Working Environment : Python #1

Transcription

1 Working Environment : Python #1 Serdar ARITAN Biomechanics Research Group, Faculty of Sports Sciences, and Department of Computer Graphics Hacettepe University, Ankara, Turkey 1

2 Physics has several aspects: Physics as a science represents the quest to understand the basic laws of nature. Physics provides the tools to understand the processes occurring in nature on all time and length scales. Physics also provides the conceptual and theoretical background for developing new technologies. Mechanics is the part of physics that addresses the motion of objects. However, in order to predict motion, we need quantitative tools to describe motion. Our main tool is calculus and associated analytical and numerical methods. The study of motion is traditionally called kinematics, which is in many ways closer to mathematics than to physics. 2

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4 The Mars Climate Orbiter was a 338 kilogram robotic space probe launched by NASA on December 11, 1998 to study the Martian climate, atmosphere, and surface changes and to act as the communications relay in the Mars Surveyor '98 program for Mars Polar Lander. However, on September 23, 1999, communication with the spacecraft was lost as the spacecraft went into orbital insertion, due to ground-based computer software which produced output in non-si units of pound-seconds instead of the metric units of newton-seconds specified in the contract between NASA and Lockheed. 4

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6 TIOBE has been collecting data on programming language popularity for many years 6

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9 Distinct Features of Python Extensible (packages) Embeddable into applications Functional programming Object-Oriented programming Rapid Prototyping Great for readability and presentation White space is significant Low maintenance costs Exception handling Free (open source) 9

10 The core philosophy of the language is summarized by the document "PEP 20 (The Zen of Python)", Beautiful is better than ugly. Explicit is better than implicit. Simple is better than complex. Complex is better than complicated. Readability counts. Try; >>>import this 10

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13 There are no units in your program, so you need to keep track of the units yourself. If you calculate the velocity of a car driving 20.0 m in 1.0 s, you find: >>> x = 20.0 >>> t = 1.0 >>> v = x/t >>> v 20 It is your job to know that the units for the answer is m/s. In your script, you could write: x = 20.0 # m t = 1.0 # s This does not solve the problem, but this is a habit that makes it easier for you to spot the units. 13

14 Your program tends to return a long sequence of numbers irrespective of the number of digits you put in: It does not care about significant numbers as do we: >>> x = 10.0; >>> t = 3.0; >>> x/t It is therefore always your job not to report the numbers you get out directly, but to apply the rules of the number of significant digits before you report the calculated value. 14

15 Digital Representation of Numbers The numbers you use in your program can be of various types and are stored digitally in various ways. Numbers in computers are stored in units of 4 or 8 bytes, which is the size most efficiently handled by the hardware. One byte corresponds to an 8-bit binary number. One bit is a binary number, which means that it is either 0 or 1. With 8 bits we can make 2 8 different numbers. For example, we could enumerate the numbers from 0 to 255 = You may use binary as a data type in your programs, but you will usually use either integers, called integers, or real numbers, called floating point numbers. 15

16 Your program tends to return a long sequence of numbers irrespective of the number of digits you put in: It does not care about significant numbers as do we: >>> x = 10.0; >>> t = 3.0; >>> x/t It is therefore always your job not to report the numbers you get out directly, but to apply the rules of the number of significant digits before you report the calculated value. 16

17 Integers Integers are usually either represented by 4 or 8 bytes of computer memory. This means that there is a maximum number that can be represented. If you use a 4-byte integer the integers can range from 2 31 = to = However, in Python your programming language will sort out problems if you exceed this number by automatically switching to a data type such as a 8 byte integer. 17

18 Floating-Point Numbers While floating-point numbers are supposed to represent real numbers, they also have a finite digital precision, corresponding to the number of possible values that can be spanned by the 4 or 8 bytes used to represent the floating-point number. Floating point numbers are represented by the (significant) digits, and an exponent, using the method that we called scientific notation. For a 4-byte floating-point number, the IEEE4 standard determines that 23 bits are used for the digits and 9 bits are used for the exponents - including the signs for both. 4-byte floating point numbers therefore have 6-9 (significant) digits, and cover a range from to for negative numbers, and from to for positive numbers. 18

19 Floating-Point Numbers However, you will not encounter 4-byte numbers when programming in Python since by default the floating-point numbers are 8-byte. In this case, numbers have (significant) digits, and cover a range from to for negative numbers, and from to for positive numbers. With 8-byte floating-point numbers you will seldom encounter cases where you are limited by the numerical precision. 19

20 Floating-Point Numbers >>> import sys >>> sys.float_info sys.float_info(max= e+308, max_exp=1024, max_10_exp=308, min= e-308, min_exp=-1021, min_10_exp=-307, dig=15, mant_dig=53, epsilon= e- 16, radix=2, rounds=1) >>> sys.int_info sys.int_info(bits_per_digit=30, sizeof_digit=4) >>> # Guess what is the answer? >>> # it must be ZERO!! 20

