PHS3XXX Computational Physics

Transcription

1 1 PHS3XXX Computational Physics Lecturer: Mike Wheatland Lecture and lab schedule (SNH Learning Studio 4003) 1 There are 10 weeks of lectures and labs Lectures: Mon 9am Labs: Wed 1pm-3pm The labs start in week 1 First lab: this Wednesday Online pre-lab quizzes to be completed each week in elearning consisting of 5 multiple-choice Qs 1 For more details see the outline.

2 2 Topics covered 2 Week 1: Review of MATLAB, numerical error Ordinary Differential Equations (ODEs) Week 2: Euler and midpoint methods projectile motion Week 3: Verlet methods planetary motion Week 4: Runge-Kutta motion of a pendulum Week 5: Relaxation method steady-state quantum physics Partial Differential Equations (PDEs) Week 6: FTCS heat diffusion Week 7: Lax, Lax-Wendroff propagation of waves Week 8: Nonlinear Lax-Wendroff traffic flow Week 9: Jacobi relaxation etc. electrostatics Week 10: Crank-Nicolson time-dependent quantum physics 2 For more details see the outline.

3 3 Computational Physics Week 1 Overview of MATLAB Types of numerical error floating point errors representation of numbers truncation error numerical method Review of MATLAB (assumed knowledge) elements of the language array/matrix features a script with examples (matlab_examples.m) an example function (dice.m) some tips on debugging a list of resources Codes: matlab_examples.m, dice.m

4 4 Overview of MATLAB MATLAB: a language and programming environment with integrated graphics, extensive function libraries emphasizing array (vector/matrix) operations with dynamic allocation of memory for arrays MATLAB may be used interactively or via.m files interactively: at the command line.m files: scripts (sets of commands) or functions MATLAB codes are interpreted they may be slow compared with compiled code (C, Fortran) MATLAB has some unusual features e.g. data are always matrices (arrays) scalars are 1 1 arrays e.g. type defaults to data double precision floating point

5 5 Numerical Errors: 1. Floating point errors Floating point 3 is an approximate representation of the reals IEEE double precision is the standard used by MATLAB this uses base 2 with 64 bits (0s or 1s) per number there are infinitely many reals but a finite number of floats the approximation produces range error and rounding error Range error is due to there being largest and smallest floats underflow: x replaced by 0 overflow: x replaced by Inf Rounding error: is due to the discrete spaces between floats results are rounded to the nearest float the fractional error is 1 2 ɛ m ɛ m = is the machine epsilon/precision this is the built-in constant eps in MATLAB but this may be important, e.g *eps-1 gives zero Undefined operations (e.g. 0/0) produce NaN (not a number) NaNs propagate (e.g. NaN+x gives NaN for all x) 3 For more info see: What Every Computer Scientist Should Know About Floating-Point Arithmetic by David Goldberg and 20.1 of Numerical Recipes or OLEx1622 Numbers and Numerics

6 6 Numerical Errors: 2. Truncation errors Truncation error: approximations in the numerical method e.g. the forward difference approximation to the derivative: f f (x + h) f (x) (x) (in general for h x) (1) h to quantify the approximation consider the Taylor series: f (x + h) = f (x) + hf (x) + 1 2! h2 f (x) +... (2) f f (x + h) f (x) (x) = 1 h 2 hf (x) +... (3) the largest error term is proportional to h and we write: f (x) = f (x + h) f (x) h we say the local truncation error is order h Truncation error limits accuracy i.e. how close you are to the correct answer + O(h) (4)

7 7 Review of MATLAB (assumed knowledge) General: (The MATLAB examples in this section are in matlab_examples.m) y=[1 2 3]; z=y ; disp(y); disp(z); y^2 % A common error y.^2 z*y v=1:10 w=pi format long; w format long; w These examples illustrate: matrix operations are used by default the product of L M and M N arrays is size L N the. (full stop) enforces element-by element operations there are handy built-in constants the full number of digits of precision is seen with format long row and column vectors are different row and column vectors of length L: 1 L and L 1 arrays to convert between them: the transpose operator (a prime) a semi-colon at the end of a line suppresses output the : (colon) operator is a simple way to generate vectors

