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The Profiler Preallocate arrays Vectorize the operation MEX

Matlab

March 26, 2013

Matlab

A numerical computing environment and fourth-generationprogramming language.

Matrix manipulations, Plotting of functions and data, Implementationof algorithms, Creation of user interfaces, Interfacing with programswritten in other languages, including C, C++, Java, and Fortran

Xin Zhang (STAT 598) Lecture 11 March 26th,2013 2 / 30

Display Windows

Command WindowEnter commands after the ”>>” symbolResults show here

Command History Window display commands used.

Windows which list all the variables you are using.

Graphic (Figure) WindowDisplays plots and graphsCreated in response to graphics commands.

M-file editor/debugger windowCreate and edit scripts of commands called M-files.


Getting Help

Type one of the following commands in the command window.

helpList all the help topics

help commandProvides help for the specified command


Basic Commands:

for FOR variable = expr, statement, ..., statement END

if IF expression

statements

ELSEIF expression

statements

ELSE

statements

END

while WHILE expression

statements

END

switch SWITCH switch_expr

CASE case_expr,

statement, ..., statement

CASE {case_expr1, case_expr2, case_expr3,...}


...

OTHERWISE,


END

Plotting Commands:

x, y, z : vectors

plot(x,y) : 2-d plot

mesh(x, y, z) 3-d plot xlabel(’label for x axis’)

title(’plot title’)

2

Vectors & Matrix

Create a vector(or Matrix)>> a=[0 2 4 6 8 10]>> a=0:2:10>> A=[1 2 3; 4 5 6; 7 8 9]>> eye(n)

Matrix operations+, -, *.*inv(A)A.^p


MA/STAT 598W Lecture 1: MATLAB

Basics:

A = [1, 2, 3; 3, 4, 5; 5, 6, 7 ] Matrix Definition

A (1, 2) (1,2)-element

A (1, 2:end) First line, elements from 2 - end

Aˆ2 A square

A + 2 Adds 2 to all the elements of A

A.̂ 2 Squares each element of A

A’ Transpose of A

diag (A) Diagonal of A

triu (A) Upper triangular part of A

rank(A) Rank of A

inv(A) Inverse of A

eig(A) Eigenvalues of A

zeros(n,m) nxm matrix of zeros

ones(n,m) nxm matrix of ones

eye(n) nxn Identity matrix

A±B Matrix addition/subtraction

A*B Matrix multiplication

A.*B Element-by-element multiplication

a = 1:n Vector with elements from 1 to n

b = 1:k:n Vector with elements from 1 to n with step k

1

Solutions to systems of linear equations

Example: a system of 3 linear equations with 3 unknows.

3x1 + 2x2 � x3 = 10

�x1 + 3x3 + 2x3 = 5

x1 � x3 � x3 = �1

(1)

>> A=[3 2 -1; -1 3 2;1 -1 -1];

>> b=[10; 5; -1];

>> x=inv(A)*b



Vectorized Computations

MATLAB is specifically designed to operate on vectors andmatrices, so it is usually quicker to perform operations on vectorsor matrices, rather than using a loop. For example:tic;index=0;for time=0:0.001:60000;index=index+1;waveForm(index)=cos(time);end;toc;Elapsed time is 17.873571 seconds.

tic;time=0:0.001:60000;waveForm=cos(time);tocElapsed time is 1.145038 seconds.


Vectorized Computations with per-element operators

function d = minDistance(x,y,z)%find the distance between a set of points and the originnPoints = length(x);d = zeros(nPoints,1); % Preallocatefor k = 1:nPoints % Compute distance for every pointd(k) = sqrt(x(k)2 + y(k)2 + z(k)2);endd = min(d); % Get the minimum distance

Vectorize the code by:d = sqrt(x .^2 + y .^2 + z .^2);


Some useful operations for vector

any() ) returns true if any element is non-zero.

size() ) returns a vector containing the number of elementsin an array along each dimension

find() ) returns the indices of any non-zero elements. To getthe non-zero values themselves you can use something likea(find(a))

cumsum() ) returns a vector containing the cumulative sumof the elements in its argument vector, e.g. cumsum([0:5])returns [0 1 3 6 10 15].

sum() ) returns the sum of all the elements in a vector, e.g.sum([0:5]) returns 15.

min, max, repmat, meshgrid, di↵, prod, cumprod, accumarray.


