Computer Arithmetic. MATH 375 Numerical Analysis. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Computer Arithmetic

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1 Computer Arithmetic MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Fall 2013

2 Machine Numbers When performing arithmetic on a computer (laptop, desktop, mainframe, cell phone, etc.) we will primarily work with two types of numbers: Integers: whole numbers in a specified range. Floating point: approximations to real numbers.

3 2 s Complement Integers Suppose an integer is represented using n bits (e.g., n = 32) bits in 2 s complement format. Index the bits from right to left. Bit (2 31 ) If the ith bit is set, the quantity in the ith column is added.

4 Example (1 of 2) Sign x = = = 35

5 Example (2 of 2) Sign x = (2 31 ) = =

6 Range of Integers 1 What are the smallest and largest integers which can be represented in 32-bit 2 s complement format?

7 Range of Integers 1 What are the smallest and largest integers which can be represented in 32-bit 2 s complement format? min 32 = (2 31 ) = max 32 = 30 i=0 2 i =

8 Range of Integers 1 What are the smallest and largest integers which can be represented in 32-bit 2 s complement format? min 32 = (2 31 ) = max 32 = 30 i=0 2 i = What are the smallest and largest integers which can be represented in 64-bit 2 s complement format?

9 Range of Integers 1 What are the smallest and largest integers which can be represented in 32-bit 2 s complement format? min 32 = (2 31 ) = max 32 = 30 i=0 2 i = What are the smallest and largest integers which can be represented in 64-bit 2 s complement format? min 64 = (2 63 ) = max 64 = 62 i=0 2 i =

10 Floating Point Numbers Consider representing a real number like π in some binary format.

11 Floating Point Numbers Consider representing a real number like π in some binary format. π is transcendental (non-repeating, non-terminating decimal number)

12 Floating Point Numbers Consider representing a real number like π in some binary format. π is transcendental (non-repeating, non-terminating decimal number) In any finite number of binary digits, π can only be approximated by some rational number. There will be round-off error.

13 Floating Point Numbers Consider representing a real number like π in some binary format. π is transcendental (non-repeating, non-terminating decimal number) In any finite number of binary digits, π can only be approximated by some rational number. There will be round-off error. Round-off error will be present when representing any real number which is not a power of 2.

14 Binary Floating Point Format The Institute for Electrical and Electronic Engineers (IEEE) published the Binary Floating Point Arithmetic Standard which specified storage and transmission formats for floating point numbers and algorithms for rounding arithmetic operations. Consider the 64-bit representation. s: sign bit c: characteristic, 11-bit exponent with base 2, according to IEEE , 1 c 2046 always f : mantissa, 52-bit binary fraction x = ( 1) s 2 c 1023 (1 + f )

15 Example s c f This number is positive since ( 1) 0 = 1. c = = = 1029 [ ] 1 1 [ ] 1 4 [ ] 1 5 [ ] 1 6 [ ] 1 7 f = ( x = ( 1) ) = = [ ] 1 9 =

16 Interval of Real Numbers The previous example actually represents an interval of numbers. Consider the next smaller and next larger floating point numbers. s c f x s = x = x l =

17 Interval of Real Numbers The previous example actually represents an interval of numbers. Consider the next smaller and next larger floating point numbers. s c f x s = x = x l = Upon rounding x represents all real numbers in the interval [ , ).

18 Floating Point Limits Smallest, non-zero positive floating point number: Largest floating point number:

19 Floating Point Limits Smallest, non-zero positive floating point number: ɛ = ( 1) (1 + 0) Largest floating point number:

20 Floating Point Limits Smallest, non-zero positive floating point number: ɛ = ( 1) (1 + 0) Largest floating point number: = ( 1) ( )

21 Decimal (Base-10) Floating Point Numbers We will express floating point numbers in base-10 form for simplicity. If x is a non-zero real number, then x can be represented as ±0.d 1 d 2... d k 1 d k d k n where 1 d 1 9 and 0 d k 9 for k > 1.

22 Decimal (Base-10) Floating Point Numbers We will express floating point numbers in base-10 form for simplicity. If x is a non-zero real number, then x can be represented as ±0.d 1 d 2... d k 1 d k d k n where 1 d 1 9 and 0 d k 9 for k > 1. The k-digit decimal machine number corresponding to x will be denoted fl (x) and is determined by rounding. To round we may perform chopping by ignoring all the decimal digits beyond the k th, fl (0.d 1 d 2... d k 1 d k d k n ) = 0.d 1 d 2... d k 1 d k 10 n or we may perform rounding in the kth decimal place. fl (0.d 1 d 2... d k 1 d k d k n ) = 0.δ 1 δ 2... δ k 1 δ k 10 n

23 Example Example Determine the 5-digit representations for e using 1 chopping, 2 rounding.

24 Example Example Determine the 5-digit representations for e using e chopping, 2 rounding.

25 Example Example Determine the 5-digit representations for e using e chopping, fl (e) = rounding.

26 Example Example Determine the 5-digit representations for e using e chopping, fl (e) = rounding. fl (e) =

27 Approximation Errors Definition Suppose that ˆp is an approximation to p. The absolute error is p ˆp. p ˆp The relative error is provided p 0. p

28 Approximation Errors Definition Suppose that ˆp is an approximation to p. The absolute error is p ˆp. p ˆp The relative error is provided p 0. p Remark: the relative error is generally preferred as a measure of accuracy, since it takes into consideration, the magnitude of the number being approximated.

