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Writing Fast MATLAB Code：编写快速的MATLAB代码

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Writing Fast MATLAB Code

Pascal Getreuer, June 2006

Contents

The Proﬁler

Array Preallocation

Vectorization

3.1 Vectorized Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Vectorized Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Inlining Simple Functions

Numerical Integration

5.1 One-Dimensional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 Multidimensional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Signal Processing

Referencing Operations

Miscellaneous Tricks

8.1 Clip a value without using if statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.2 Convert any array into a column vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.3 Find the min/max of a matrix or N-d array . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.4 Vector Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.5 Flood ﬁlling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.6 Vectorized use of set on GUI objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Further Reading

Introduction

The Matlab programming language is parsed; code is interpreted during runtime. Languages like

C and Fortran are faster because they are ﬁrst compiled into the computer’s native language. The

advantages of parsing in realtime are greater platform independence, greater language ﬂexibility, and

easier debugging. The disadvantages are slower speed, increased overhead, and limited low-level control.

This article discusses strategies for improving the speed of Matlab code. Keep in mind that Matlab

has gone through many versions and that it is available on many platforms. The fastest method on one

system may not be the fastest on another. This article provides methods that are generally fast, but

makes no claim on what is fastest.

Caution!

Learn the language ﬁrst. Optimization requires comfort with the syntax and functions of the

language. This article is not a tutorial on MATLAB.

Use comments. Optimized code tends to be terse and cryptic. Help others and yourself by

remembering to comment.

Don’t optimize code before its time. Is optimization worth the eﬀort? If the code will soon

be revised or extended, it will be rewritten anyway.

Only optimize where necessary. Focus your eﬀorts on bottlenecks.

1 The Proﬁler

Matlab 5.0 and newer versions include a tool called the “proﬁler” that helps determine where the

bottlenecks are in a program. Consider the following function:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

function result = example1(Count)

for k = 1:Count

result(k) = sin(k/50);

if result(k) < −0.9

result(k) = gammaln(k);

end

To analyze the eﬃciency this function, ﬁrst enable the proﬁler and clear any old proﬁler data:

>> profile on

>> profile clear

Now run the program. Change the input argument higher or lower so that it takes about a second.

>> example1(5000);

Then enter

>> profile report

The proﬁler generates an HTML report on the function and launches a browser window. Depending on

the system, proﬁler results may be a little diﬀerent from this example.

MATLAB Proﬁle Report: Summary

Report generated 30-Jul-2004 16:57:01

Total recorded time:

Number of M-functions:

Clock precision:

3.09 s

0.016 s

Function List

Name

Time

3.09 100.0%

Time Time/call Self time

Location

example1

gammaln

profile

3562

3.094000 2.36 76.3% ~/example1.m

0.73

0.00

23.7%

0.0%

0.000206 0.73 23.7% ../toolbox/matlab/specfun/gammaln.m

0.000000 0.00

0.0% ../toolbox/matlab/general/profile.m

0.0% ../toolbox/matlab/general/profreport.m

profreport 0.00

Clicking the “example1” link gives more details:

100% of the total time in this function was spent on the following lines:

3: for k = 1:Count

2.11 68% 4:

result(k) = sin(k/50);

0.14

5% 6:

if result(k) < -0.9

result(k) = gammaln(k);

end

0.84 27% 7:

The most time-consuming lines are displayed, along with time, time percentage, and line number. The

most costly lines are the computations on lines 4 and 7.

2 Array Preallocation

Matlab’s matrix variables have the ability to dynamically augment rows and columns. For example,

>> a = 2

a =

>> a(2,6) = 1

a =

Matlab automatically resizes the matrix. Internally, the matrix data memory must be reallocated with

larger size. If a matrix is resized repeatedly—like within a for loop—this overhead can be signiﬁcant.

To avoid frequent reallocations, “preallocate” the matrix with the zeros command. Consider the code:

a(1) = 1;

b(1) = 0;

for k = 2:8000

a(k) = 0.99803

b(k) = 0.06279

a(k − 1) − 0.06279

a(k − 1) + 0.99803

b(k − 1);

end

The proﬁler timed this code to take 0.47 seconds. After the for loop, both arrays are row vectors of

length 8000, thus to preallocate, create empty a and b row vectors each with 8000 elements.

a = zeros(1,8000);

b = zeros(1,8000);

a(1) = 1;

% Preallocation

b(1) = 0;

for k = 2:8000

a(k) = 0.99803

b(k) = 0.06279

end

a(k − 1) − 0.06279

a(k − 1) + 0.99803

b(k − 1);

With this modiﬁcation, the code takes only 0.14 seconds (over three times faster). Preallocation is often

easy to do, in this case it was only necessary to determine the right preallocation size and add two lines.

What if the ﬁnal array size can vary? Then use the upper bound on the array size and cut the excess

after the loop:

a = zeros(1,10000);

count = 0;

% Preallocate

for k = 1:10000

v = exp(rand(1) rand(1));

if v > 0.5

count = count + 1;

a(count) = v;

% Conditionally add to array

end

a = a(1:count);

% Trim the result

The average run time of this program is 0.42 seconds without preallocation and 0.18 seconds with it.

Preallocation can also be done with cell arrays, using the cell command to create the desired size.

Using preallocation with a frequently resizing cell array is even more beneﬁcial than with double arrays.

3 Vectorization

To “vectorize” a computation means to replace parallel operations with vector operations. This strategy

often improves speed ten-fold. Good vectorization is a skill that must be developed; it requires comfort

with Matlab’s language and creativity.