8 8 Program control: Conditionals e.g. if-else statements x=input( Value of integer x? ); if x==0 disp( x is equal to zero ); else if x>0 disp( x is positive ); else disp( x is negative ); end end note the use of the relational operators == and > note also the match between the indenting and the logic Other control structures we will often use for loops e.g. tau=0.1; LENGTH=10; for n=1:length time(n)=tau*n; end the loop should include only statements depending on n you need also to be familiar with nested loops we will occasionally use while and switch

9 9 Array operations: MATLAB started as a MATrix LABoratory operations are array (matrix) operations by default array operations are relatively fast rather than our loop use: tau=0.1; LENGTH=50000; time2=tau*(1:length); MATLAB has built-in array generation functions linspace() and logspace() generate row vectors ones(), zeros, eye(), diag() generate simple arrays The operator find is very useful it locates indices of array elements matching a condition 0.5 x1=-0.5:0.05:0.5; x2=x1; L=length(x1); x2(find(abs(x1)<0.25))=0; plot(1:l,x1, o,1:l,x2, + ); xlabel( Array index ); ylabel( Array value ); Array value Array index

10 10 The dot operator forces element-wise operations e.g. x.^2 this command squares the components of the vector x whereas x^2 gives an error if x is not a square array The colon operator is useful to define vectors the colon also identifies all indices in a dimension e.g. x=ones(5); % x is a 5 x 5 matrix of ones y=x(:,1); % y is a 5 x 1 vector of ones the semi-colon can define column vectors e.g. x=[1;2;3]; as well as suppressing output (at the end of a statement)

11 11 Linear algebra: An example of solution of a linear system: A=rand(3,3) % This is the same as rand(3) b=rand(3,1) x=a\b res=a*x-b max(res) % Compare with the machine precision/epsilon eps... disp([ eps =,num2str(eps)]); this example sets up and solves a 3 3 linear system Ax = b where A and b contain random numbers between 0 and 1 the x=a\b command solves the system in MATLAB the residual res is comparable to the machine precision 4 the determinant and the eigenvectors of A are given by eig(): det(a) eig(a) [V,D]=eig(A) 4 The fractional spacing of floating point numbers see later in this lecture.

12 12 Plotting: The basic command is plot(x,y): % Plotting example x=0:0.1:2*pi; f1=sin(x); f2=cos(x); plot(x,f1,x,f2); xlabel( x ) ylabel( sin(x), cos(x) ) 1 A simple plot in MATLAB... always label your axes! 0.5 y x

13 13 Plotting: Also: semilogx(x,y), semilogy(x,y), loglog(x,y) use these if there is a large range of values in x/y/both use semilogy(x,y) if you have exponential behaviour in y use loglog(x,y) if you have power-law behaviour For plotting a surface use e.g. contourf or surfc 1 colormap(hot); y=-pi:0.1:pi; z=sin(x) *cos(y); surfc(x,y,z); axis([min(x) max(x) min(y)... max(y)]); xlabel( x ); ylabel( y ); zlabel( z ); z y x 4 6

14 14 Writing functions in MATLAB: A simple function (dice.m) % dice.m % Function returning a specified number of random integers, each % from one to six. % % Inputs: % ndice - number of random integers required % Outputs: % ran6 - row vector with the integers function ran6 = dice(ndice) ran6=floor(6*rand(1,ndice))+1; this code goes in the separate file dice.m example of interactive use of dice.m in MATLAB: N=10000; test=dice(n); x=1:6; hist(test,x) xlabel( Value ); ylabel( Number ); % Generate throws of a die % Horizontal axis values % Histogram of results

15 Number Value Example of defining and calling an anonymous function x=[0:0.1:10]; cauchy=@(x,a) 1./pi./(1+(x-a).^2); plot(x,cauchy(x,5)); xlabel( x ); ylabel( y ); y x this code can be written at the command line or put a script

16 16 MATLAB dos and don ts: Keep your lines short use the ellipsis... to break long lines also shorten lines by defining sub-expressions separately Understand the basics of order of operations operations generally occur L to R use this correctly and avoid over-bracketing Use the e scientific notation in your code e.g. e=1.602e+19 in your code but use in writing Write clear code include comments as required

