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MATLAB Tutorial Series Tuning MATLAB for Better Performance Kadin Tseng

MATLAB Tutorial Series Tuning MATLAB for Better Performance Kadin Tseng Scientific Computing and Visualization, IS&T Boston University. Topics Covered. 1. Performance Issues 1.1 Memory Allocations 1.2 Vector Representations 1.3 Compiler 1.4 Other Considerations

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MATLAB Tutorial Series Tuning MATLAB for Better Performance Kadin Tseng

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  1. MATLAB Tutorial Series Tuning MATLAB for Better Performance Kadin Tseng Scientific Computing and Visualization, IS&T Boston University

  2. Topics Covered 1. Performance Issues 1.1 Memory Allocations 1.2 Vector Representations 1.3 Compiler 1.4 Other Considerations 2. Multiprocessing with MATLAB

  3. 1. Performance Issues 1.1 Memory Access 1.2 Vector Representations 1.3 Compiler 1.4 Other Considerations

  4. 1.1 Memory Access Memory access patterns often affect computational performance. Some effective ways to enhance performance in MATLAB : • Allocate array memory before using it • For-loops Ordering • Compute and save array in-place where applicable.

  5. How Does MATLAB Allocate Arrays ? • MATLAB arrays are allocated in contiguous address space. Without pre-allocation x = 1; for i=2:4 x(i) = i; end

  6. How … Arrays ? Examples • MATLAB arrays are allocated in contiguous address space. • Pre-allocate arrays enhance performance significantly. n=5000; tic for i=1:n x(i) = i^2; end toc Wallclock time = 0.00046 seconds n=5000;x = zeros(n,1); tic for i=1:n x(i) = i^2; end toc Wallclock time = 0.00004 seconds The timing data are recorded on Katana. The actual times may vary depending on the processor.

  7. Passing Arrays Into A Function MATLAB uses pass-by-reference if passed array is used without changes; a copy will be made if the array is modified. MATLAB calls it “lazy copy.” Consider the following example: function y = lazyCopy(A, x, b, change) If change, A(2,3) = 23; end % change forces a local copy of a y = A*x + b; % use x and b directly from calling program pause(2) % keep memory longer to see it in Task Manager On Windows, can use Task Manager to monitor memory allocation history. >> n = 5000; A = rand(n); x = rand(n,1); b = rand(n,1); >> y = lazyCopy(A, x, b, 0); >> y = lazyCopy(A, x, b, 1);

  8. For-loop Ordering • Best if inner-most loop is for array left-most index, etc. (column-major) • For a multi-dimensional array, x(i,j), the 1D representation of the same array, x(k), follows column-wise order and inherently possesses the contiguous property n=5000; x = zeros(n); for i=1:n % rows for j=1:n % columns x(i,j) = i+(j-1)*n; end end Wallclock time = 0.88 seconds n=5000; x = zeros(n); for j=1:n % columns for i=1:n % rows x(i,j) = i+(j-1)*n; end end Wallclock time = 0.48 seconds for i=1:n*n x(i) = i; end x = 1:n*n;

  9. Compute In-place • Compute and save array in-place improves performance (and reduce memory usage) x = rand(5000); tic y = x.^2; toc Wallclock time = 0.30 seconds x = rand(5000); tic x = x.^2; toc Wallclock time = 0.11 seconds Caveat: May not be worthwhile if it involves data type or size change …

  10. OtherConsiderations • Generally, better to use function instead of script • m-file • Script m-file is loaded into memory and evaluate one line at a time. Subsequent uses require reloading. • Function m-file is compiled into a pseudo-code and is loaded on first application. Subsequent uses of the function will be faster without reloading. • Function is modular; self cleaning; reusable. • Global variables are expensive; difficult to track. • Physical memory is much faster than virtual mem. • Avoid passing large matrices to a function and modifying only a handful of elements.

  11. Other Considerations (cont’d) • load and save are efficient to handle whole data file; textscan is more memory-efficient to extract text meeting specific criteria. • Don’t reassign array that results in change of data type or shape. • Limit m-files size and complexity. • Computationally intensive jobs often require large memory … • Structure of array more memory-efficient than array of structures.

