KURSUS MATLAB ONLINE Skripsi, Tesis, DISERTASI 081219449060

EXERCISE 2          3X3 MATRIX DIAGONALISATION (part 1) A' = S-1 . A . S                                                                                                                                             (7) The transformation matrix S turns A into a diagonal matrix A'.  Once the transformation matrix S has been found, the eigenvectors of A are contained in the columns of the transformation matrix on the right in Eq. 7 and in the rows of its inverse in Eq. 7, S-1. The eigenvalues A are the diagonal elements of A' and the eigenvalue in the nth diagonal element corresponds to the eigenvector in the nth column of S.  Numerical diagonalisation is achieved by bringing the matrix A gradually towards diagonal form using a sequence of similarity transformations.  For symmetric matrices a sequence of orthogonal similarity transformations is used. One such procedure is called Jacobi transformation.  The routine jacobi_trans.c  performs just such a transformation. Its input is the matrix A, the outputs are the eigenvalues in array e[i] and the eigenvectors in matrix v[i][j].  Note that typedef statements have been used for declaration of scalars, matrices and vectors. (For further details, see your course JS 3065 Computational Methods).

12           EXERCISE 3          3X3 MATRIX DIAGONALISATION Write a programme which calls jacobi and solves the eigenvalue problem for Calculate the eigenvalues and eigenvectors of this matrix by hand and compare your numerical results with your analytical result.

12.1.1    Exercise 4            Nearly Free Electron Energy Band Structure The question of what determines whether a material is an electrical conductor, insulator or semi-conductor is of great importance in solid state physics.  The free electron model, where electrons are treated as an ideal Fermi gas, gives good insight into heat capacity and thermal conductivity but fails to explain the electrical conductivity of solids.  It is found that the crystal structure is most important in this problem as Bragg reflections of electron waves within the crystal lead to the formation of energy band gaps, which in turn determine the electrical conductivity of the solid.  The Schrödinger equation for electrons in a ‘1-D crystal’ is                                                                                                          (8)  is the eigenfunction for wavenumber k, are the allowed energies and V(x) is a periodic potential energy.  A plot of versus k is called the band structure.  Since the potential energy operator and the eigenfunctions both contain periodic functions they can be expanded in Fourier series.  We will use the complex form of Fourier series to expand the periodic function f(x) = f(x+na)  (a = period, n  is 0, 1, 2, ..)                                                                                                                         (9) In the nearly free electron problem we choose to limit the expansion for the potential and wavefunction such that only coefficients with –2  n  2 are nonzero.  Solutions to the Schrödinger equation using such a limited expansion for the wave function are quite accurate provided the magnitudes of the potential energy Fourier components in the problem are much less than the kinetic energies of the electrons.  The expansions for the potential energy and wave functions are                                              (10a)                                                                                                          (10b) The right hand equality in Eq. 10a is valid provided v1 = v-1 and v2 = v-2.  We also choose v0 = 0.  The factor exp[ikx] multiplying each term in Eq. 10b is necessary to make k a Bloch eigenfunction.  (See lecture notes from course JS 3014 Thermal and Electronic Properties).  When the potential and eigenfunction are substituted into Eq. 8, the resulting set of linear equations is Inspection of the matrix equation above shows that its diagonal elements depend on wave number k.  An energy band calculation using these equations consists of repeated construction of the matrix and computation of its eigenvalues.  The notation Gn in the equations above means 2n/a.  In 1-D the reciprocal lattice consists of points given by k = Gn with n integer.  The equations are given in atomic units in which h bar = e = me = 1.  It will be helpful if you choose units for wavenumbers to be in units of /a so that k = ±1 correspond to the Brillouin zone boundaries for the problem and Gn =2n.

