Chapter 3. Errors and numerical stability
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1 Chapter 3 Errors and numerical stability 1 Representation of numbers Binary system : micro-transistor in state off 0 on 1 Smallest amount of stored data bit Object in memory chain of 1 and Byte 8 bits
2 On most micro-computers and workstations : word 4 bytes 32 bits double precision word 8 bytes 64 bits operations in double precision On certain super-computers (Cray) : word 8 bytes 64 bits double precision on Cray quadruple precision on a PC or workstation!
3 Integers An integer is stored in a word 4 bytes 32 bits In n-bit systems : i = s n 1 2 n 1 + s n 2 2 n s s s s k = 0 or 1 [s n 1, s n 2... s 2, s 1, s 0 ] stored in memory Largest representable integer : 2 n 1 In a 32-bit system : = If n > overflow
4 Signed integers What about negative integers? Binary arithmetic on words of fixed length arithmetic on a finite cyclic group Add 1 to the largest representable integer : = First 1 lost 0 In a n-bit system : addition is defined modulo 2 n
5 Binary representation of a negative integer? Inverse each bit (0 1 and 1 0), then add 1 Two s complement representation Examples : in a 3-bit system Integer 101 Add Add = 0 Opposite of = 011
6 Examples : in an 8-bit system -2 : : : : : Conclusions The first bit indicates the sign of the integer : 1. if first bit 0 : integer > 0 (0 i 2 n 1 1) 2. if the first bit is 1 : integer <0 ( 2 n 1 i < 0)
7 Real numbers Floating point representation Scientific notation base (10) 2. exponent (24) 3. mantissa (0.6022)
8 In computers, base 2 only exponent and mantissa are stored x = s 2 e s sign (0 for positive, 1 for negative) e exponent (integer) f 1 = 1 for x 0 f 1 = 0 for x = 0 f k = 0 or 1 for k > 1 p f k 2 k Single precision real one word = 32 bits k=1 sign exponent mantissa
9 8-bit exponent 128 e 127 (2 7 = 128) 23-bit mantissa 1 2 M 1 ( 1 2) Upper limit : 23 M = = 2 k = k=1 ( ( ) 23 ) = 1 ( ) Largest representable real : ( 1 Smallest non-zero representable real ( ) )
10 In single precision, accuracy on real numbers 6 digits (1/2) In double precision, real number 2 words = 64 bits 11-bit exponent 52-bit mantissa x = Accuracy 15 digits N.B. : Number of bits in exponent range Number of bits in mantissa precision
11 2 Consequences 1. Every number is not necessarily representable Rounding error Example : (0.1) 10 = ( ) 2 2. ɛ > 0 smallest representable number If 0 < x < ɛ x replaced by 0 Underflow Example : If a > 2 ɛ a 1 < 2 ɛ = 0 a 1 a = 0
12 3. Non-homogeneous distribution of real numbers Higher density near 0 Example : 16-bit system with 7-bit mantissa, 8-bit exponent Distance between 2 successive values of mantissa ( ) 2 = 2 7 (0.0078) 10 Small exponent (e.g. -100) distance between 2 successive reals in floating point representation = Large exponent (e.g. +100) distance between 2 successive reals in floating point representation
13 3 Numerical errors Arithmetic with a finite number of digits x approximation of x Absolute error ɛ a = x x Relative error ɛ r = ɛ a /x = x/x 1 Possible errors Example : Distributivity and associativity do not necessarily hold!!
14 Examples associativity of multiplication in 2-digit representation : ( ) 0.54 = = ( ) = = 0.20 associativity of addition in 6-digit representation : ( ) = = = ( ) = = distributivity in 6-digit representation : ( )/ = / / = =
15 3 classes of errors 1. Initial error 2. Truncation error 3. Rounding error Examples of truncation error Calculate e x using e x = 1 + x + x2 2! + x3 3! + x4 4! + O(x5 ) O(x n ) number of the same order as x n Calculate defined integral using quadrature rule b a dxf(x) n w i f(x i ) i=1
16 A numerical calculation comprises many steps! Examples : Suppose x = x + δ, ȳ = y + η z = x y z = ( x ȳ) = x + δ (y + η) + ɛ ɛ rounding error z = z + δ η + ɛ Depending on the sign of δ, η, ɛ, the error can be large or small. If x et y are almost equal, z is almost zero and the error is large z = x/y z = ( x/ȳ) = x + δ y + η + ɛ z z + 1 y δ x y 2 η + ɛ Error on z includes errors from x and y rounding error ɛ Si y is small, the error is large
17 It is important to understand how errors can propagate in a calculation 1. Small variations in initial data can give rise to big differences in final results ill-conditioned problem Examples : weather forecast, butterfly effect 2. Truncation errors usually depend on a parameter N if N : calculated solution exact solution Reduce truncation error by choosing N larger, might not be practical 3. Rounding errors accumulate randomly, and often cancel each other In certain cases, they can increase rapidly instability
18 Examples : Calculate e 1/3 in a 4-digit representation ( exact value = ) initial error = Propagated error e e 1/3 = e (1 e ) Calculate e x using expansion e x = 1 + x + x2 2! + x3 3! + x4 4! + O(x5 ) with x = If O(x 5 ) terms are neglected, truncation error ( ) ! 6! Summing truncated quantities = In a 10-digit representation : result =
19 Important source of rounding errors : Cancellation error in particular when two very close numbers are subtracted Example : In a 3-digit representation = = Exact value :
20 Example : Solve x 2 178x + 2 = 0 x ± = b ± b 2 4ac 2a b 2 >> 4ac x implies the subtraction of two very close numbers x + = x = Compare the number of significant digits! Improve accuracy by using x + x = c/a : x = 2 =
21 Example : Recurrence relations Particularly sensitive to propagations of initial and rounding errors Recurrence relation of Bessel function J n (x) J n+1 (x) = 2n x J n(x) J n 1 (x) If n > x, 2n/x > 1 multiplies errors in J n huge loss of accuracy For example, in a 6-digit representation J 0 (1) = , J 1 (1) = J 7 (1) = instead of !! Set J 8 (1) = 0, J 7 (1) = k, use backward recurrence relation and renormalise result using J 0 (x) + 2J 2 (x) + 2J 4 (x) +... = 1
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