Computational modeling

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1 Computational modeling Lecture 3 : Random variables Theory: 1 Random variables Programming: 1 Implicit none statement 2 Modules 3 Outputs 4 Functions 5 Conditional statement Instructor : Cedric Weber Course : 4CCP1000

2 P. Higgs wins the Nobel prize LHC collider * Peter Higgs shares the Nobel prize for Physics with professor Francois Englert * KCL BSc '50, MSc '52, PhD '54 2

Schedule Class/Week Chapter Topic Milestones 1 Monte Carlo UNIX system / Fortran 2 Monte Carlo Fibonacci sequence 3 Monte Carlo Random variables 4 Monte Carlo Central Limit Theorem 5 Monte Carlo

3 Schedule Class/Week Chapter Topic Milestones 1 Monte Carlo UNIX system / Fortran 2 Monte Carlo Fibonacci sequence 3 Monte Carlo Random variables 4 Monte Carlo Central Limit Theorem 5 Monte Carlo Monte Carlo integration Milestone 1 6 Differential equations The Pendulum 7 Differential equations A Quantum Particle in a box 8 Differential equations The Tacoma bridge Milestone 2 9 Linear Algebra System of equations 10 Linear Algebra Matrix operations Milestone 3 3

4 What did you learn last time? 1. Compilation and execution of a program " " "ý " " 2. do loop " " " " ý " 3. Reading a number from the keyboard " " ý " 4. Understanding the limitations of a computer" " "ý " 5. Sequences / Finding a fix point with pen & paper ý " 4

week 1 week 2 week 3 Milestone 1 : Monte

5 Where are we going? 100% * Lecture 1: * Introduction 90% 80% 70% * Lecture 2: * Fibonacci sequence / do loop * Lecture 3: * Today: another type of sequence the random number generator 60% 50% 40% 30% 20% 10% 0% week 1 week 2 week 3 Milestone 1 : Monte Carlo integration week 4 week 5 week 6 week 7 week 8 week 9 week 10 5

6 Part 1 : Theory Random variables 6

Jacob Bernoulli (1654-1705): Game Theory q The Bernoulli family, originally form Holland, established in Switzerland, were probably the most

nephew, Daniel Bernoulli (1700-1782) became famous for his work in hydrodynamics" q Simple idea: lets define the outcome of an experiment X i, and

When I repeat the experiment, I sometimes get X i =0, and sometimes I obtain X i =1" q Let s think of X i as a coin, which is flipped (head/tail).

7 Jacob Bernoulli ( ): Game Theory q The Bernoulli family, originally form Holland, established in Switzerland, were probably the most predominant math family of Europe" q Jacob discovered the Euler constant" q Famous for his work on Game theories The Art of Conjecture" q His nephew, Daniel Bernoulli ( ) became famous for his work in hydrodynamics" q Simple idea: lets define the outcome of an experiment X i, and the result of the experiment is either X i =0 or X i =1. When I repeat the experiment, I sometimes get X i =0, and sometimes I obtain X i =1" q Let s think of X i as a coin, which is flipped (head/tail). " q Experiment: flip the coin 1000 times" q how many tail, how many head?" q Definition of a Probability: " q P(head) = N(head) / total P(tail) = N(tail) / total " q Reference: The Life and Times of the Central Limit Theorem, William J. Adams" 7 = 500 = 500

q I give you a number R q R is any rational number between 0 and 1 q R is random R Flipping a coin q How could you use R to describe the coin flipping experiment?

8 q I give you a number R q R is any rational number between 0 and 1 q R is random R Flipping a coin q How could you use R to describe the coin flipping experiment? q Write down the relation between R and the coin configurations (head/tail) q Example, I obtain : q R=0.121 Corresponds to : [FILL IN] q R=0.551 Corresponds to : [FILL IN] 8

9 How can we obtain a random R? * We ll get to that in a moment First a short recap on the modulo * X mod Y : remainder of the division X/Y * Examples : * 1 mod 5 = 1 (remains 1 from the division 1/5) * 3 mod 5 = 3 * 5 mod 5 = 0 * 6 mod 5 = [FILL IN] * 10 mod 5 = [FILL IN] * Question: which one of the following statement is true? q X mod Y < Y q X mod Y = Y q X mod Y > Y 9