Vectorized Logic

>> [1,5,3] < [2,2,4]ans = 1 0 1>> find([1,5,3] < [2,2,4]) % Find indices of nonzero elementsans = 1 3>> any([1,5,3] < [2,2,4])% True if any element of a vector isnonzero (or per-column for a matrix)ans = 1>> all([1,5,3] < [2,2,4]) %True if all elements of a vector arenonzero (or per-column for a matrix)ans = 0

Vectorized logic also works on arrays.>> find(eye(3) == 1) % Find which elements == 1 in the 3x3identity matrixans = 1 5 9 % (element locations given as indices)

Functions

MATLAB functions are similar to C functions.

MATLAB programs are stored as plain text in m-files. Each m-filecontains exactly one MATLAB function. Thus, a collection ofMATLAB functions can lead to a large number of relatively small files.

function [y1,...,yN] = myfun(x1,...,xM)

Exercise: Return the sum and di↵erence of two variables



Inlining Simple Functions

Every time an M-file function is called, the Matlab interpreterincurs some overhead. Additionally, many M-file functions beginwith conditional code that checks the input arguments for errors ordetermines the mode of operation.Example:y = zeros(size(x)); % Preallocatefor k = 3:length(x)-2

y(k) = median(x(k-2:k+2));end

Inlining a function means to replace a call to the function with thefunction code itself. The median function can be inclined.

tmp = sort(x(k-2:k+2));y(k) = tmp(3);

Matlab Programing Example

ReferenceAn Algorithmic Introduction to the Numerical Simulation of StochasticDi↵erential Equations

AuthorDesmond J. Higham


Simulation of Brownian motion

W(0)=0 with Probability 1

W

j

= W

j�1 + dW

j

, j = 1, 2, . . . ,N (2)

Each dWj is an independent random variable of the formp�tN(0, 1)]

random number generator randn – each call to randu produces an”pseudorandom” number

Notice: Matlab starts arrays from index 1 and not index 0.


Simulation of Brownian motion

%BPATH1 Browniam path simulation

randn(’state’,100) % set the state of randnT=1; N=500; dt=T/N;dW=zeros(1,N); % preallocate arraysW=zeros(1,N); % for e�ciency

dW(1)=sqrt(dt)*randn; % first approximation outside the loopW(1)=dW(1); % since W(0)=0 is not allowedfor j=2:NdW(j)=sqrt(dt)*randn; % general incrementW(j)=W(j-1)+dW(j);

end

plot([0:dt:T],[0,W],’r-’); %plot W against txlabel(’t’,’FomtSize’,16)ylabel(’W(t),’FontSize’,16,’Rotation’,0)


Simulation of Brownian motion:vectorized

Instead of loop, we can perform the same computation more elegantlyand e�ciently using higher level ”vectorized” commands.

Function randn(i,N) creates a 1-by-N array of independent N(0,1)samples.

Function cumsum() computes the cummulative sum of its argument.


Simulation of Brownian motion:vectorized

%BPATH2 Brownian path simulation: vectorized

randn(’state’,100) % set the state of randnT = 1; N = 500; dt = T/N;

dW = sqrt(dt)*randn(1,N); % incrementsW = cumsum(dW); % cumulative sum

plot([0:dt:T],[0,W],’r-’) % plot W against txlabel(’t’,’FontSize’,16)ylabel(’W(t)’,’FontSize’,16,’Rotation’,0)


Approximation of Stochastic integrals

Given a suitable function h, consider two sums

Ito integralN�1X

i=0

h(tj

)(W (tj+1)�W (t

j

)) (3)

Stratonovich integral

N�1X

i=0

h(t

j

+ t

j+1

2)(W (t

j+1)�W (tj

)) (4)


Example when h = W

%STINT Approximate stochastic integrals%% Ito and Stratonovich integrals of WdW

randn(state,100) % set the state of randnT = 1; N = 5 0; dt = T/N;

dW = sqrt(dt)*randn(1,N); % incrementsW = cumsum(dW); % cumulative sum

ito = sum([0,W (1 : end � 1)]. ⇤ dW )strat = sum((0.5 ⇤ ([0,W (1 :end � 1)] +W ) + 0.5 ⇤ sqrt(dt) ⇤ randn(1,N)). ⇤ dW )

itoerr = abs(ito � 0.5 ⇤ (W (end)^2� T ))straterr = abs(strat � 0.5 ⇤W (end)^2)


Euler-Maruyama Method

A SDE is given by

Integral form

X (t) = X0 +

Zt

0f (X (s))ds +

Zt

0g(X (s))dW (s), 0 t T . (5)

Or di↵erential equation form

dX (t) = f (X (t))dt + g(X (t))dW (t), X (0) = X0, 0 t T .(6)

The Euler-Maruyama(EM) method takes the form

X

j

= X

j�1+f (Xj�1)�t+g(X

j�1)(W (⌧j

)�W (⌧j�1)), j = 1, 2, . . . , L.