Kami ada di Jakarta Selatan. KAMI MEMBERIKAN KURSUS MATLAB ONLINE - HUBUNGI MASTER ENGINEERING EXPERT (MEE) 081219449060.  Kami membuka kursus Matlab untuk pemula dan mahasiswa atau insinyur yang ingin memperdalam Matlab dan menerapkan dalam bidang teknikal, engineering, rekayasa, dsb. Format bimbingannya tugas-tugas yang bisa membantu Skripsi, Tesis, DISERTASI Bimbingan dilakukan secara online bisa lewat WA atau email Dijamin Bisa, atau bisa mengulang kembali. Kami juga dapat membantumembuatkan aplikasi atau program matlab/lainnya. Anda akan dilatih oleh Tim Profesional - HUBUNGI MASTER ENGINEERING EXPERT (MEE) 081219449060.   Email: kursusmatlab@gmail.com 13           EXERCISE 4 Write a programme which contains a function which constructs the 5x5 matrix above as a function of k.  The programme should also call the diagonalisation routine to obtain the eigenvalues of the matrix.

Use your programme to obtain the bandstructure of a solid where all of the Fourier components of the potential are weak (v-2 = v-1 = v1  = v2  =  0.1) and for a solid where some of the Fourier components of the potential are strong (v-2 = 0.3  v-1 = 0.7 v1  = v2  =  0.1).  Compare the band gaps calculated in your programme at the Brillouin zone centre and at its edges with the result from perturbation theory that the nth band gap is twice the nth Fourier component of the periodic potential, e.g. the first gap at k = 1 (in units of /a) is equal to 2 v1.

