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So ultimately in order to understand nature it may be necessary to have a deeper understanding. of mathematical relationships But the real reason is that the subject is enjoyable and although we. humans cut nature up in different ways and we have different courses in different departments such. compartmentalization is really artificial and we should take our intellectual pleasures where we find. them Richard Feynman The Laws of Thermodynamics, Why a preface you may ask Isn t that just a mere exposition of a raison d e tre of an. author s choice of material preferences biases teaching philosophy etc To a large extent I. can answer in the affirmative to that A preface ought to be personal Indeed what you will. see in the various chapters of these notes represents how I perceive computational physics. should be taught, This set of lecture notes serves the scope of presenting to you and train you in an algorith. mic approach to problems in the sciences represented here by the unity of three disciplines. physics mathematics and informatics This trinity outlines the emerging field of computa. tional physics, Our insight in a physical system combined with numerical mathematics gives us the rules. for setting up an algorithm viz a set of rules for solving a particular problem Our under. standing of the physical system under study is obviously gauged by the natural laws at play. the initial conditions boundary conditions and other external constraints which influence the. given system Having spelled out the physics for example in the form of a set of coupled. partial differential equations we need efficient numerical methods in order to set up the final. algorithm This algorithm is in turn coded into a computer program and executed on available. computing facilities To develop such an algorithmic approach you will be exposed to several. physics cases spanning from the classical pendulum to quantum mechanical systems We will. also present some of the most popular algorithms from numerical mathematics used to solve. a plethora of problems in the sciences Finally we will codify these algorithms using some of. the most widely used programming languages presently C C and Fortran and its most. recent standard Fortran 20031 However a high level and fully object oriented language like. Python is now emerging as a good alternative although C and Fortran still outperform. Python when it comes to computational speed In this text we offer an approach where one. can write all programs in C C or Fortran We will also show you how to develop large. programs in Python interfacing C and or Fortran functions for those parts of the program. which are CPU intensive Such an approach allows you to structure the flow of data in a high. level language like Python while tasks of a mere repetitive and CPU intensive nature are left. to low level languages like C or Fortran Python allows you also to smoothly interface your. program with other software such as plotting programs or operating system instructions. Throughout this text we refer to Fortran 2003 as Fortran implying the latest standard Fortran 2008 will. only add minor changes to Fortran 2003,vi Preface, A typical Python program you may end up writing contains everything from compiling and. running your codes to preparing the body of a file for writing up your report. Computer simulations are nowadays an integral part of contemporary basic and applied re. search in the sciences Computation is becoming as important as theory and experiment In. physics computational physics theoretical physics and experimental physics are all equally. important in our daily research and studies of physical systems Physics is the unity of theory. experiment and computation2 Moreover the ability to compute forms part of the essen. tial repertoire of research scientists Several new fields within computational science have. emerged and strengthened their positions in the last years such as computational materials. science bioinformatics computational mathematics and mechanics computational chemistry. and physics and so forth just to mention a few These fields underscore the importance of sim. ulations as a means to gain novel insights into physical systems especially for those cases. where no analytical solutions can be found or an experiment is too complicated or expensive. to carry out To be able to simulate large quantal systems with many degrees of freedom. such as strongly interacting electrons in a quantum dot will be of great importance for future. directions in novel fields like nano techonology This ability often combines knowledge from. many different subjects in our case essentially from the physical sciences numerical math. ematics computing languages topics from high performace computing and some knowledge. of computers, In 1999 when I started this course at the department of physics in Oslo computational.