  12. Memory Management • Maximize memory availability. • 32-bit systems < 2 or 3 GB • 64-bit systems running 32-bit MATLAB < 4GB • 64-bit systems running 64-bit MATLAB < 8TB • (93 GB on some Katana nodes) • Minimize memory usage. (Details to follow …)

  13. Minimize Memory Usage • Use clear, pack or other memory saving means when possible. If double precision (default) is not required, the use of ‘single’ data type could save substantial amount of memory. For example, >> x=ones(10,'single'); y=x+1; % y inherits single from x • Use sparse to reduce memory footprint on sparse matrices >> n=5000; A = zeros(n); A(3,2) = 1; B = ones(n); >> C = A*B; >> As = sparse(A); >> Cs = As*B; % it can save time for low density >> A2 = sparse(n,n); A2(3,2) = 1;

  14. Minimize Memory Usage (Cont’d) • Use “matlab –nojvm …” saves lots of memory – if warranted • Memory usage query For Unix: Katana% top For Windows: >> m = feature('memstats'); % largest contiguous free block Use MS Windows Task Manager to monitor memory allocation. • Distribute memory among multiprocessors via MATLAB Parallel Computing Toolbox.

  15. Special Functions for Real Numbers MATLAB provides a few functions for processing real, noncomplex, data specifically. These functions are more efficient than their generic versions: • realpow – power for real numbers • realsqrt – square root for real numbers • reallog – logarithm for real numbers • realmin/realmax – min/max for real numbers n = 1000; x = 1:n; x = x.^2; tic x = sqrt(x); toc Wallclock time = 0.00022 seconds n = 1000; x = 1:n; x = x.^2; tic x = realsqrt(x); toc Wallclock time = 0.00004 seconds • isreal reports whether the array is real • single/double converts data to single-/double-precision

  16. Vector Operations • MATLAB is designed for vector and matrix operations. The use of for-loop, in general, can be expensive, especially if the loop count is large or nested. • Without array pre-allocation, its size extension in a for-loop is costly as shown before. • From a performance standpoint, in general, vector representation should be used in place of for-loops. i = 0; for t = 0:.01:100 i = i + 1; y(i) = sin(t); end Wallclock time = 0.1069seconds t = 0:.01:100; y = sin(t); Wallclock time =0.0007seconds

  17. Vector Operations of Arrays >> A = magic(3) % define a 3x3 matrix A A = 8 1 6 3 5 7 4 9 2 >> B = A^2; % B = A * A; >> C = A + B; >> b = 1:3 % define b as a 1x3 row vector b = 1 2 3 >> [A, b'] % add b transpose as a 4th column to A ans = 8 1 6 1 3 5 7 2 4 9 2 3

  18. Vector Operations >> [A; b] % add b as a 4th row to A ans = 8 1 6 3 5 7 4 9 2 1 2 3 >> A = zeros(3) % zeros generates 3*3 array of 0’s A = 0 0 0 0 0 0 0 0 0 >> B = 2*ones(2,3) % ones generates 2 * 3 array of 1’s B = 2 2 2 2 2 2 Alternatively, >> B = repmat(2,2,3) % matrix replication

  19. Vector Operations >> y = (1:5)’; >> n = 3; >> B = y(:, ones(1,n)) % B = y(:, [1 1 1]) or B=[y y y] B = 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 Again, B can be generated via repmat as >> B = repmat(y, 1, 3);

  20. Vector Operations >> A = magic(3) A = 8 1 6 3 5 7 4 9 2 >> B = A(:, [1 3 2]) % switch 2nd and third columns of A B = 8 6 1 3 7 5 4 2 9 >> A(:, 2) = [ ] % delete second column of A A = 8 6 3 7 4 2

  21. Vector Operation Example n = 3000; x = zeros(n); for j=1:n for i=1:n x(i,j) = i+(j-1)*n; x(i,j) = x(i,j)^2; end end Wallclock time = 0.128seconds n = 3000; i = (1:n) '; I = repmat(i,1,n); % replicate along j x = I + (I'-1)*n; x = x.^2; Wallclock time = 0.127 seconds • Notes on the vector form of the computations : • To eliminate the for-loops, all values of i and j must be made available at once. The ones or repmat utilities can be used to replicate rows or columns. In this special case, J = I' is used to save computations and memory. • Often, there are trade-offs between efficiency and memory using the vector form. Here, the creation of I adds to the memory and compute time. However, as more works can be leveraged against I, efficiency improves.

