Computation for Physics 計算物理概論

1. Computation for Physics計算物理概論 數位資料表示法

2. Numeral systems計數系統

3. Bit=Binary digit • Bit=0,1 • b, bit • Byte=Eight bits • B • K=Kilo=1000=103 or 1024=210 • M=Mega=106or 1024=220 • G=Giga=109or 1024=230 • T=Tera=1012or 1024=240 • KB=kilobyte, MB=kilobyte, GB=gigabyte

4. Integer

5. Unsigned integer • Map N-bits to 0,1,…,2N-1=2N possible numbers • Unsigned byte=8 bits • 0,…,255 • Unsigned short integer=16 bits • 0,…,65535 • Unsigned integer=32 bits • 0,…,4,294,967,295 • Unsigned long integer=64 bits • 0,…, 18,446,744,073,709,551,615

6. Most and leastsignificant bit a 8-bits number Most significant bit=msb Least significant bit=lsb

7. Signed integer How to represent negative numbers in N-bits without “-” ? • Sign-and-magnitude method • Use msbas “sign”, N-1 bits as “magnitude” • Two representations for zero • One’s complement • Use “bit complement” as the arithmetic negative • Two representations for zero • Two’s complement • Use “two’s complement of the absolute value” • One representation for zero

8. Sign-and-magnitude sign magnitude = = = =

9. Sign-and-magnitude • Addition of x and y • If x and y have the same sign  sgn(x)(x+y) • If x and y have different sign • If |x|>|y|  sgn(x)*(x-y) • If |y|>|x|  sgn(y)*(y-x) • Subtraction x and y=Addition of x and (-y) • If overflow return error message

10. One’s complement = = = =

11. One’s complement

12. One’s complementAddition

13. One’s complementend-around carry

14. One’s complementnegative zero

15. TWO’s complement = = = = =

16. two’s complement

17. Two’s ComplementAddition

18. Two’s ComplementAddition ignore

19. Two’s ComplementAddition

20. Two’s ComplementAddition ignore

21. Two’s ComplementAddition Positive+Positive=Negative! overflow  error message

22. Most and leastsignificant bit a 8-bits number Most significant bit=msb Least significant bit=lsb

23. Logic gates

24. Logic gates邏輯閘

25. And Gate

26. OR Gate

27. NOT Gate

28. NAND Gate

29. NOR Gate

30. XOR Gate

31. XNOR Gate

32. Adder Full Adder Half Adder

33. Real number

34. Radix point小數點 • Base 10 notation  decimal point • Base 2 notation  binary point Radix point

35. Scientific notation科學記號

36. Floating pointrepresentation浮點數表示法 exponent significand (mantissa)

37. IEEE 754 • IEEE • Institute of Electrical and Electronics Engineers • IEEE 754 • IEEE Standard for Floating-Point Arithmetic • Arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers (including signed zeros and subnormal numbers), infinities, and special "not a number" values (NaNs) • Interchange formats: encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form • Rounding rules: properties to be satisfied when rounding numbers during arithmetic and conversions • Operations: arithmetic and other operations on arithmetic formats • Exception handling: indications of exceptional conditions (such as division by zero, overflow, etc.)

38. IEEE 754 binary16Half precision • Sign bit: 1 bit • Exponent width: 5 bits • Significand precision: 11 (10 explicitly stored) • Exponent encoding • Offset=15, Emin=-14,Emax=15 • Minimum positive value=2^-14 • Maximum positive value=(2-2^-10)2^15=65504 • Minimum subnormal value=2^-24≈5.96×10^-5

39. IEEE 754 binary32 single precision • Sign bit: 1 bit • Exponent width: 8 bits • Significand precision: 24 (23 explicitly stored) • Exponent encoding • Offset=127, Emin=-126,Emax=127 • Minimum positive value≈2.2 ×10^-308 • Maximum positive value≈1.8 ×10^308 • Minimum subnormal value≈4.9 ×10^-324

40. IEEE 754 binary64Double precision • Sign bit: 1 bit • Exponent width: 11 bits • Significand precision: 53 (52 explicitly stored) • Exponent encoding • Offset=1023, Emin=-1022,Emax=1023 • Minimum positive value=2^-126≈1.18 ×10^-38 • Maximum positive value=(2-2^-23)2^127≈3.4 ×10^38 • Minimum subnormal value=2^-149≈1.4 ×10^-45

41. Representation error

42. Rounding Errors • In base-10 system • ½=0.5 • 1/3=0.333333333333333333333333333333333333333 • In base-2 system • Terminating iff denominators are powers of 2 (1/2, 3/16)

43. IEEE Rounding Modes • Truncation: • Keep the desired number of digits unchanged, removing all less-significant digits; also called rounding toward zero. • 0.142857 ≈ 0.142 (All digits less significant than the third removed). • Round to Nearest: (Default) • Round to the nearest valid representation. Break ties by rounding to an even digit • +23.524, +24.524; -23.5-24, -24.5 -24 (symmetry between +/- numbers) • Round to Nearest: • Round to the nearest valid representation. Break ties by rounding away from zero. • +23.524,+24,525; -23.5-24, -24.5 -25 (symmetry between +/- numbers) • Round to −∞: • Round to a value less than or equal to the original number. If the original number is positive, this is equivalent to truncation. • Round to +∞: • Round to a value greater than or equal to the original number. If the original number is negative, this is equivalent to truncation.

44. IEEE 754Special values • Positive infinity • Negative infinity • (positive zero=ordinary zero) • Negative zero: -0 • NaNs: “Not a number” values • Subnormal numbers

45. Signed zero • Sign=0+ or 1- • Exponent=0 • Significand(Mantissa)=0 • Arithmetic • , , • , • , • NaN, NaN

46. Signed infinity • Sign=0+ or 1- • Exponent=maximum value • 111112 • FFH • 7FF16 • Significand(Mantissa)=0

47. NaNs • Sign  quiet, signaling • Exponent=maximum value • 111112 • FFH • 7FF16 • Significand(Mantissa)≠0 • Creation • NaN, NaN,NaN • NaN, =NaN • =1 or NaN • Square root or logarithm of negative number • Inverse sine or cosine of a number with absolute value greater than 1

48. Subnormal numbers • Sign=0+ or 1- • Exponent=0 • 000002 • 00H • 00016 • Significand(Mantissa)≠0

49. Floating pointaddition • x=123456.7 = 1.234567 × 10^5 • y=101.7654 = 1.017654 × 10^2 = 0.001017654 × 10^5 • x+y=(1.234567+0.001017654) × 10^5 • x+y=1.235584654 × 10^5 • x+y≈1.235585 × 10^5 • Round-off error!

50. Floating point addition of a small number • x=1.234567× 10^5 • y=9.876543× 10^-3=0.00000009876543 × 10^5 • x+y=(1.234567+0.00000009876543) × 10^5 • x+y=1.23456709876543 × 10^5 • x+y≈1.234567 × 10^5=x • Round-off error!