# Numerical Matrix Analysis

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1 Numerical Matrix Analysis Lecture Notes #10 Conditioning and / Peter Blomgren, Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA Spring 2016 Peter Blomgren, / (1/23)

2 Outline 1 Finite Precision IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors 2 Theorem and Notation Fundamental Axiom of Example 3 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Peter Blomgren, / (2/23)

3 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors Finite Precision A 64-bit real number, double The Binary Standard (IEEE The Institute for Electrical and Electronics Engineers) standard specified the following layout for a 64-bit real number: Where sc 10 c 9... c 1 c 0 m 51 m m 1 m 0 Symbol Bits Description s 1 The sign bit 0=positive, 1=negative c 11 The characteristic (exponent) m 52 The mantissa r = ( 1) s 2 c 1023 (1+f), c = 10 k=0 c n 2 n, f = 51 k=0 m k 2 52 k Peter Blomgren, / (3/23)

4 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors IEEE Special Signals In order to be able to represent zero, ±, and NaN (not-a-number), the following special signals are defined in the IEEE standard: Type S (1 bit) C (11 bits) M (52 bits) signaling NaN u 2047 (max).0uuuuu u (*) quiet NaN u 2047 (max).1uuuuu u negative infinity (max) positive infinity (max) negative zero positive zero (*) with at least one 1 bit. From Peter Blomgren, / (4/23)

5 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors Examples: Finite Precision r = ( 1) s 2 c 1023 (1+f), c = Example #1 10 k=0 c n 2 n, f = 51 k=0 m k 2 52 k 0, , ( r 1 = ( 1) ) = = 3.0 Example #2 (The Smallest Positive Real Number) 0, , r 2 = ( 1) (1+2 52) Peter Blomgren, / (5/23)

6 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors Examples: Finite Precision r = ( 1) s 2 c 1023 (1+f), c = 10 k=0 c n 2 n, f = 51 k=0 m k 2 52 k Example #3 (The Largest Positive Real Number) 0, , ( r 3 = ( 1) ) 2 52 ( = ) Peter Blomgren, / (6/23)

7 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors That s Quite a Range! In summary, we can represent { ±0, ± , ± , ±, NaN } and a whole bunch of numbers in ( , ) ( , ) Bottom line: Over- or under-flowing is usually not a problem in IEEE floating point arithmetic. The problem in scientific computing is what we cannot represent. Peter Blomgren, / (7/23)

8 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors Fun with Matlab......Integers ( ) 2 53 = 2 ( ) ( ) = 2 ( ) 2 53 = (2 53 1) = 1 realmax = realmin = eps = The smallest not-exactly-representable integer is ( ) = 9,007,199,254,740,993. Peter Blomgren, / (8/23)

9 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors Something is Missing Gaps in the Representation 1 of 3 There are gaps in the floating-point representation! Given the representation 0, , for the value v 1 = ( ), the next larger floating-point value is 0, , i.e. the value v 2 = ( ) The difference between these two values is = ( ). Any number in the interval (v 1,v 2 ) is not representable! Peter Blomgren, / (9/23)

10 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors Something is Missing Gaps in the Representation 2 of 3 A gap of doesn t seem too bad... However, the size of the gap depend on the value itself... Consider r = 3.0 0, , and the next value 0, , Here, the difference is = 2 51 ( ). In general, in the interval [2 n,2 n+1 ] the gap is 2 n 52. Peter Blomgren, / (10/23)

11 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors Something is Missing Gaps in the Representation 3 of 3 At the other extreme, the difference between 0, , and the next value 0, , is = That s a fairly significant gap!!! (A number large enough to comfortably count all the particles in the universe...) See, e.g. mrob/pub/math/numbers-10.html Peter Blomgren, / (11/23)

12 IEEE Binary Floating Point (from Math 541) Non-representable Values a Source of Errors The Relative Gap It makes more sense to factor the exponent out of the discussion and talk about the relative gap: Exponent Gap Relative Gap (Gap/Exponent) Any difference between numbers smaller than the local gap is not representable, e.g. any number in the interval [ 3.0, ) 2 51 is represented by the value 3.0. Peter Blomgren, / (12/23)

13 Theorem and Notation Fundamental Axiom of Example The Floating Point Theorem ǫ mach Theorem Floating point numbers represent intervals! Notation: We let fl(x) denote the floating point representation of x R. The relative gap defines ǫ mach x R, there exists ǫ with ǫ ǫ mach, such that fl(x) = x(1+ǫ). In 64-bit floating point arithmetic ǫ mach In matlab, the command eps returns this value. Let the symbols,,, and denote the floating-point operations: addition, subtraction, multiplication, and division. Peter Blomgren, / (13/23)