  22. Functions Useful in Vectorizing

  23. Laplace Equation Laplace Equation: (1) Boundary Conditions: (2) Analytical solution: (3)

  24. Discrete Laplace Equation Discretize Equation (1) by centered-difference yields: (4) where n and n+1 denote the current and the next time step, respectively, while (5) For simplicity, we take

  25. Computational Domain y, j x, i

  26. Five-point Finite-Difference Stencil Interior cells. Where solution of the Laplace equation is sought. x x x o x x (i, j) Exterior cells. Green cells denote cells where homogeneous boundary conditions are imposed while non-homogeneous boundary conditions are colored in blue. x x o x x

  27. SOR Update Function How to vectorize it ? Remove the for loops Define i = ib:2:ie; Define j = jb:2:je; Use sum for del % original code fragment jb = 2; je = n+1; ib = 3; ie = m+1; for i=ib:2:ie for j=jb:2:je up = ( u(i ,j+1) + u(i+1,j ) + ... u(i-1,j ) + u(i ,j-1) )*0.25; u(i,j) = (1.0 - omega)*u(i,j) +omega*up; del = del + abs(up-u(i,j)); end end % equivalent vector code fragment jb = 2; je = n+1; ib = 3; ie = m+1; i = ib:2:ie; j = jb:2:je; up = ( u(i ,j+1) + u(i+1,j ) + ... u(i-1,j ) + u(i ,j-1) )*0.25; u(i,j) = (1.0 - omega)*u(i,j) + omega*up; del = del + sum(sum(abs(up-u(i,j)))); More efficient way ?

  28. Solution Contour Plot

  29. SOR Update Function

  30. An integration of the cosine function between 0 and π/2 • Integration scheme is mid-point rule for simplicity. • Several parallel methods will be demonstrated. Integration Example Worker 1 mid-point of increment Worker 2 a = 0; b = pi/2; % range m = 8; % # of increments h = (b-a)/m; % increment p = numlabs; n = m/p; % inc. / worker ai = a + (i-1)*n*h; aij = ai + (j-1)*h; cos(x) h Worker 3 Worker 4 x=a x=b

  31. function intOut = Integral(fct, a, b, n) %function intOut = Integral(fct, a, b, n) % performs mid-point rule integration of "fct" % fct -- integrand (cos, sin, etc.) % a -- starting point of the range of integration % b –- end point of the range of integration % n -- number of increments % Usage example: >> Integral(@cos, 0, pi/2, 500) % 0 to pi/2 h = (b – a)/n; % increment length intOut = 0.0; % initialize integral for j=1:n % sum integrals aij = a +(j-1)*h; % function is evaluated at mid-interval intOut = intOut + fct(aij+h/2)*h; end Integration Example — the Kernel Vector form of the function: function intOut = Integral(fct, a, b, n) h = (b – a)/n; aij = a + (0:n-1)*h; intOut = sum(fct(aij+h/2))*h;

  32. % serial integration tic m = 10000; a = 0; b = pi*0.5; intSerial = Integral(@cos, a, b, m); toc Integration Example — Serial Integration

  33. Integration Example Benchmarks • Timings (seconds) obtained on a quad-core Xeon X5570 • Computation linearly proportional to # of increments. • FORTRAN and C timings are an order of magnitude faster.

  34. Compiler mcc is a MATLAB compiler: • It compiles m-files into C codes, object libraries, or stand-alone executables. • A stand-alone executable generated with mcc can run on compatible platforms without an installed MATLAB or a MATLAB license. • Many MATLAB general and toolbox licenses are available. Infrequently, MATLAB access may be denied if all licenses are checked out. Running a stand-alone requires NO licenses and no waiting.