INTRODUCTION TO MATLAB        10 WHAT IS MATLAB?           11 HISTORY OF MATLAB       12 AN INITIAL EXAMPLE       13 Population Growth           13 SOME FEATURES              15 RUNNING MATLAB          16 Startup Under Unix          16 Startup Under Windows 95/98     16 To Exit   16 HELP & DEMOS  17 Through the Command Window  17 Through the Help Window            17 Through the Help Desk    17 Demonstrations 17 CONTROLLING THE ENVIRONMENT          18 A 2ND EXAMPLE               19 Final Approach   19 MAP GENERATION           20 MAP SMOOTHING            21 MAP CODE          22 mapGen               22 MapPlateau        23 PRINCIPLES OF  “GOOD” PROGRAMS        25 REVIEW               26 MATLAB SYNTAX              27 SOME SIMPLE CALCULATIONS     28 BASIC MATHEMATICAL OPERATORS          29 EXPRESSIONS     30 VARIABLES & ASSIGNMENT          31 THE ANS VARIABLE          32 Examining a Variable’sValue         32 SEMICOLON, COMMA & PERIOD 33 Semicolon           33 Comma 33 Period   33 LAYOUT CONVENTIONS 35 COMMENTS        36 USING COMMENTS          37 COMMENTS & MATLAB HELP       38 SOME USEFUL MATHEMATICAL FUNCTIONS          39 SCRIPT M-FILES  40 Basic Approach using M-Files       40 Naming 40 MATLAB NAME SPACE     41 Matlab Search Path          41 REVIEW               42 MATLAB VARIABLES & DATA TYPES            43 VARIABLES          44 IDENTIFIERS       45 MEANINGFUL IDENTIFIERS           46 Some Examples  46 WEAK TYPING    47 IMPLICATIONS OF WEAK TYPING               48 Advantages         48 Disadvantages    48 SPECIAL (BUILT-IN) CONSTANTS & VARIABLES       49 COMPLEX NUMBERS       50 NAN & INF          51 STRINGS              52 MANAGING VARIABLES   53 clear      53 MANAGING WORKSPACE: LOAD  54 MANAGING WORKSPACE: SAVE   55 REVIEW               56 I/O IN MATLAB  57 BASIC INPUT       58 The Input Statement        58 INPUTTING STRINGS       59 BEHAVIOUR OF INPUT STATEMENT           60 MICRO: AMOEBA EXPANSION      61 MICRO RUN        62 BASIC OUTPUT   63 The disp Statement          63 THE FORMAT COMMAND              64 FORMAT EXAMPLES         65 MORE    66 FILES IN GENERAL            67 Opening & Closing Files   67 FPRINTF               68 CONTROLLING FPRINTF  69 FSCANF 70 PROJECTILE EXAMPLE     71 PROJECTILE CODE            72 PROJECTILE RUN              74 PROJECTILE PLOT             75 REVIEW               76 ARRAYS IN MATLAB 1 – VECTORS               77 MOTIVATION     78 VISUALISING A 1D ARRAY (VECTOR)          79 MATLAB & ARRAYS          80 VECTOR CREATION          81 VECTOR CREATION EXAMPLE       82 ADDRESSING THE ELEMENTS OF A VECTOR            84 COLUMN VS. ROW VECTORS        85 COLUMN VS ROW EXAMPLE         86 SCALAR-ARRAY MATHEMATICS    87 EXAMPLE CEL2FAR.M      88 CEL2FAR RUN     89 ARRAY-ARRAY MATHEMATICS     90 EXAMPLE – BOUNCING.M             91 BOUNCING.M CODE        92 REVIEW               94 MATLAB ARRAYS 2 – MATRICES   95 2D(+) ARRAYS – MOTIVATION      96 VISUALISING 2D ARRAYS               97 VISUALISING 2D ARRAYS EXAMPLE            98 CREATION           99 CREATION EXAMPLE        100 MATHEMATICS OF MATRICES      101 MATHEMATICS EXAMPLE              102 ADDRESSING & MANIPULATION  105 ADDRESSING & MANIPULATION EXAMPLE             106 EXAMPLE – NETWORK TRAFFIC   109 TRAFFIC CODE    110 TRAFFIC EXAMPLE PLOTS              112 SPECIAL MATRICES          113 SPECIAL MATRICES EXAMPLE       114 ARRAY SIZES       115 ARRAY SIZES EXAMPLE    116 MULTI-DIMENSIONAL ARRAYS    117 LINEAR EQUATIONS        118 LINEAR EQUATIONS EXAMPLE     119 REVIEW               120 MATLAB RELATIONAL OPERATORS & BASES           121 MOTIVATION - RELATIONAL OPERATORS               122 BOOLEAN OPERATIONS  123 EXAMPLE – PIECEWISE FUNCTIONS           127 PIECEWISE FUNCTION EXAMPLE 128 NAN & EMPTY ARRAYS    129 BASE CONVERSION          130 BASE CONVERSION EXAMPLE       131 BIT LEVEL OPERATIONS  132 REVIEW               133 MATLAB SELECTION STATEMENTS (IF & SWITCH) 134 MOTIVATION     135 SIMPLE IF            136 EXAMPLES – SIMPLE IF    137 IF-ELSE  138 EXAMPLE – IF-ELSE          139 IF-ELSEIF-…         140 EXAMPLE – IF-ELSEIF…    142 TRUTH TABLES & COMPOUND IFS             143 SWITCH               145 EXAMPLE –SWITCH          147 REVIEW               148 LOOPS IN MATLAB           149 MOTIVATION     150 DEFINITE ITERATION (FOR LOOP)               151 FOR LOOP SCHEMATIC    152 FOR LOOP EXAMPLE        153 FOR LOOP ADDITIONAL  154 INDEFINITE ITERATION (WHILE LOOP)      155 WHILE LOOP SCHEMATIC              156 WHILE LOOP EXAMPLES 157 WHILE VS IF        158 GENERAL LOOPS & BREAK             159 BREAK SCHEMATIC          160 COMBINED (MAJOR) EXAMPLE    161 WORMS RUN     164 LOOP USAGE GUIDELINES             165 LOOP CONTROL 166 COMMON TYPES OF LOOPS          167 LOOPS VS IMPLICIT VECTORISATION         168 LOOPS VS. VECTOR EXAMPLE       169 REVIEW               170 EFFICIENCY & ERRORS    171 MOTIVATION     172 NUMERIC LIMITATIONS 173 Overflow             173 Underflow           174 ROUNDING & CANCELLATION     175 Rounding (Precision)        175 Cancelation (Order of Precedence)            175 WORK-AROUNDS FOR NUMERIC LIMITATIONS     176 For Instance: Comparing Two Numbers    176 ERRORS & DEBUGGING IN MATLAB           177 MATLAB DEBUGGER        178 ORDER OF AN ALGORITHM           179 TIMING IN MATLAB         180 PERFORMANCE PROFILING IN MATLAB    181 PROFILER EXAMPLE         182 GUIDELINES FOR EFFICIENCY       183 Special Cases and Redundant Checking     183 EFFICIENCY: REDUNDANCY & FUNCTIONS             184 Avoid Redundant Computations  184 Minimise Costly Function Usage  184 EFFICIENCY: ARRAYS       185 Minimise Array Referencing          185 Time vs Space     185 EFFICIENCY: LOOP TERMINATION             186 Avoid Late Termination of Loops (Needless Calculations)  186 EFFICIENCY – COMPLICATING ISSUES       187 LOOPING VS IMPLICIT VECTORISATION    188 TIMING EXAMPLE             189 SHELL1 – LOOPING WITHOUT EFFICIENCY              190 SHELL2 – LOOPING EFFICIENTLY 191 SHELL3 – IMPLICIT VECTORISATION          192 REVIEW               193 BASICS OF MATLAB FUNCTIONS  194 MOTIVATION     195 CALLING FUNCTIONS      196 OVERVIEW OF FUNCTION CALL & RETURN PROCESS          197 ON MATLAB'S IN-BUILT FUNCTIONS         198 WHAT MAKES A GOOD FUNCTION – SOFTWARE ENGINEERING PRINCIPLES              199 FUNCTION DECLARATION SYNTAX            200 A FIRST FUNCTION           201 EXAMPLE: TIMING OF FUNCTIONS            202 COMMENTS & FUNCTIONS           203 TOP-DOWN VIEW OF FUNCTION DESIGN               204 BOTTOM-UP VIEW OF FUNCTIONS            205 THE BLACK BOX PARADIGM          206 PARAMETERS – INPUT CHANNEL               207 RETURNED VALUES – OUTPUT CHANNEL               208 EXAMPLE – DESCRIPTION              209 EXAMPLE RESULTS           210 EXAMPLE – DAILYTEMPS               211 EXAMPLE – GETTEMPS    212 EXAMPLE – PROCESSTEMPS          213 EXAMPLE – TEMPTABLE  215 EXAMPLE – TEMPPLOT    217 REVIEW               218 FURTHER MATLAB FUNCTIONS   219 FORMAL VS. ACTUAL PARAMETERS & OUTPUTS   220 PARAMETER ASSOCIATION           221 PARAMETER ASSOCIATION EXAMPLE        222 PASS BY VALUE & PASS BY REFERENCE     224 PASS BY VALUE