21 Functions A user defined function acts just like a predefined mathematical function such as sin or exp. def convertf(tc): # Converts from centigrade to Fahrenheit TF = 9.0/5.0*TC return TF We call our new function by typing: >> convertf(45.0) Notice that Python requires each such used-defined function to either be in the same file as the script using the function, of the function must be in a separate file, and that the file must be in the search path for Python. 21

22 Plotting Data-Sets Python not only works as a numerical calculator, it also has advanced data visualization capabilities from pylab import * m = array([1.0,2.0,4.0,6.0,9.0,11.0]) V = array([0.13,0.26,0.50,0.77,1.15,1.36]) plot(m, V, o ) xlabel( m (kg) ) ylabel( V (l) ) 22

23 Plotting a Function using SymPy SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python and does not require any external libraries. from sympy import symbols from sympy.plotting import plot x = symbols('x') p1 = plot(x*x) p2 = plot(x) 23

24 Plotting a Function We must first generate a sequences of numbers that sequence for the x es and one sequence for the corresponding values of (x) and (x 2 ), and then plot the two sequences. from pylab import * x0, x1, dx = -10.0, 10.0, 0.1 n = int(ceil((x1-x0)/dx) + 1) x = zeros((n,1),float) for i in range(n): x[i] = x0 + i*dx plot(x, x, '-'), show() plot(x, x*x, '-'), show() 24

25 Vectorization We can make a sequence of x es in several ways using functions that are built into Python instead of using a loop. For example, the function linspace generates a sequence of equidistant numbers from 0.0 to 10.0:. >>> x = linspace(0,10,10) >>> print(x) [ ] In this case we generated 10 numbers, but you are free to choose your own resolution. 25

26 Vectorization An alternative to specifying the number of points you want, as we do with linspace, is to specify the step size. The expression ([0.0:10.0:0.3]) returns an array starting at the value 0 and ending at 10.0 in steps of 0.3. >>> x = arange(0.0, 10.0, 0.3) >>> x array([ 0., 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3., 3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6., 6.3, 6.6, 6.9, 7.2, 7.5, 7.8, 8.1, 8.4, 8.7, 9., 9.3, 9.6, 9.9]) 26

27 Random Numbers Sometimes you need randomness to enter your physical simulation. For example, you may want to model the motion of a tiny dust of grain in the air bouncing about due to random hits by the air molecules, so called Brownian motion. Unfortunately, we cannot generate really random numbers, but most programs have decent pseudo-random number generators, which generate a sequence of numbers that appear to be random. In Python we can simulate the throw of a dice using >>> randint(6) where randint(n) generates a random integer between 0 and n 1. 27

28 Conditions if we return to discuss the motion of the grain of dust, we want to model its motion according to a simple rule: I throw a dice. If I get between 1 and 3, the grain moves a step forward, otherwise it moves a step backward. Every time we throw the dice, the grain moves a distance dx = ±1. If the grain is at position x i at step i, the grain will be at a position x i + 1 = x i + dx at step i + 1. from pylab import * n = 1000 x = zeros(n, float) for i in range(n - 1): if (randint(6)+1<=3): dx = -1 else: dx = 1 x[i+1] = x[i] + dx 28

29 Conditions plot(x), xlabel( i ), ylabel( x(i) ) show() 29

30 Conditions Pythonic formulation of the random walk, which you may have fun trying to understand. from pylab import * x = cumsum(2*(randint(1,7,1000)<=3)-1) plot(x), show() You can research help about cumsum by pressing CTRL+I in Ipython console. 30

31 Reading Real Data When you work with physics you need to handle real data. Fortunately, it is very simple to read such as file into Python. It is done by a single command: >>> runner = loadtxt("run100m.txt") We split the data into two arrays t and x by: >>> t = runner[:,0] >>> x = runner[:,1] 31

32 Summary Direct calculations are done on the command line >>> 10.0*sin(pi/3)+4.0**3 Defining and reusing variables >>> a = 2.0, b = 4.5, c = a**2 + b**2 Vectorized plotting of functions >>> x=linspace(0,10,0.01), y=exp(-x)*sin(x),plot(x,y),show() A function can be part of a script of stored in a separate.py file. It has the syntax def myfunction(a,b,c): v = a*b*c d = v**2 y = 2.0*d return y 32

33 Summary Plotting You plot two arrays t and x versus each other by plot(t,x, -b ), xlabel( t (s) ), ylabel( x (m) ) Plotting several data-sets in the same plot: plot(t1,x1, -b,t2,x2, -r ) 33

34 Summary Loops for-loops run a counter sequentially through a list of values: for i in range(100): x[i] = sin(i/100.0) while-loops run until a given expression is true i = 0 while (i<100): i = i + 1 x[i] = sin(i/100.0) 34

35 Seconds Exercises (a)write a script that calculates the number of seconds, s, given the number of hours, h, according to the formula s = 3600 h. (b) Use the script to find the number of seconds in 1.5, 12 and 24 h. Spherical mass (a) Write a script that calculates the mass of a sphere given its radius r and mass density ρ according to the formula m = (4π/3) ρr 3. (b) Use the script to find the mass of a sphere of steel of radius r = 1mm, r = 1m, and r = 10 m. 35