Exercise(s) Solution(s) to the exercise(s)

Transcription

1 Exercise(s) Problem 1. Counting configurations Consider two different types of atoms, say A and B (represented by red and blue, respectively in the figures). Let A atoms and B atoms be distributed on N ( ) lattice sites. How many different ways can these atoms be arranged? This is an important question. Let the number of different arrangements be called. Then, As we shall see later, the natural logarithm of multiplied by a constant (called the Boltzmann constant) is known as the configurational entropy ( ) and plays a key role in the thermodynamics of binary, substitutional alloys. Write a C program that reads and from a file, and returns. Test the code for, ; and,,. Solution(s) to the exercise(s) Solution to Problem 1: Counting configurations A solution to Problem 1 is the code given below. We would like to draw your attention to the function factorial which calls itself in a recursive fashion. / Cprogramming/Exercise1.c Copyright (C) 2012 Prita Pant and M P Gururajan This program is a free software; you can redistribute it and/or modify it under the terms of the GNU General Public License published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY of FITMESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA , USA

2 / / Include the standard libraries and library for string manipulations / #include<stdio.h> #include<stdlib.h> / Function to calculate factorial / int factorial(int n); / The main function of C program always returns an integer; if the returned integer is zero, the Operating system knows that the program execution was successful. In our case, the main program does not take any input parameters; so, we write void / int main(void){ FILE fp; / File pointer for reading data / int Na,Nb; / Number of A and B atoms respectively / int N; / Total number of lattice points / double omega; / Number of arrangements / / Condition opening of files; if opening of files is not successful for any reason, the code will inform the user so / if( (fp=fopen("nanb.dat","r")) == NULL){ printf("unable to open elements.dat. \n"); printf("exiting\n"); exit(0); else{ fp = fopen("nanb.dat","r"); / Read the input data from the file "NaNb.dat"; note that we use fgets for reading the string / fscanf(fp,"%d%d",&na,&nb); / Close the opened file / fclose(fp); / Calculate the total number of atoms, hence, the lattice points /

3 N = Na+Nb; / Calculate omega by calling the factorial function / omega = (double)(factorial(n))/( (double) (factorial(na))(double) (factorial(nb)) ); / Output the number of configurations / printf("%lf is the total number of configurations, given\n",omega); printf("na = %d; Nb= %d; N = %d\n",na,nb,n); printf("%d\n",factorial(na)); printf("%d\n",factorial(nb)); printf("%d\n",factorial(n)); / The job is done; return 0 to the operating system signaling the successful completion of the program / return 0; / The function to calculate factorial / int factorial(int n){ / Note the calculation of factorial using a recursive algorithm / if(n == 1) return 1; else return nfactorial(n-1); When this program is run with the values of and, we do get the right answer of 70 (as can be checked by manual calculation). However, the code obviously fails for and. This is because, on the computer, for the storage of numbers, there are only a finite amount of memory. Hence, if a number is bigger than the largest number that the computer can store, handle and manipulate, we will not be able to deal with them on the computer. In this exercise, one can use Stirling's formula, namely to overcome this problem. A code that uses Stirling's formula is given below. Note that the factorial function no longer returns an integer value. / Cprogramming/Exercise1StirlingApprox.c Copyright (C) 2012 Prita Pant and M P Gururajan This program is a free software; you can redistribute it and/or modify it under the terms of the GNU General Public License published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

4 This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY of FITMESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA , USA / / Include the standard libraries and library for string manipulations / #include<stdio.h> #include<stdlib.h> #include<math.h> / Function to calculate factorial / double factorial(int n); / The main function of C program always returns an integer; if the returned integer is zero, the Operating system knows that the program execution was successful. In our case, the main program does not take any input parameters; so, we write void / int main(void){ FILE fp; / File pointer for reading data / int Na,Nb; / Number of A and B atoms respectively / int N; / Total number of lattice points / double omega; / Number of arrangements / / Condition opening of files; if opening of files is not successful for any reason, the code will inform the user so / if( (fp=fopen("nanb.dat","r")) == NULL){ printf("unable to open elements.dat. \n"); printf("exiting\n");

5 exit(0); else{ fp = fopen("nanb.dat","r"); / Read the input data from the file "NaNb.dat"; note that we use fgets for reading the string / fscanf(fp,"%d%d",&na,&nb); / Close the opened file / fclose(fp); / Calculate the total number of atoms, hence, the lattice points / N = Na+Nb; / Calculate omega by calling the factorial function / omega = (double)(factorial(n))/( (double) (factorial(na))(double) (factorial(nb)) ); / Output the number of configurations / printf("%lf is the total number of configurations, given\n",omega); printf("na = %d; Nb= %d; N = %d\n",na,nb,n); printf("%le\n",factorial(na)); printf("%le\n",factorial(nb)); printf("%le\n",factorial(n)); / The job is done; return 0 to the operating system signaling the successful completion of the program / return 0; / The function to calculate factorial / double factorial(int n){ / Note the calculation of factorial using a recursive algorithm / if(n == 1) return 1.; else return sqrt(2m_pi)pow(n,(double)n+0.5)exp(-n); In a similar manner, real numbers, again, can only be represented on a computer upto some finite accuracy. This can also lead to difficulties. For example, if two roots are very close, the usual formula for calculating the roots of a quadratic equation,, namely,

6 will fail to return accurate results. Please see Numerical recipes in C for a detailed discussion on these issues.