EXAMPLE              225 SCOPE RULES & WORKSPACES     226 SCOPE EXAMPLE               227 RUN-TIME STRUCTURE & THE STACK        228 RETURN, ERROR & WARNING      229 NARGIN & VARIABLE INPUTS       230 NARGOUT & VARIABLE OUTPUTS              231 NARGIN & NARGOUT EXAMPLE   232 GLOBAL VARIABLES         237 EFFICIENCY ISSUES          238 VARARGIN & VARARGOUT            239 SUB-FUNCTIONS              240 REVIEW               241 GRAPHICS & OTHER MATLAB FEATURES   242 MOTIVATION     243 2D GRAPHICS –PLOT        244 AXES & LABELS   245 PRINTING FIGURES          246 MULTIPLE FIGURES & SUB-PLOTS              247 OTHER 2D PLOTS              248 MULTIPLE PLOTS EXAMPLE           249 2D PLOTS - OUTPUT        252 ADDING TEXT     254 3D GRAPHICS – LINE        255 3D GRAPHICS – SURFACE               256 3D SURFACES EXAMPLE  257 POLYNOMIALS   258 INTERPOLATION & SPLINES          260 OPTIMISATION  262 INTEGRATION, DIFFERENTIATION & ORDINARY DIFFERENTIAL EQUATIONS             263 DATA-STRUCTURES         264 OBJECT ORIENTED           266 GUI DESIGN        267 TOOLBOXES        268 REVIEW               269 3D MAPS IN MATLAB, A CASE STUDY         270 PROBLEM SPECIFICATION             271 BASIC APPROACH & DATA STRUCTURE     272 MAP CREATION – ALGORITHM    274 MAP CREATION – EXAMPLE          275 CREATED MAP    276 MAP CREATION – CODE  277 MAP SMOOTHING/WEATHERING – ALGORITHM   279 MAP SMOOTHING – EXAMPLE     280 SMOOTHED MAP              281 MAP SMOOTHING – CODE            282 CRATER ADDITION – ALGORITHM              286 CRATER ADDITION – EXAMPLE    287 CRATERED MAP 288 CRATER ADDITION – CODE           289 RIVER ADDITION – ALGORITHM  292 RIVER ADDITION – EXAMPLE        293 MAP WITH RIVER             294 RIVER ADDITION – CODE               295 REVIEW               301 14           INTRODUCTION TO MATLAB • MATLAB is a powerful engineering environment and language -              powerful tool in engineering problem solving, data analysis, modeling and visualisation -              useful throughout your degree & used in other courses • Matlab is the vehicle used by this course to teach: -              problem solving by computer -              programming • Today’s lecture is introductory: -              “driving” Matlab -              illustration by example Reference:                                                                      For Engineers (Ch. 1 & 2)                                                             Mastering (Ch. 2, 3 & 34)                                                             User’s (Ch. 1, 23  & “A Quick Overview”) 14.1       What is Matlab? MATLAB:                            MATrix LABoratory • “High performance language for technical computing” • Interactive environment incorporating: -              programming language -              mathematical calculations (computation) -              data visualisation -              large number of inbuilt routines corresponding to many mathematical, engineering & science problems • Typical uses include: -              mathematical calculations -              engineering & scientific problem solving -              modeling & simulation -              data analysis & visualisation • Basic data structure (unit of computation) of an array/matrix -              vector operations -              natural expression of many engineering & scientific problems 14.2       History of Matlab • First introduced at Stanford Uni in 1979 -              initially an interactive shell from which to call FORTRAN routines • Later MathWorks was formed to market Matlab.