14 Theorem and Notation Fundamental Axiom of Example ǫ mach All floating-point operations are performed up to some precision, i.e. x y = fl(x +y), x y = fl(x y), x y = fl(x y), x y = fl(x/y) This paired with our definition of ǫ mach gives us Axiom (The Fundamental Axiom of ) For all x,y F (where F is the set of floating point numbers), there exists ǫ with ǫ ǫ mach, such that x y = (x +y)(1+ǫ), x y = (x y)(1+ǫ), x y = (x y)(1+ǫ), x y = (x/y)(1+ǫ) That is every operation of floating point arithmetic is exact up to a relative error of size at most ǫ mach. Peter Blomgren, / (14/23)

15 Theorem and Notation Fundamental Axiom of Example Example: Floating Point Error Scaled by Consider the following polynomial on the interval [1.92, 2.08]: p(x) = (x 2) 9 = x 9 18x x 7 672x x x x x x p(x) = x 9 18x x 7 672x x x x x x 512 p(x) as above p(x) = (x 2) Peter Blomgren, / (15/23)

16 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Peter Blomgren, / (16/23)

17 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward : Introduction 1 of 3 With the knowledge that (floating point) errors happen, we have to re-define the concept of the right answer. Previously, in the context of conditioning we defined a mathematical problem as a map f : X Y where X C n is the set of data (input), and Y C m is the set of solutions. Peter Blomgren, / (17/23)

18 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward : Introduction 2 of 3 We now define an implementation of an algorithm on a floating-point device, where F satisfies the fundamental axiom of floating point arithmetic as another map f : X Y i.e. f( x) Y is a numerical solution of the problem. Wiki-History: Pentium FDIV bug ( 1994) The Pentium FDIV bug was a bug in Intel s original Pentium FPU. Certain FP division operations performed with these processors would produce incorrect results. According to Intel, there were a few missing entries in the lookup table used by the divide operation algorithm. Although encountering the flaw was extremely rare in practice (Byte Magazine estimated that 1 in 9 billion FP divides with random parameters would produce inaccurate results), both the flaw and Intel s initial handling of the matter were heavily criticized. Intel ultimately recalled the defective processors. Peter Blomgren, / (18/23)

19 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward : Introduction 3 of 3 The task at hand is to make useful statements about f( x). Even though f( x) is affected by many factors roundoff errors, convergence tolerances, competing processes on the computer, etc; we will be able to make (maybe surprisingly) clear statements about f( x). Note that depending on the memory model, the previous state of a memory location may affect the result in e.g. the case of cancellation errors: If we subtract two 16-digit numbers with 13 common leading digits, we are left with 3 digits of valid information. We tend to view the remaining 13 digits as random. But really, there is nothing random about what happens inside the computer (we hope!) the randomness will depend on what happened previously... Peter Blomgren, / (19/23)

20 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Accuracy The absolute error of a computation is and the relative error is f( x) f( x) f( x) f( x) f( x) this latter quantity will be our standard measure of error. If f is a good algorithm, we expect the relative error to be small, of the order ǫ mach. We say that f is accurate if x X f( x) f( x) f( x) = O(ǫ mach ) Peter Blomgren, / (20/23)

21 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Interpretation: O(ǫ mach ) Since all floating point errors are functions of ǫ mach (the relative error in each operation is bounded by ǫ mach ), the relative error of the algorithm must be a function of ǫ mach : f( x) f( x) f( x) = e(ǫ mach ) The statement e(ǫ mach ) = O(ǫ mach ) means that C R + such that e(ǫ mach ) Cǫ mach, as ǫ mach 0 In practice ǫ mach is fixed, and the notation means that if we were to decrease ǫ mach, then our error would decrease at least proportionally to ǫ mach. Peter Blomgren, / (21/23)

22 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward If the problem f : X Y is ill-conditioned, then the accuracy goal f( x) f( x) f( x) may be unreasonably ambitious. = O(ǫ mach ) Instead we aim for stability. We say that f is a stable algorithm if x X f( x) f( x) = O(ǫ mach ) f( x) for some x with x x x = O(ǫ mach ) A stable algorithm gives approximately the right answer, to approximately the right question. Peter Blomgren, / (22/23)

23 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Backward For many algorithms we can tighten this somewhat vague concept of stability. An algorithm f is backward stable if x X f( x) = f( x) for some x with x x x = O(ǫ mach ) A backward stable algorithm gives exactly the right answer, to approximately the right question. Next time: Examples of stable and unstable algorithms; of Householder triangularization. Peter Blomgren, / (23/23)

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