(7)



Example: f (X ) = �X and g = µX

dX (t) = �X (t)dt + µX (t)dW (t) (8)

The exact solution to this SDE is

X (t) = X (0)exp((�� 1

2µ2)t + µW (t)) (9)

� = 2, µ = 1 and X(0)=1



%EM Euler-Maruyama method on linear SDE%% SDE is dX = lambda ⇤ Xdt +mu ⇤ XdW , X(0) = Xzero,% where lambda = 2, mu = 1 and Xzero = 1.%% Discretized Brownian path over [0,1] has dt = 2^ � 8.% Euler-Maruyama uses timestep R ⇤ dt.

randn(state,100)lambda = 2; mu = 1; Xzero = 1; % problem parametersT = 1;N = 2^8; dt = 1/N;dW = sqrt(dt) ⇤ randn(1,N); % Brownian incrementsW = cumsum(dW); % discretized Brownian pathXtrue = Xzero ⇤ exp((lambda� 0.5 ⇤mu

^2) ⇤ ([dt : dt : T ]) +mu ⇤W );plot([0:dt:T],[Xzero,Xtrue],’m-’), hold on



R = 4;Dt = R ⇤ dt; L = N/R ; % L EM steps of size Dt = R*dtXem = zeros(1,L); % preallocate for e�ciencyXtemp = Xzero;for j = 1:LWinc = sum(dW (R ⇤ (j � 1) + 1 : R ⇤ j));Xtemp = Xtemp + Dt ⇤ lambda ⇤ Xtemp +mu ⇤ Xtemp ⇤Winc ;Xem(j) = Xtemp;

end

plot([0:Dt:T],[Xzero,Xem],’r–*’), hold o↵xlabel(’t’,’FontSize’,12)ylabel(’X’,’FontSize’,16,’Rotation’,0,’HorizontalAlignment’,’right’)

emerr = abs(Xem(end)� Xtrue(end))


Strong Convergence of the EM Method

A method is said to have strong order of convergence equal to � if thereexists a constant C such that

E |Xn

� X (⌧)| C�t

� (10)

for any fixed ⌧ = n�t 2 [0,T ] and �t e�ciently small.

We lete

Strong

�t

:= E |XL

� X (T )|, L�t = T (11)

The inequality (10) is equivalent to

e

Strong

�t

C�t

� (12)

for su�ciently small �t.



Let’s look at the Strong convergence of EM for the SDE in our lastexample em.m with order � = 1/2.

Step 1: Compute 1000 di↵erent discretized Brownian paths over [0,1]with �t = 2�9.

Step 2: For each path, EM is applied with 5 di↵erent stepsize:�t = 2p�1�t for 1 p 5.

Step 3: Compute the endpoint error in the sth sample path for thepth stepsize and store it in Xerr(s,p)

Step 4: mean(Xeer(s,p)) will give us the approximation to e

Strong

�t

for5 di↵erent stepsize.



If the inequality holds with approximation equality, then, taking logs, wehave

log eStrong�t

⇡ logC + 1/2 log�t (13)



%EMSTRONG Test strong convergence of Euler-Maruyama%% Solves dX = lambda ⇤ Xdt +mu ⇤ XdW , X(0) = Xzero,% where lambda = 2, mu = 1 and Xzer0 = 1.%% Discretized Brownian path over [0,1] has dt = 2^ � 9.% E-M uses 5di↵erent timesteps: 16dt, 8dt, 4dt, 2dt, dt.% Examine strong convergence at T = 1 : E |X

L

� X (T )|.

randn(state,100)lambda = 2; mu = 1; Xzero = 1; % problem parametersT = 1; N = 2^9; dt = T/N;M = 1000; % number of paths sampled



Xerr = zeros(M,5); % preallocate arrayfor s = 1:M, % sample over discrete Brownian pathsdW = sqrt(dt) ⇤ randn(1,N); % Brownian incrementsW = cumsum(dW); % discrete Brownian pathXtrue = Xzero ⇤ exp((lambda� 0.5 ⇤mu