• An evolved language -              initially a collection of things felt to be necessary -              little top-down design -              “Like every other scripting language, Matlab began as a simple way to do powerful things, and it has become a not-so-simple way to do very powerful things.”, Webb & Wilson, Dr. Dobb’s Journal, Jan 1999 • Heavily used by universities world wide in their engineering and science faculties for teaching - (see links off course homepage) • Used by scientists and engineers in research, development and design 14.3       An Initial Example 14.3.1    Population Growth A colony of micro-organisms grows at such a rate its population doubles every 14.3 hours.

Given an initial population of 1000 organisms, calculate the population on an hourly basis for the period of two days.

Kami ada di Jakarta Selatan. KAMI MEMBERIKAN KURSUS MATLAB ONLINE - HUBUNGI MASTER ENGINEERING EXPERT (MEE) 081219449060.  Kami membuka kursus Matlab untuk pemula dan mahasiswa atau insinyur yang ingin memperdalam Matlab dan menerapkan dalam bidang teknikal, engineering, rekayasa, dsb. Format bimbingannya tugas-tugas yang bisa membantu Skripsi, Tesis, DISERTASI Bimbingan dilakukan secara online bisa lewat WA atau email Dijamin Bisa, atau bisa mengulang kembali. Kami juga dapat membantumembuatkan aplikasi atau program matlab/lainnya. Anda akan dilatih oleh Tim Profesional - HUBUNGI MASTER ENGINEERING EXPERT (MEE) 081219449060.   Email: kursusmatlab@gmail.com %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % amoeba.m %                           Matlab example to show problem solving. A %                           species doubles its population every 14.3 %                           hours. Given an initial population of 1000 %                           determine its population hourly %                           for the first two days % Author: Spike % Date:   10/2/1999 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% % Initial “setup” work % %%%%%%%%%%%%%%%%%%%%%%%% initialNum = 1000;                           % Initial population doublePeriod = 14.3;                      % Time in hours to                   % double population interval = 0:48;                  % Two days on an hourly                                                                                % basis %%%%%%%%%%%%%%%%%%%%%%% % Perform calculation % %%%%%%%%%%%%%%%%%%%%%%% population = initialNum*2.0.^(interval/doublePeriod); %%%%%%%%%%%%%%%%% % Output values % %%%%%%%%%%%%%%%%% combined = [interval ; population]; disp('Micro Organism Population'); disp('Hour            Population'); fprintf('%d\t%6.0f\n',combined); 1st Example (Cont) %%%%%%%%%%%%%%%% % Plot results % %%%%%%%%%%%%%%%% plot(interval,population,'-+'); xlabel('Hours'); ylabel(‘Population’); >> amoeba Micro Organism Population Hour      Population 0               1000 1               1050 2               1102 14             1971 15           2069 47             9759 48           10244 14.4       Some Features • Powerful computation engine • 3rd generation scripting (programming) language -              functions & argument passing -              iteration & slection • Implicitly vectorised operation • Powerful 2D and 3D plotting capabilities • Diverse and powerful inbuilt functions -              linear algebra -              polynomials -              fourier analysis -              integration & differentiation -              differential equations • Add-on toolboxes for specific problem domains • multi-media capabilities -              GUI builder -              Movies & sound 14.5       Running Matlab 14.5.1    Startup Under Unix Type matlab at the shell prompt:                $ matlab Or through the workspace menu on octarine 14.5.2    Startup Under Windows 95/98 From program menu select Student Edition of Matlab 14.5.3    To Exit Type quit (or exit) in the command window:                >> quit Or through Matlab’s File menu.

14.6       Help & Demos • Matlab has a extremely comprehensive help 14.6.1    Through the Command Window help       (e.g., help mean). Most general form of help. Without options lists all help topics. If option then provides help for that particular command. Always start with this.

lookfor  (e.g., lookfor 3d) Look for commands that are relevant to the topic specified.

Kami ada di Jakarta Selatan. KAMI MEMBERIKAN KURSUS MATLAB ONLINE - HUBUNGI MASTER ENGINEERING EXPERT (MEE) 081219449060.  Kami membuka kursus Matlab untuk pemula dan mahasiswa atau insinyur yang ingin memperdalam Matlab dan menerapkan dalam bidang teknikal, engineering, rekayasa, dsb. Format bimbingannya tugas-tugas yang bisa membantu Skripsi, Tesis, DISERTASI Bimbingan dilakukan secara online bisa lewat WA atau email Dijamin Bisa, atau bisa mengulang kembali. Kami juga dapat membantumembuatkan aplikasi atau program matlab/lainnya. Anda akan dilatih oleh Tim Profesional - HUBUNGI MASTER ENGINEERING EXPERT (MEE) 081219449060.   Email: kursusmatlab@gmail.com 14.6.2    Through the Help Window • Window based version of the above • Invoked from the help menu or by typing helpwin in the command window.

14.6.3    Through the Help Desk • Web browser based help, invoked from menu or by entering helpdesk or doc in the command window.

14.6.4    Demonstrations • Large series of examples: demo command 14.7       Controlling the Environment • A list of “meta” commands useful for controlling the Matlab environment and execution <cntrl-c>                             Terminate currently executing command. Useful if stuck in an infinite loop.

clc                                        Clear the screen and home the cursor.