physics and computational science in general were still perceived by the majority of physi. cists and scientists as topics dealing with just mere tools and number crunching and not as. subjects of their own The computational background of most students enlisting for the course. on computational physics could span from dedicated hackers and computer freaks to people. who basically had never used a PC The majority of undergraduate and graduate students. had a very rudimentary knowledge of computational techniques and methods Questions like. do you know of better methods for numerical integration than the trapezoidal rule were not. uncommon I do happen to know of colleagues who applied for time at a supercomputing. centre because they needed to invert matrices of the size of 104 104 since they were using. the trapezoidal rule to compute integrals With Gaussian quadrature this dimensionality was. easily reduced to matrix problems of the size of 102 102 with much better precision. Less than ten years later most students have now been exposed to a fairly uniform introduc. tion to computers basic programming skills and use of numerical exercises Practically every. undergraduate student in physics has now made a Matlab or Maple simulation of for example. the pendulum with or without chaotic motion Nowadays most of you are familiar through. various undergraduate courses in physics and mathematics with interpreted languages such. as Maple Matlab and or Mathematica In addition the interest in scripting languages such. as Python or Perl has increased considerably in recent years The modern programmer would. typically combine several tools computing environments and programming languages A typ. ical example is the following Suppose you are working on a project which demands extensive. visualizations of the results To obtain these results that is to solve a physics problems like ob. taining the density profile of a Bose Einstein condensate you need however a program which. is fairly fast when computational speed matters In this case you would most likely write a. We mentioned previously the trinity of physics mathematics and informatics Viewing physics as the trinity. of theory experiment and simulations is yet another example It is obviously tempting to go beyond the. sciences History shows that triunes trinities and for example triple deities permeate the Indo European. cultures and probably all human cultures from the ancient Celts and Hindus to modern days The ancient. Celts revered many such trinues their world was divided into earth sea and air nature was divided in animal. vegetable and mineral and the cardinal colours were red yellow and blue just to mention a few As a curious. digression it was a Gaulish Celt Hilary philosopher and bishop of Poitiers AD 315 367 in his work De. Trinitate who formulated the Holy Trinity concept of Christianity perhaps in order to accomodate millenia of. human divination practice,Preface vii, high performance computing program using Monte Carlo methods in languages which are. tailored for that These are represented by programming languages like Fortran and C. However to visualize the results you would find interpreted languages like Matlab or script. ing languages like Python extremely suitable for your tasks You will therefore end up writing. for example a script in Matlab which calls a Fortran or C program where the number. crunching is done and then visualize the results of say a wave equation solver via Matlab s. large library of visualization tools Alternatively you could organize everything into a Python. or Perl script which does everything for you calls the Fortran and or C programs and. performs the visualization in Matlab or Python Used correctly these tools spanning from. scripting languages to high performance computing languages and vizualization programs. speed up your capability to solve complicated problems Being multilingual is thus an advan. tage which not only applies to our globalized modern society but to computing environments. as well This text shows you how to use C and Fortran as programming languages. There is however more to the picture than meets the eye Although interpreted languages. like Matlab Mathematica and Maple allow you nowadays to solve very complicated problems. and high level languages like Python can be used to solve computational problems compu. tational speed and the capability to write an efficient code are topics which still do matter. To this end the majority of scientists still use languages like C and Fortran to solve sci. entific problems When you embark on a master or PhD thesis you will most likely meet. these high performance computing languages This course emphasizes thus the use of pro. gramming languages like Fortran Python and C instead of interpreted ones like Matlab. or Maple You should however note that there are still large differences in computer time be. tween for example numerical Python and a corresponding C program for many numerical. applications in the physical sciences with a code in C or Fortran being the fastest. Computational speed is not the only reason for this choice of programming languages An. other important reason is that we feel that at a certain stage one needs to have some insights. into the algorithm used its stability conditions possible pitfalls like loss of precision ranges. of applicability the possibility to improve the algorithm and taylor it to special purposes etc. etc One of our major aims here is to present to you what we would dub the algorithmic. approach a set of rules for doing mathematics or a precise description of how to solve a. problem To device an algorithm and thereafter write a code for solving physics problems. is a marvelous way of gaining insight into complicated physical systems The algorithm you. end up writing reflects in essentially all cases your own understanding of the physics and. the mathematics the way you express yourself of the problem We do therefore devote quite. some space to the algorithms behind various functions presented in the text Especially in. sight into how errors propagate and how to avoid them is a topic we would like you to pay. special attention to Only then can you avoid problems like underflow overflow and loss of. precision Such a control is not always achievable with interpreted languages and canned. functions where the underlying algorithm and or code is not easily accesible Although we. will at various stages recommend the use of library routines for say linear algebra3 our. belief is that one should understand what the given function does at least to have a mere. idea With such a starting point we strongly believe that it can be easier to develope more. complicated programs on your own using Fortran C or Python. We have several other aims as well namely, We would like to give you an opportunity to gain a deeper understanding of the physics. you have learned in other courses In most courses one is normally confronted with simple. systems which provide exact solutions and mimic to a certain extent the realistic cases. Many are however the comments like why can t we do something else than the particle in. Such library functions are often taylored to a given machine s architecture and should accordingly run faster. than user provided ones,viii Preface, a box potential In several of the projects we hope to present some more realistic cases. to solve by various numerical methods This also means that we wish to give examples of. how physics can be applied in a much broader context than it is discussed in the traditional. physics undergraduate curriculum, To encourage you to discover physics in a way similar to how researchers learn in the. context of research, Hopefully also to introduce numerical methods and new areas of physics that can be stud.

ied with the methods discussed, To teach structured programming in the context of doing science. The projects we propose are meant to mimic to a certain extent the situation encountered. during a thesis or project work You will tipically have at your disposal 2 3 weeks to solve. numerically a given project In so doing you may need to do a literature study as well. Finally we would like you to write a report for every project. Our overall goal is to encourage you to learn about science through experience and by asking. questions Our objective is always understanding and the purpose of computing is further. insight not mere numbers Simulations can often be considered as experiments Rerunning. a simulation need not be as costly as rerunning an experiment. Python is now emerging as a good alternative although C and Fortran still outperform Python when it comes to computational speed In this text we offer an approach where one can write all programs in C C or Fortran We will also show you how to develop large programs in Python interfacing C and or Fortran functions for those parts of the program which are CPU intensive Such an approach

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