^2) +mu ⇤W (end));for p = 1:5R = 2(p � 1);Dt = R ⇤ dt; L = N/R ;Xtemp = Xzero;for j = 1:LWinc = sum(dW (R ⇤ (j � 1) + 1 : R ⇤ j));Xtemp = Xtemp + Dt ⇤ lambda ⇤ Xtemp +mu ⇤ Xtemp ⇤Winc ;

endXerr(s, p) = abs(Xtemp � Xtrue); % store the error at t = 1

endend



Dtvals = dt ⇤ (2.^([0 : 4]));loglog(Dtvals,mean(Xerr),0 b ⇤ �0); hold onloglog(Dtvals, (Dtvals.^(.5)),0 r ��0); hold o↵ % reference slope of 1/2axis([1e-3 1e-1 1e-4 1])xlabel(’\Delta t’);ylabel(’Sample average of|X (T )� X

L

|’)title(emstrong.m,FontSize,10)

%%%% Least squares fit of error = C ⇤ Dtq %%%%A = [ones(5, 1), log(Dtvals)]; rhs = log(mean(Xerr));sol = A \rhs; q = sol(2)resid = norm(A ⇤ sol � rhs)


Weak Convergence of the EM Method

A method is said to have weak order of convergence equal to � if thereexists a constant C such that for all functions p in some class

|Ep(Xn

)� Ep(X (⌧))| C�t

� (14)

for any fixed ⌧ = n�t 2 [0,T ] and �t e�ciently small. We focus on thecase where p is identity function

We lete

Weak

�t

:= |EXL

� EX (T )|, L�t = T (15)

The inequality (14) is equivalent to

e

Weak

�t

C�t

� (16)

for su�ciently small �t.



Let’s look at the Weak convergence of EM for the SDE in our lastexample em.m with order � = 1.

Step 1: Compute 1000 di↵erent discretized Brownian paths over [0,1]with �t = 2�9.

Step 2: For each path, EM is applied with 5 di↵erent stepsize:�t = 2p�1�t for 1 p 5.

Step 3: Compute the endpoint error in the sth sample path for thepth stepsize and store it in Xerr(s,p)

Step 4: mean(Xeer(s,p)) will give us the approximation to e

Strong

�t

for5 di↵erent stepsize.



If the inequality holds with approximation equality, then, taking logs, wehave

log eStrong�t

⇡ logC + log�t (17)



%EMWEAK Test weak convergence of Euler-Maruyama%% Solves dX = lambda ⇤ Xdt +mu ⇤ XdW , X(0) = Xzero,% where lambda = 2, mu = 1 and Xzer0 = 1.%% E-M uses 5di↵erent timesteps: 2^(p � 10), p = 1,2,3,4,5.% Examine weak convergence at T = 1 : |E (X

L

)� E (X (T ))|.%% Di↵erent paths are used for each E-M timestep.% Code is vectorized over paths.%% Uncommenting the line indicated below gives the weak E-M method.

randn(state,100);lambda = 2; mu = 0.1; Xzero = 1; T = 1; % problem parametersM = 50000;



Xem = zeros(5,1); % preallocate arraysfor p =1:5 % take various Euler timestepsDt = 2^(p � 10); L = T/Dt; % L Euler steps of size DtXtemp = Xzero*ones(M,1);for j = 1:LWinc = sqrt(Dt)*randn(M,1)Xtemp = Xtemp + Dt ⇤ lambda ⇤ Xtemp +mu ⇤ Xtemp. ⇤Winc ;

endXem(p) = mean(Xtemp);

endXerr =abs(Xem � exp(lambda));



Dtvals = 2.^([1 : 5]� 10);loglog(Dtvals,Xerr ,0 b ⇤ �0), hold onloglog(Dtvals,Dtvals,0 r ��0), hold o↵ % reference slope of 1axis([1e-3 1e-1 1e-4 1])xlabel(’\Delta t’), ylabel(’|E (X (T ))� Sample average of X

L

|’)title(’emweak.m’,’FontSize’,10)

%%%% Least squares fit of error = C ⇤ dtq %%%%A = [ones(p, 1), log(Dtvals)]; rhs = log(Xerr);sol = A\ rhs; q = sol(2)resid = norm(A*sol - rhs)


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