  35. Compiler (Cont’d) • Some compiled codes may run more efficiently than m-files because they are not run in interpretive mode. • A stand-alone enables you to share it without revealing the source. www.bu.edu/tech/research/training/tutorials/matlab/vector/miscs/compiler/

  36. Compiler (Cont’d) Input arguments MATLAB root How to build a standalone executable >> mcc –o ssor2Dijc –m ssor2Dij How to run ssor2Dijc on Katana Katana% run_ssor2Dijc.sh /usr/local/apps/matlab_2009b 256 256 Details: • The m-file is ssor2Dij.m • Input arguments to code are processed as strings by mcc. Convert with str2num if need be. ssor2Dij.m requires 2 inputs; m, n if isdeployed, m=str2num(m); end • Output cannot be returned; either save to file or display on screen. • The executable is ssor2Dijc • run_ssor2Dijc.sh is the run script generated by mcc. • None of SOR codes benefits, in runtime, from mcc.

  37. MATLAB Programming Tools • profile - profiler to identify “hot spots” for performance enhancement. • mlint - for inconsistencies and suspicious constructs in M-files. • debug - MATLAB debugger. • guide - Graphical User Interface design tool.

  38. MATLAB profiler To use profile viewer, DONOT start MATLAB with –nojvm option >> profile on –detail 'builtin' –timer 'real' >> % run code to be >> % profiled here >> % >> % >> profile viewer % view profiling data >> profile off % turn off profiler Profiling example. >> profile on >> ssor2Dij % profiling the SOR Laplace solver >> profile viewer >> profile off Turns on profiler. –detail 'builtin' enables MATLAB builtin functions; -timer 'real' reports wallclock time.

  39. How to Save Profiling Data Two ways to save profiling data: • Save into a directory of HTML files Viewing is static, i.e., the profiling data displayed correspond to a prescribed set of options. View with a browser. 2. Saved as a MAT file Viewing is dynamic; you can change the options. Must be viewed in the MATLAB environment.

  40. Profiling – save as HTML files Viewing is static, i.e., the profiling data displayed correspond to a prescribed set of options. View with a browser. >> profile on >> plot(magic(20)) >> profile viewer >> p = profile('info'); >> profsave(p, ‘my_profile') % html files in my_profile dir

  41. Profiling – save as MAT file Viewing is dynamic; you can change the options. Must be viewed in the MATLAB environment. >> profile on >> plot(magic(20)) >> profile viewer >> p = profile('info'); >> save myprofiledata p >> clear p >> load myprofiledata >> profview(0,p)

  42. MATLAB “grammar checker” • mlint is used to identify coding inconsistencies and make coding performance improvement recommendations. • mlint is a standalone utility; it is an option in profile. • MATLAB editor provides this feature. • Debug mode can also be invoked through editor.

  43. Running MATLAB in Command Line Mode and Batch Katana% matlab -nodisplay –nosplash –r “n=4, myfile(n); exit” Add –nojvm to save memory if Java is not required For batch jobs on Katana, use the above command in the batch script. Visit http://www.bu.edu/tech/research/computation/linux-cluster/katana-cluster/runningjobs/ for instructions on running batch jobs.

  44. Comment Out Block Of Statements On occasions, one wants to comment out an entire block of lines. If you use the MATLAB editor: • Select statement block with mouse, then • press Ctrl R keys to insert % to column 1 of each line. • press Ctrl T keys to remove % on column 1 of each line. If you use another editor: • %{ n = 3000; x = rand(n); %} • if 0 n = 3000; x = rand(n); end

  45. Multiprocessing With MATLAB • Explicit parallel operations MATLAB Parallel Computing Toolbox Tutorial www.bu.edu/tech/research/training/tutorials/matlab-pct/ • Implicit parallel operations • Require shared-memory computer architecture (i.e., multicore). • Feature on by default. Turn it off with katana% matlab –singleCompThread • Specify number of threads with maxNumCompThreads To be deprecated. Still supported on Version 2010b • Activated by vector operation of applications such as hyperbolic or trigonometric functions, some LaPACK routines, Level-3 BLAS. • See “Implicit Parallelism” section of the above link.

  46. Useful SCV Info Please help us do better in the future by participating in a quick survey: http://scv.bu.edu/survey/spring11tut_survey.html • SCV home page (http://scv.bu.edu/) • Resource Applications (https://acct.bu.edu/SCF) • Help • Web-based tutorials (http://scv.bu.edu/) • (MPI, OpenMP, MATLAB, IDL, Graphics tools) • HPC consultations by appointment • Kadin Tseng (kadin@bu.edu) • Doug Sondak (sondak@bu.edu) • help@twister.bu.edu, help@katana.bu.edu

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