3.1 Vectorized Computations

Many standard Matlab functions are “vectorized,” they can operate on an array as if the function had

been applied individually to every element.

>> sqrt([1,4;9,16])

>> abs([0,1,2,−5,−6,−7])

ans =

Consider the following function:

function d = minDistance(x,y,z)

% Find the min distance between a set of points and the origin

nPoints = length(x);

d = zeros(nPoints,1);

% Preallocate

for k = 1:nPoints

% Compute distance for every point

d(k) = sqrt(x(k)ˆ2 + y(k)ˆ2 + z(k)ˆ2);

end

d = min(d);

% Get the minimum distance

For every point, the distance between it and the origin is computed and stored in d. The minimum

distance is then found with min. To vectorize the distance computation, replace the for loop with

vector operations:

function d = minDistance(x,y,z)

% Find the min distance between a set of points and the origin

d = sqrt(x.ˆ2 + y.ˆ2 + z.ˆ2);

d = min(d);

% Compute distance for every point

% Get the minimum distance

The modiﬁed code performs the distance computation with vector operations. The x, y and z arrays

are ﬁrst squared using the per-element power operator, .^ (the per-element operators for multiplication

and division are .* and ./). The squared components are added with vector addition. Finally, the

square root of the vector sum is computed per element, yielding an array of distances.

The ﬁrst version of the minDistance program takes 0.73 seconds on 50000 points. The vectorized

version takes less than 0.04 seconds, more than 18 times faster.

Some useful functions for vectorizing computations:

min, max, repmat, meshgrid, sum, cumsum, diff, prod, cumprod, filter

3.2 Vectorized Logic

The previous section shows how to vectorize pure computation. Bottleneck code often involves condi-

tional logic. Like computations, Matlab’s logic operators are vectorized:

>> [1,5,3] < [2,2,4]

ans =

Two arrays are compared per-element. Logic operations return “logical” arrays with binary elements.

How is this useful? Matlab has a few powerful functions for operating on logical arrays:

find: Find indices of nonzero elements.

any: True if any element of a vector is nonzero (or per-column for a matrix).

all: True if all elements of a vector are nonzero (or per-column for a matrix).

>> find([1,5,3] < [2,2,4])

>> find(eye(3) == 1)

ans =

The find function returns the indices where the vector logic operation returns true. In the ﬁrst example,

1 < 2 is true, 5 < 2 is false, and 3 < 4 is true, so find reports that the ﬁrst and third comparisons are

true. In the second example, find returns the indices where the identity matrix is equal to one. The

indices 1, 5, and 9 correspond to the diagonal of a 3 by 3 matrix.

The any and all functions are simple but occasionally very useful. For example, any(x(:) < 0)

returns true if any element of x is negative.

Example 1: Removing elements

The situation often arises where array elements must be removed on some per-element condition. For

example, this code removes all NaN and inﬁnite elements from an array x:

i = find(isnan(x)

x(i) = [];

isinf(x));

% Find bad elements in x

% and delete them

Alternatively,

i = find(˜isnan(x) & ˜isinf(x)); % Find elements that are not NaN and not infinite

x = x(i); % Keep those elements

Both of these solutions can be further streamlined by using logical indexing:

x(isnan(x)

isinf(x)) = [];

% Delete bad elements

x = x(˜isnan(x) & ˜isinf(x));

% Keep good elements

Example 2: Piecewise functions

The sinc function has a piecewise deﬁnition,

(

sin(x)/x, x = 0

1, x = 0

sinc(x) =

This code uses find with vectorized computation to handle the two cases separately:

function y = sinc(x)

% Computes the sinc function per−element for a set of x values.

y = ones(size(x));

% Set y to all ones, sinc(0) = 1

% Find nonzero x values

i = find(x ˜= 0);

y(i) = sin(x(i)) ./ x(i);

% Compute sinc where x ˜= 0

An interesting alternative is y = (sin(x) + (x == 0))./(x + (x == 0)).

Example 3: Drawing images with meshgrid

The meshgrid function takes two input vectors and converts them to matrices by replicating the ﬁrst

over the rows and the second over the columns.

>> [x,y] = meshgrid(1:5,1:3)

x =

y =

The matrices above work like a map for a width 5, height 3 image. For each pixel, the x-location can

be read from x and the y-location from y. This may seem like a gratuitous use of memory as x and y

simply record the column and row positions, but this is useful. For example, to draw an ellipse,

% Create x and y for a width 150, height 100 image

[x,y] = meshgrid(1:150,1:100);

% Ellipse with origin (60,50) of size 15 x 40

Img = sqrt(((x−60).ˆ2 / 15ˆ2) + ((y−50).ˆ2 / 40ˆ2)) > 1;

% Plot the image

imagesc(Img); colormap(copper);

axis image; axis off;

Drawing lines is almost the same, just a change in the formula.

[x,y] = meshgrid(1:150,1:100);

% The line y = x 0.8 + 20

Img = (abs((x 0.8 + 20) − y) > 1);

imagesc(Img); colormap(copper);

axis image; axis off;

Polar functions can be drawn by ﬁrst converting x and y variables with the cart2pol function.

[x,y] = meshgrid(1:150,1:100);

[th,r] = cart2pol(x − 75,y − 50); % Convert to polar

% Spiral centered at (75,50)

Img = sin(r/3 + th);

imagesc(Img); colormap(hot);

axis image; axis off;

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