ErrorAnalysisPartTheBasics.pptVIP

Error Analysis Part 1 The Basics Key Concepts Analytical vs. numerical Methods Representation of floating-point numbers Concept of significant digits Distinguishing different kinds of errors Round-off / chopping / truncation errors True/approximate absolute and relative errors Acceptable errors Analytical vs. Numerical Methods Find the intersection of y1 = 2x + 3 y2 = x + 2 Find the intersection of y1 = x y2 = cos(x) Analytical vs. Numerical Methods Analytical Methods Accurate solution Difficult and not always possible Numerical Methods Approximation of true solution What method to use? How good is our approximation? (Error Analysis) How efficient is our method? (Algorithm design, Convergence rate) Does our methods always work? (Convergence) Number Representation Do machines represent integers and floating-point numbers using the same representation? How does computer represent integers? How does computer represent floating-point numbers? Representation of Integers 13 as 8-bit unsigned integers (no negative #) 1310 = 000011012 = 0 x 27 + 1 x 26 + 0 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 = 8 + 4 + 0 + 1 Exercise What is the equivalent decimal number represented by the following binary number? 1101012 = ? Representation of Floating-point Numbers 156.78 = 0.15678 x 103 in an imaginary base-10 floating-point system Normalized Representation(and notations used in this course) σ is the sign β is the base, e is the exponent binary : β =2 decimal : β =10 1/β ≤ m 1 (i.e., a1 ≠ 0) binary: 0.5 ≤ m 1 decimal: 0.1 ≤ m 1 Representation of Floating-point Numbers Exercise What is the normalized floating-point representation of 12.7510 (for β = 2)? What is the normalized floating-point representation of 2.210 (for β = 2)? What is the equivalent decimal value of (0.110111)2 x 23 ? Implication of floating-point representations Only limited range of quantities may be represented Number too larger ? overflow Number too