29 Example Example Determine the absolute and relative error present in each of the 5-digit approximations to e.

30 Example Example Determine the absolute and relative error present in each of the 5-digit approximations to e. Chopping ê = : e ê and Rounding ê = : e ê and e ê e e ê e

31 Significant Digits Definition The number ˆp is said to approximate p to t significant digits if t is the largest non-negative integer for which p ˆp p 5 10 t.

32 Example Suppose that ˆp agrees with p to 5 significant digits. p ˆp 0.01 ( , ) 0.1 ( , ) 1 ( , ) 10 (9.9995, ) 100 (99.995, ) 1000 (999.95, )

33 Chopping and Relative Error x fl (x) x = 0.d 1 d 2... d k 1 d k d k n 0.d 1 d 2... d k 1 d k 10 n 0.d 1 d 2... d k 1 d k d k n = 0.d k n k 0.d 1 d 2... d k 1 d k d k n = 0.d k d 1 d 2... d k 1 d k d k k k = 10 k+1

34 Machine Arithmetic The performance of basic arithmetic operations on a computing device also results in approximations. We will define the following machine arithmetic operations: x y = fl (fl (x) + fl (y)) x y = fl (fl (x) fl (y)) x y = fl (fl (x) fl (y)) x y = fl (fl (x) fl (y))

35 Example Using 5-digit chopping arithmetic with x = 2/3 and y = 3/7, compute the following quantities and the errors involved in their calculation. Operation Result Actual Abs. Err. Rel. Err. x y x y x y x y

36 Example Using 5-digit chopping arithmetic with x = 2/3 and y = 3/7, compute the following quantities and the errors involved in their calculation. Operation Result Actual Abs. Err. Rel. Err. x y x y x y x y

37 Example Using 5-digit chopping arithmetic with x = 2/3 and y = 3/7, compute the following quantities and the errors involved in their calculation. Operation Result Actual Abs. Err. Rel. Err. x y x y x y x y

38 Example Using 5-digit chopping arithmetic with x = 2/3 and y = 3/7, compute the following quantities and the errors involved in their calculation. Operation Result Actual Abs. Err. Rel. Err. x y x y x y x y

39 Example Using 5-digit chopping arithmetic with x = 2/3 and y = 3/7, compute the following quantities and the errors involved in their calculation. Operation Result Actual Abs. Err. Rel. Err. x y x y x y x y

40 Example Using 5-digit chopping arithmetic with x = 2/3 and y = 3/7, compute the following quantities and the errors involved in their calculation. Operation Result Actual Abs. Err. Rel. Err. x y x y x y x y Remark: the relative errors are small, so the machines results can be trusted in these cases.

41 Example Suppose y = 3/7, v = , and w = Using 5-digit chopping arithmetic compute the following results and the associated errors. Operation Result Actual Abs. Err. Rel. Err. y v (y v) w

42 Example Suppose y = 3/7, v = , and w = Using 5-digit chopping arithmetic compute the following results and the associated errors. Operation Result Actual Abs. Err. Rel. Err. y v (y v) w

43 Example Suppose y = 3/7, v = , and w = Using 5-digit chopping arithmetic compute the following results and the associated errors. Operation Result Actual Abs. Err. Rel. Err. y v (y v) w

44 Example Suppose y = 3/7, v = , and w = Using 5-digit chopping arithmetic compute the following results and the associated errors. Operation Result Actual Abs. Err. Rel. Err. y v (y v) w Remark: in this example the subtraction of nearly equal quantities leads to larger relative errors.

45 Subtraction of Nearly Equal Quantities Suppose x > y and the k-digit representations of x and y are respectively fl (x) = 0.d 1 d 2... d p a p+1 a p+2... a k 10 n fl (y) = 0.d 1 d 2... d p b p+1 b p+2... b k 10 n.

46 Subtraction of Nearly Equal Quantities Suppose x > y and the k-digit representations of x and y are respectively fl (x) = 0.d 1 d 2... d p a p+1 a p+2... a k 10 n fl (y) = 0.d 1 d 2... d p b p+1 b p+2... b k 10 n. Since the first p decimal digits of x and y are the same, then where x y = 0.c p+1 c p+2... c k 10 n p 0.c p+1 c p+2... c k = 0.a p+1 a p+2... a k 0.b p+1 b p+2... b k.