10 How can we obtain a random R? * How can we implement a random process in a computer when the only thing a computer can do is carry out pre- defined calculations? * Answer: we generate a long sequence of numbers, each of this number can be seen as one realization of the experiment (coin flipping) * Example - throwing a dice : we would generate a long sequence of numbers between 1 and 6, each number corresponds to one experiment * Condition : the numbers in the sequence have to be independent. Counter example : the sequence is not random. * Random number generator: they generate numbers which seem independent. More precisely, if we generate a sequence of random numbers, we don t want some pattern to repeat and start over with the same sub- sequence of numbers (cycle length, e.g ) 10

11 Random generator * In a computer, we generate a sequence (mod stands for the modulo of): x i =(ax i 1 + c) modm * a and c are chosen parameters * We will generate a sequence of numbers: 41,3,213,60,300 * We have the property : x i < m * How can we obtain a random number R in the interval [0,1] from the obtained xi? * Answer : Ri = [FILL IN] 11

12 One bad choice of parameters * a=12 * c=0 * m=143 * x 0 =1 x i =(ax i 1 + c) modm * Let s try it : x 1 = 12x 0 mod 143 = 12 x 2 = 12x 1 mod 143 = 144mod 143 = 1 x 3 = 12x 2 mod 143 = 12 * Random sequence obtained : 1, 12, 1, 12, 1, 12, 1, 12 12

13 Part 2 : Programming Implicit none statement Outputs and plots Modules Function Conditional statement 13

14 1. Implicit none statement 2. Outputs and plotting tool 3. Modules 4. Function 5. Conditional statement 14

15 Spot the mistake 1. Program something 2. Integer(4) :: var1 3. do i = 0, 100 Ø There is one problem in this code, can you find it? 4. var1 = var1 + i 5. write(*,*) var1 =, var1 6. end do 7. end program 15

16 Implicit none statement 1. Program something 2. Implicit none 3. Integer(4) :: var1 4. do i = 0, var1 = var1 + i 6. write(*,*) var1 =, var1 7. end do Ø Implicit none statement comes right after the block definition (program, function, subroutine, ) Ø It prevents to use undeclared variables 8. end program 16

17 1. Implicit none statement 2. Outputs and plotting tool 3. Modules 4. Function 5. Conditional statement " 17

18 IO operations : Inputs 1. Program something 2. Implicit none 3. Integer(4) :: var1 4. write(*,*) enter the variable to the keyboard and hit enter: 5. read(*,*) a 6. write(*,*) you entered :,a * To read a variable from the keyboard, use the command: * read(*,*) variable_name * Why did we add the write(*,*) statement at line 4? 7. end program 18

19 IO operations : File output 1. Program something 2. Implicit none 3. Real(8) :: a, b 4. Integer(4) :: i 5. do i=1, a=(3.1428/360.0) * dble(i) write(100,*) a, cos(a) 9. end do 10. end program * Replacing the first * of the write statement by a number, redirects the output into a file named fort.xxx * To write a list of variables to a file, use the command: * write(100,*) a, b * This appends a line to the end of the file named fort.100 with the values of a and b 19

IO operations : File output 1. Program something 2. Implicit none 3. Real(8) :: a, b 4. Integer(4) :: i 5. do i=1,360 6. a=(3.

end program * Replacing the first * of the write statement by a number, redirects the output into a file named fort.

20 IO operations : File output 1. Program something 2. Implicit none 3. Real(8) :: a, b 4. Integer(4) :: i 5. do i=1, a=(3.1428/360.0) * dble(i) write(100,*) a, cos(a) 9. end do 10. end program * Replacing the first * of the write statement by a number, redirects the output into a file named fort.xxx * To write a list of variables to a file, axis" use the command: * write(100,*) a, b Horizontal Vertical axis" * This appends a line to the end of the file named fort.100 with the values of a and b 20

21 Plotting tool : xmgrace command u In the terminal, type : xmgrace fort

22 Plotting tool : xmgrace command u In the terminal, type : xmgrace fort

23 1. Implicit none statement 2. Outputs and plotting tool 3. Modules 4. Function 5. Conditional statement " 23

24 Modules * A module contains all your functions, it is your library * It forms an independent block, and needs to be above your program * To access an element from the module in your program, after the line program test, you need to specify use mylibrary, to tell the code that you want to make the functions contained in the module mylibrary available in your 1. Module mylibrary 2. Contains 3. Function1 4. Function2 5. Function3 6. End module 7. program test 8. use mylibrary 9. implicit none 10. Real(8) ::. 11. [..] 12. end program program section 24