17 17 Resources for learning MATLAB: Read the documentation available within MATLAB: doc plot opens a graphical browser with information whos x in-terminal information on a variable help plot in-terminal information on a command Use the e-texts on the Resources page on elearning: Hahn & Valentine Essential MATLAB for Engineers Chapter 1 of Garcia Numerical Methods for Physics Refer to the MATLAB web site (mathworks.com) as well as other web sites on MATLAB You need to be on top of the MATLAB programming! Lab 1 is a MATLAB review/refresher based on Chapter 1 of Hahn & Valentine the lab question sheets are all on elearning

18 18 Mike s debugging tips: 1. Pay attention to error messages they are meaningful think about what the messages mean 2. Remove unnecessary parts of your code strip it back to simple clear code 3. Check variable values by printing them to screen in interactive mode type the variable name for variables in functions use e.g. disp([ x =,num2str(x)]) or remove semicolons! if there are large numbers of values, plot them think about what the values mean 4. Halt program execution prematurely when there are errors halt a loop with a break statement halt a function with a return statement use this to determine when an error occurs 5. Think again: the machine is just following your instructions

19 19 Appendix: Codes % matlab examples.m % % An example of a script: the MATLAB examples from lecture 1. This % script is not intended to be run independently (although it will % run without errors excepting an intended error). % Clear memory and only show a few digits clear all; format short; % Data x=1; whos x; y=[1 2 3]; z=y ; disp(y); disp(z); % Array operations, constants, digits of precision yˆ2 % A common error y.ˆ2 z y v=1:10 w=pi format long w format short % Use of ellipsis to line break x=... 2;

20 20 % Graphical help browser doc % Program control % If else statements x=input( Value of integer x? ); if x==0 disp( x is equal to zero ); else if x>0 disp( x is positive ); else disp( x is negative ); end end % Loops tau=0.1; LENGTH=50000; for n=1:length time(n)=tau n; end % Array version use this! time2=tau (1:LENGTH); % Find elements of an array matching a condition find((time > 1) & (time < 1.2)) % Example in lecture x1= 0.5:0.05:0.5; x2=x1; L=length(x1); x2(find(abs(x1)<0.25))=0;

21 21 plot(1:l,x1, o,1:l,x2, + ); xlabel( Array index ); ylabel( Array value ); % Matrix algebra A=rand(3,3) % This is the same as rand(3) b=rand(3,1) x=a\b res=a x b max(res) % Compare with the machine precision/epsilon eps... disp([ eps =,num2str(eps)]); % What is the machine precision? disp([ eps 1 =,num2str(1+0.5 eps 1)]); % Example of calculating determinant, eigenvectors and values det(a) eig(a) [V,D]=eig(A) % 1 D plotting example x=0:0.1:2 pi; f1=sin(x); f2=cos(x); plot(x,f1,x,f2); xlabel( x ) ylabel( y ) title( A simple plot in MATLAB... always label your axes! ); % 2 D plotting examples colormap(hot); % Avoid the rainbow colormap! y= pi:0.1:pi; z=sin(x) cos(y);

22 22 % Filled contours... contourf(x,y,z); axis([min(x) max(x) min(y)... max(y)]); xlabel( x ); ylabel( y ); % Or a surface surfc(x,y,z); axis([min(x) max(x) min(y) max(y)]); xlabel( x ); ylabel( y ); zlabel( z ); % Functions see the file dice.m N=10000; test=dice(n); % Generate throws of a die x=1:6; % Horizontal axis values hist(test,x) % Histogram of results xlabel( Value ); ylabel( Number ); % Defining and calling an anonymous function x=[0:0.1:10]; cauchy=@(x,a) 1./pi./(1+(x a).ˆ2); plot(x,cauchy(x,5)); xlabel( x ); ylabel( y );

23 23 % dice.m % Function returning a specified number of random integers, each % from one to six. % % Inputs: % ndice number of random integers required % Outputs: % ran6 row vector with the integers function ran6 = dice(ndice) ran6=floor(6 rand(1,ndice))+1;