University:
Boston University
Course:
MET CS 201 | Introduction to Programming
Academic year:

2024
Views:

9

Pages:

20
Author:

carburi1xj0

Code Tuning and Optimization Bad: Good: for i=1:N x(i) = i; end for i=1:N y(i) = rand(); end for i=1:N x(i) = i; y(i) = rand(); end  Reduces for-loop overhead  More important, improve chances of pipelining  Loop fisssion splits statements into multiple loops Code Tuning and Optimization Avoid if statements within loops Bad: if has overhead cost and may inhibit pipelining for i=1:N if i == 1 %perform i=1 calculations else %perform i>1 calculations end end Good: %perform i=1 calculations for i=2:N %perform i>1 calculations end Code Tuning and Optimization Divide is more expensive than multiply • Intel x86 clock cycles per operation • add 3-6 • multiply 4-8 • divide 32-45 • Bad: c = 4; for i=1:N x(i)=y(i)/c; end • Good: s = 1/c; for i=1:N x(i) = y(i)*s; end Code Tuning and Optimization Function Call Overhead for i=1:N Bad: do stuff myfunc(i); end Good: function myfunc(i) myfunc2(N); end function myfunc2(N) for i=1:N do stuff end end Function m-file is precompiled to lower overhead for repeated usage. Still, there is an overhead . Balance between modularity and performance. Code Tuning and Optimization Minimize calls to math & arithmetic operations Bad: for i=1:N z(i) = log(x(i)) + log(y(i)); v(i) = x(i) + x(i)^2 + x(i)^3; end Good: for i=1:N z(i) = log(x(i) * y(i)); v(i) = x(i)*(1+x(i)*(1+x(i))); end Tuning MATLAB for Better Performance Special Functions for Real Numbers MATLAB provides a few functions for processing real number 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 n = 1000; x = 1:n; x = x.^2; tic x = realsqrt(x); toc Wallclock time = 0.00022 seconds Wallclock time = 0.00004 seconds square_root.m real_square_root.m  isreal reports whether the array is real  single/double converts data to single-, or double-precision Tuning MATLAB for Better Performance Vector Is Better Than Loops  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 and nested.  Without array pre-allocation, its size extension in a for-loop is costly as shown before.  When possible, use vector representation instead of for-loops. i = 0; for t = 0:.01:100 i = i + 1; y(i) = sin(t); end t = 0:.01:100; y = sin(t); Wallclock time = 0.1069 seconds Wallclock time = 0.0007 seconds for_sine.m vec_sine.m Tuning MATLAB for Better Performance 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 Tuning MATLAB for Better Performance 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 x 3 array of 0’s A= 0 0 0 0 0 0 0 0 0 >> B = 2*ones(2,3) % ones generates 2 x 3 array of 1’s B= 2 2 2 2 2 2 Alternatively, >> B = repmat(2,2,3) % matrix replication Tuning MATLAB for Better Performance 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); Tuning MATLAB for Better Performance 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 Tuning MATLAB for Better Performance Vector Utility Functions Function Description all Test to see if all elements are of a prescribed value any Test to see if any element is of a prescribed value zeros Create array of zeroes ones Create array of ones repmat Replicate and tile an array find Find indices and values of nonzero elements diff Find differences and approximate derivatives squeeze Remove singleton dimensions from an array prod Find product of array elements sum Find the sum of array elements cumsum Find cumulative sum shiftdim Shift array dimensions logical Convert numeric values to logical sort Sort array elements in ascending /descending order Tuning MATLAB for Better Performance function in range of 0 to 𝜋/2 • Equals to sum of all rectangles (width times height of bars) • Integral is area under b m a i 1  cos( x)dx    a ih a ( i 1) h cos(x) h m cos( x )dx   cos(a  (i  12 )h)h i 1 ; % range ; % # of increments ; % increment mid-point of increment Tuning MATLAB for Better Performance % integration with for-loop tic m = 100; a = 0; % lower limit of integration b = pi/2; % upper limit of integration h = (b – a)/m; % increment length integral = 0; % initialize integral for i=1:m x = a+(i-0.5)*h; % mid-point of increment i integral = integral + cos(x)*h; end toc h a X(1) = a + h/2 b X(m) = b - h/2 Tuning MATLAB for Better Performance % integration with vector form tic m = 100; a = 0; % lower limit of integration b = pi/2; % upper limit of integration h = (b – a)/m; % increment length x = a+h/2:h:b-h/2; % mid-point of m increments integral = sum(cos(x))*h; toc h a X(1) = a + h/2 b X(m) = b - h/2 Tuning MATLAB for Better Performance Integration Example Benchmarks increment m for-loop Vector 10000 0.00044 0.00017 20000 0.00087 0.00032 40000 0.00176 0.00064 80000 0.00346 0.00130 160000 0.00712 0.00322 320000 0.01434 0.00663  Timings (seconds) obtained on Intel Core i5 3.2 GHz PC  Computational effort linearly proportional to # of increments. Tuning MATLAB for Better Performance u u  2 0 2 x y 2 2 Boundary Conditions: u( x,0 )  sin(x ) 0 u( x,1)  sin(x )e  u(0, y )  u(1, y )  0 0 x1 x1 0 y1 Analytical solution: u( x, y )  sin(x )e y 0  x  1; 0  y  1 Tuning MATLAB for Better Performance Finite Difference Numerical Discretization Discretize equation by centered-difference yields: uin, 1 j  uin1,j  uin1,j  ui,jn 1  ui,jn 1 4 i  1,2, ,m; j  1,2, ,m where n and n+1 denote the current and the next time step, respectively, while ui,jn  u n(xi ,y j ) i  0,1,2, ,m  1; j  0,1,2, ,m  1  u n(ix, jy ) For simplicity, we take x  y  1 m1 Tuning MATLAB for Better Performance Computational Domain y, j u(x,1)  sin(x )e  x, i u(1,y)  0 u(0,y)  0 u(x,0)  sin(x) uin, j 1  n n uin1,j  uin1,j  ui,j  u 1 i,j 1 4 i  1,2,  ,m; j  1,2,  ,m Tuning MATLAB for Better Performance Five-point Finite-difference Stencil Interior cells. Where solution of the Laplace equation is sought. x x o x x 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