25 Optional: split the module from your program file Content of file prog.f90 : Content of file library.f90 : 1. program test 2. Use mylibrary 3. Implicit none 4. Real(8) ::. 5. [..] 6. end program 1. Module mylibrary 2. Contains 3. Functions.. 4. End module Compilation : The order matters! gfortran [space] o [space] prog_name.out [space] library.f90 [space] prog.f90

26 1. Implicit none statement 2. Outputs and plotting tool 3. Modules 4. Function 5. Conditional statement " 26

Functions 4.1" 2.1" Function block 1. Function some_name ( input_variable_name ) 2. Implicit none 3. real(8) :: input_variable_name 4. real(8) :: some_name 5. some_name = input_variable_name + 2.

The computer stores the input value to the variable input_variable_name 4. The computer stores the output value to the variable some_name.

27 Functions 4.1" 2.1" Function block 1. Function some_name ( input_variable_name ) 2. Implicit none 3. real(8) :: input_variable_name 4. real(8) :: some_name 5. some_name = input_variable_name End function 1. A function is defined by a block function end function 2. It is a black- box which receives an input value (2.1) and returns an output values (4.1) 3. The computer stores the input value to the variable input_variable_name 4. The computer stores the output value to the variable some_name. This is the name that you want to give to your function (volume_of_a_sphere) 5. Within the block of the function, you can do any calculation (define more variables if needed). The value given to input_variable_name 27 is obtained when the function is executed in your program.

28 Example : volume of a sphere BLOCK 1 BLOCK 2 1. Function volume_of_a_sphere ( radius ) 2. real(8) :: radius 3. real(8) :: volume_of_a_sphere 4. volume_of_a_sphere = 3.14*radius**3*4.0/ End function 6. program test 7. Real(8) :: vol 8. vol = volume_of_a_sphere ( 2.0 ) 9. write(*,*) volume is :, vol 10. end program * Blocks do not communicate, variables in Block 1 only exist in this block, same for variables in Block 2 * Same variable names can be used in the block 1&2, they are still different * The computer replaces in your program the function by its numerical evaluation obtained with the given argument 28

29 Example : volume of a sphere BLOCK 1 BLOCK Function volume_of_a_sphere ( radius ) 2. real(8) :: radius 3. real(8) :: volume_of_a_sphere 4. volume_of_a_sphere = 3.14*radius**3*4.0/ End function 6. program test 7. Real(8) :: vol 8. vol = volume_of_a_sphere ( 2.0 ) 9. write(*,*) volume is :, vol 10. end program * Function is followed by a name of your choice, it is how you want to name this new function * The chosen name of the function also has to appear in the argument list, the value returned by the function will be the value of this variable * Radius is a chosen variable name. The value of this variable is attributed when the function is called from the program section. In this example, radius takes the value 2.0 * you can use in your program this function in the same way you would use the Fortran functions (sqrt, exp, sin, ).

30 Spot the mistakes. 1. Function volume_of_a_sphere ( radius ) 2. real(8) :: radius 3. volume_of_a_sphere = 3.14 * (radius**3) * 4.0/ End function 5. program test 6. real(8) :: vol 7. vol = volume_of_a_sphere ( 2.0 ) 8. write(*,*) volume is :, vol 9. write(*,*) radius is :, radius 10. end program Mistake 1 : line? Mistake 2 : line? 30

31 1. Implicit none statement 2. Outputs and plotting tool 3. Modules 4. Function 5. Conditional statement " 31

32 BLOCK 1 BLOCK 2 1. Program something 2. Implicit none 3. Real(8) :: a 4. a=0.2 Conditional statement 5. If ( a < 1.0 ) then 6. write(*,*) a smaller than 1 7. else 8. write(*,*) a larger than 1 9. end if 10. end program * The if statement executes: * the block 1 if the condition a<1.0 is satisfied * The block 2 if the condition a<1.0 is NOT satisfied * Possible logical tests are: * a < 1.0 a > 1.0 * a <= 1.0 a >= 1.0 * a==1.0 a!=

33 Practice Problems u Exercice 1 : Random number generator u Exercice 2 : Flipping a coin 33

Computational modeling

Computational modeling Lecture 5 : Monte Carlo Integration Physics: Integration The Monte Carlo method Programming: Subroutine Differences Subroutine/Function The Fortran random number subroutine Monte