multiplication • Square Root

  9. Number Representation (Notation) bigFloats used as aproximation b = 1212/343434  0.35 * 10-2 bigFloats used with error-bounds: b‘ = (0.35  0.01) * 10-2 = (35, -2, 1) (f, e, d) := f  d*Be The number is exact when d = 0

  10. Normalizing Error Digits • (35, -2, 1)2 = (1225, -4, 71) • Storing many insignificant digits is a waste of space and time Number is normalized when f doesnt start with 0 and 0  d < B Delayed normalization improves accuracy

  11. Examples for Normalization

  12. Algorithms for Maintaining bigFloat Ranges • Addition • Multiplication • Square root With a global precision bound (gpb) c‘ = a‘  b‘: the system produces the most accurate range without the exceeding global p. bound

  13. Addition A safe Algorithm: • Align the least significant bits • Add the digits and error values • Normalize (123, 0, 1) + (246, -2, 3) (12300, 0, 100) + (246, -2, 3) = (12546, 0, 103) Normalized: (125, 0, 2)

  14. Addition Problem: far more digits computed then nessesary Solution: Find the least significant bit (gpb, magnitude of both summands) and don‘t use smaller bits (123, 0, 1) + (2, -2, 1) = (125, 0, 2)

  15. Multiplication (f1, e1, d1) * (f2, e2, d2) = (f1 f2, e1+ e2, f1 d2+ f2 d1+ e1 e2) If f1, f2 positive f1 and f2 (both n digits) not exact => result doesnt have 2n but n significant digits Don‘t compute more digits then the significant ones (like in Addition)

  16. Square Root  Newton iteration: i+1½ii 1 ,2,... converges toward  i would be exact but  Might be not exact and operations can‘t be done accurate still  lies between i and i+1

  17. Composite Precision Bounds X approximates x • absolute precision(AP): |X-x|  2-a • relative precision(RP): |X-x|  |x| 2-r • composite precision(CP): |X-x|  max(2-a, |x| 2-r) written: X = x[a,r] CP  AP r =  CP  RP a = 

  18. Algorithms • Most significant bit (MSB) mx :=log|x| • mx[a,r] = MX 2Mx  |x| < 21+Mx + max{2e-r, 2a} • Computing Mx is expensive • Often a lower bound for Mx is sufficient

  19. Theory of Exact Computation • Computation of semi-algebraic problems is theoretically possible. • An algebraic number(AN) is any complex root of a polynomial (with integer coefficients) • 2 is AN, since it is root of x2-2 • Isolating interval representation: 2 = (x2-2, [1,2])

  20. Semi-algebraic Predicates (x1, ..., xn) is a Boolean combination of P(x1, ..., x) p 0 p  {=, <, >, , } Example: 0(x, y) : (x2 + y2 -1 < 0)  (x + y -5 = 0) Is the union of an open disc and a straight line

  21. Tarski Formula A first-order formula based on semi-algebraic predicates is a true Tarski sentence. Example: (x)(y)0(x,y) Sets defined by Tarski formulas are semi-algebraic.

  22. Practical Use? Theorem: A semi-algebraic problem can be solved in double-exponential time on a Turing-machine. • Mainly of theoretical interest • Practical use if some instances can be computed quickly (double exponential time is worst case)

  23. Theory of Root Bounds • Comparing values is a basic operation • It is sufficient to compare to zero  determine the sign of an expression • How can we know a number is 0 without looking at infinite signs and without using „some heuristic cut-off epsilon value“?

  24. Detecting Zero • h is a height bound on  • The height of an algebraic number is the maximum value of the coefficients in its minimal polynomial • Cauchy‘s bound says:  is non-zero iff || > 1 / (1+h) • Evaluate  to an AP of 1+log(1+h) to determine the sign

More Related