47 Subtraction of Nearly Equal Quantities Suppose x > y and the k-digit representations of x and y are respectively fl (x) = 0.d 1 d 2... d p a p+1 a p+2... a k 10 n fl (y) = 0.d 1 d 2... d p b p+1 b p+2... b k 10 n. Since the first p decimal digits of x and y are the same, then where x y = 0.c p+1 c p+2... c k 10 n p 0.c p+1 c p+2... c k = 0.a p+1 a p+2... a k 0.b p+1 b p+2... b k. Remark: the result x y has at most significance. digits of

48 Subtraction of Nearly Equal Quantities Suppose x > y and the k-digit representations of x and y are respectively fl (x) = 0.d 1 d 2... d p a p+1 a p+2... a k 10 n fl (y) = 0.d 1 d 2... d p b p+1 b p+2... b k 10 n. Since the first p decimal digits of x and y are the same, then where x y = 0.c p+1 c p+2... c k 10 n p 0.c p+1 c p+2... c k = 0.a p+1 a p+2... a k 0.b p+1 b p+2... b k. Remark: the result x y has at most k p digits of significance.

49 Subtraction of Nearly Equal Quantities Suppose x > y and the k-digit representations of x and y are respectively fl (x) = 0.d 1 d 2... d p a p+1 a p+2... a k 10 n fl (y) = 0.d 1 d 2... d p b p+1 b p+2... b k 10 n. Since the first p decimal digits of x and y are the same, then where x y = 0.c p+1 c p+2... c k 10 n p 0.c p+1 c p+2... c k = 0.a p+1 a p+2... a k 0.b p+1 b p+2... b k. Remark: the result x y has at most k p digits of significance. Any further calculations with x y will inherit only k p digits of significance.

50 Extended Example (1 of 5) Use 4-digit rounding arithmetic to solve the following quadratic equation. x x + 1 = 0

51 Extended Example (1 of 5) Use 4-digit rounding arithmetic to solve the following quadratic equation. x x + 1 = 0 If we solve the equation exactly we see that x and x

52 Extended Example (2 of 5) x 1 = b b 2 4ac 2a = b ( ) 2 ( )( )(0.10 2a = b a = b a = a = =

53 Extended Example (3 of 5) Absolute Error: = Relative Error: =

54 Extended Example (4 of 5) Suppose now we rationalize the quadratic formula so as to avoid the subtraction of nearly equal quantities. x 1 = b ( b 2 4ac b + ) b 2 4ac 2a b + b 2 4ac 2c = b + b 2 4ac = = =

55 Extended Example (5 of 5) Absolute Error: = 10 7 Relative Error: =

56 Re-arrangement of Calculations In addition to avoiding the subtraction of nearly equal results in calculations, it is generally a good idea to reduce the number of operations performed to obtain a desired result.

57 Re-arrangement of Calculations In addition to avoiding the subtraction of nearly equal results in calculations, it is generally a good idea to reduce the number of operations performed to obtain a desired result. Example Evaluate the polynomial p(x) = x 3 5.9x x at x = 7.14 using 3-digit arithmetic.

58 Re-arrangement of Calculations In addition to avoiding the subtraction of nearly equal results in calculations, it is generally a good idea to reduce the number of operations performed to obtain a desired result. Example Evaluate the polynomial p(x) = x 3 5.9x x at x = 7.14 using 3-digit arithmetic. For the sake of comparison, the exact value is p(7.14) =

59 Evaluation (1 of 2) Consider p(x) = x 3 5.9x x and intermediate results obtained using 3-digit chopping and 3-digit rounding arithmetic.

60 Evaluation (1 of 2) Consider p(x) = x 3 5.9x x and intermediate results obtained using 3-digit chopping and 3-digit rounding arithmetic. x x 2 x 3 5.9x 2 3.4x Chopping 0.714E E E E E02 Rounding 0.714E E E E E02

61 Evaluation (2 of 2) Chopping: Rounding: p(7.14) = Abs. Err. = = Rel. Err. = = p(7.14) = Abs. Err. = = Rel. Err. = =

62 Nested Polynomial Form We may reduce the number of arithmetic operations performed by re-writing the polynomial as p(x) = x 3 5.9x x = x (x(x 5.9) + 3.4)

63 Nested Polynomial Form We may reduce the number of arithmetic operations performed by re-writing the polynomial as p(x) = x 3 5.9x x = x (x(x 5.9) + 3.4) Evaluate p(7.14) using 3-digit chopping and rounding arithmetic.

64 Evaluation (Chopping) p(7.14) = 7.14 (7.14( ) + 3.4) = 7.14 (7.14(1.24) + 3.4) = 7.14( ) = 7.14(12.2) = = 89.8 Abs. Err. = = Rel. Err. = =

65 Evaluation (Rounding) p(7.14) = 7.14 (7.14( ) + 3.4) = 7.14 (7.14(1.24) + 3.4) = 7.14( ) = 7.14(12.3) = = 90.5 Abs. Err. = = Rel. Err. = =

66 Homework Read Section 1.2. Exercises: 1 5, 9, 11, 15, 17, 21, 24

Mathematical preliminaries and error analysis

Mathematical preliminaries and error analysis Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan September 12, 2015 Outline 1 Round-off errors and computer arithmetic