编写快速的MATLAB代码

Writing Fast MATLAB Code Pascal Getreuer, June 2006 Contents 1 The Profiler 2 2 Array Preallocation 3 3 Vectorization 5 3.1 Vectorized Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Vectorized Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Inlining Simple Functions 10 5 Numerical Integration 12 5.1 One-Dimensional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Multidimensional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6 Signal Processing 16 7 Referencing Operations 19 8 Miscellaneous Tricks 22 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 9 Further Reading 24 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! ˆ Learn the language first. Optimization requires comfort with the syntax and functions of the language. This article is not a tutorial on MATLAB. ˆ Use comments. Optimized code tends to be terse and cryptic. Help others and yourself by remembering to comment. ˆ 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. ˆ Only optimize where necessary. Focus your efforts on bottlenecks. 1 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 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: 3.09 s Number of M-functions: 4 Clock precision: 0.016 s Function List Name Time Time Time/call Self time Location example1 3.09 100.0% 1 3.094000 2.36 76.3% ~/example1.m gammaln 0.73 23.7% 3562 0.000206 0.73 23.7% ../toolbox/matlab/specfun/gammaln.m profile 0.00 0.0% 1 0.000000 0.00 0.0% ../toolbox/matlab/general/profile.m profreport 0.00 0.0% 1 0.000000 0.00 0.0% ../toolbox/matlab/general/profreport.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 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); 5: 0.14 5% 6: if result(k) < -0.9 0.84 27% 7: result(k) = gammaln(k); 8: end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 = 2 >> a(2,6) = 1 a = 2 0 0 0 0 0 0 0 0 0 0 1 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 * a(k − 1) − 0.06279 * b(k − 1); b(k) = 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. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a = zeros(1,8000); % Preallocation b = zeros(1,8000); a(1) = 1; b(1) = 0; for k = 2:8000 a(k) = 0.99803 * a(k − 1) − 0.06279 * b(k − 1); b(k) = 0.06279 * a(k − 1) + 0.99803 * b(k − 1); end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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); % Preallocate count = 0; for k = 1:10000 v = exp(rand(1)*rand(1)); if v > 0.5 % Conditionally add to array count = count + 1; a(count) = v; end 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. 4 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 standardMatlab 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]) ans = 1 2 3 4 >> abs([0,1,2,−5,−6,−7]) ans = 0 1 2 5 6 7 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); % Compute distance for every point d = min(d); % 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. 5 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 = 1 0 1 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: ˆ find: Find indices of nonzero elements. ˆ any: True if any element of a vector is nonzero (or per-column for a matrix). ˆ all: True if all elements of a vector are nonzero (or per-column for a matrix). >> find([1,5,3] < [2,2,4]) ans = 1 3 >> find(eye(3) == 1) ans = 1 5 9 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. 6 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) | isinf(x)); % Find bad elements in x x(i) = []; % 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 or x = x(˜isnan(x) & ˜isinf(x)); % Keep good elements Example 2: Piecewise functions The sinc function has a piecewise definition, sinc(x) = { sin(x)/x, x 6= 0 1, x = 0 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 i = find(x ˜= 0); % Find nonzero x values 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 = 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 7 y = 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 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; 8 Example 4: Polynomial interpolation Given n points x1, x2, x3, . . . xn and n corresponding function values y1, y2, y3, . . . yn, the coefficients c0, c1, c2, . . . cn−1 of the interpolating polynomial of degree n− 1 can be found by solving x1 n−1 x1n−2 · · · x12 x1 1 x2 n−1 x2n−2 · · · x22 x2 1 ... ... xn n−1 xnn−2 · · · xn2 xn 1   cn−1 cn−2 ... c0  =  y1 y2 ... yn  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . function c = polyint(x,y) % Given a set of points and function values x and y, % computes the interpolating polynomial. x = x(:); % Make sure x and y are both column vectors y = y(:); n = length(x); % n = Number of points %%% Construct the left−hand side matrix %%% xMatrix = repmat(x, 1, n); % Make an n by n matrix with x on every column powMatrix = repmat(n−1:−1:0, n, 1); % Make another n by n matrix of exponents A = xMatrix .ˆ powMatrix; % Compute the powers c = A\y; % Solve matrix equation for coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The strategy to construct the left-hand side matrix is to first make two n by n matrices of bases and exponents and then put them together using the per element power operator, .^ . The repmat function (“replicate matrix”) is used to make the base matrix xMatrix and the exponent matrix powMatrix. xMatrix =  x(1) x(1) · · · x(1) x(2) x(2) · · · x(2) ... ... x(n) x(n) · · · x(n)  powMatrix =  n− 1 n− 2 · · · 0 n− 1 n− 2 · · · 0 ... ... n− 1 n− 2 · · · 0  The xMatrix is made by repeating the column vector x over the columns n times. The powMatrix is a row vector with elements n − 1, n − 2, n − 3, . . . , 0 repeated down the rows n times. The two matrices could also have been created with [powMatrix, xMatrix] = meshgrid(n-1:-1:0, x). This function is only an example; use the standard polyfit function for serious polynomial interpola- tion. It is more general and algorithmically more efficient (see polyfit.m). 9 4 Inlining Simple Functions Every time an M-file function is called, the Matlab interpreter incurs some overhead. Additionally, many M-file functions begin with conditional code that checks the input arguments for errors or deter- mines the mode of operation. Of course, this overhead is negligible for a single function call. It should only be considered when the function being called is an M-file, the function itself is “simple,” that is, implemented with only a few lines, and called frequently from within a loop. For example, this code calls the M-file function median repeatedly to perform median filtering: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . % Apply the median filter of size 5 to signal x y = zeros(size(x)); % Preallocate for k = 3:length(x)−2 y(k) = median(x(k−2:k+2)); end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Given a 2500-sample array for x, the overall run time is 0.42 seconds. “Inlining a function” means to replace a call to the function with the function code itself. Beware that inlining should not be confused with Matlab’s “inline” function datatype. Studying median.m (type “edit median” on the console) reveals that most of the work is done using the built-in sort function. The median call can be inlined as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . % Apply the median filter of size 5 to signal x y = zeros(size(x)); % Preallocate for k = 3:length(x)−2 tmp = sort(x(k−2:k+2)); y(k) = tmp(3); % y(k) = median(x(k−2:k+2)); end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Now the overall run time for a 2500-sample input is 0.047 seconds, nearly 9 ti