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# Matrix multiplication (best computational approach)

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Julián Francisco on 14 May 2012
I have to make a coordinates transformation between two reference systems (axes). For it, three matrices (3x3) have to be multiplied due to some intermediate axes are used. I have thought two approaches to resolve this:
method #1: Making the multiplication directly, i.e. v_f = R1*R2*R3*v_i
method #2: Split into steps.
step 1: v_3i = R3*v_i
step 2: v_23 = R2*v_3i
step 3: v_f = R1*v_23
where: R1, R2 and R3 are 3x3 matrices
v_f,v_i, v_3i, v_23 are 3x1 vectors
I would like to know what method is more efficient computationally (less time) to do the transformation (this will be made a lot of times).
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### Accepted Answer

Teja Muppirala on 14 May 2012
I think you might be better off grouping your multiplications so that you maximize matrix-vector products instead of matrix-matrix products.
For example:
R1 = rand(3);
R2 = rand(3);
R3 = rand(3);
v = rand(3,1);
tic;
for n = 1:1e6, R1*R2*R3*v; end;
toc;
% Slightly faster
tic;
for n = 1:1e6, R1*(R2*(R3*v)); end;
toc;
This is especially true for larger matrices much bigger than 3x3.
R1 = rand(1000);
R2 = rand(1000);
R3 = rand(1000);
v = rand(1000,1);
tic;
R1*R2*R3*v;
toc;
% Much faster
tic;
R1*(R2*(R3*v));
toc;
Of course if the R's are all constant and you are reusing them, then you should just multiply them all together and store that result to begin with.
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Julián Francisco on 14 May 2012
@Jan Simon: Thank you again.

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### More Answers (2)

Jan on 14 May 2012
There is no difference internally. The intermediate results have to be calculated in both cases. A TIC/TOC measurement allows you to find out, which one is faster.
Usually it is possible to calculate R1*R2*R3 explicitely without a matrix multiplication.
[EDITED, appended after your comment]:
function R = RotMatrix(N, A)
% N: [1 x 3], normalized vector to rotate around
% A: [1 x 1], angle
% Calculate trigonometric values once to save time:
cA = cos(A);
sA = sin(A);
cmA = 1 - cA;
x = N(:, 1);
y = N(:, 2);
z = N(:, 3);
ysA = y .* sA; % Some optimizations to save products
zsA = z .* sA;
xsA = x .* sA;
xcmA = x .* cmA;
ycmA = y .* cmA;
yzcmA = z .* ycmA;
xycmA = x .* ycmA;
xzcmA = z .* xcmA;
R = [cA + x .* xcmA, xycmA + zsA, xzcmA - ysA; ...
xycmA - zsA, cA + y .* ycmA, yzcmA + xsA; ...
xzcmA + ysA, xsA - yzcmA, cA + z .* z .* cmA];
For a real application it could be more efficient to accept [nFrame x 3] arrays as list of vectors and a [nFrame x 1] list of angles. Then the context decides, which output dimensions are most useful, perhaps a [3 x 3 x nFrame] array or three [3 x nFrame] matrices, which allow a DOT product directly to transform vectors.
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Julián Francisco on 14 May 2012
Thank you back for your comments and answers.

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James Tursa on 14 May 2012
If you are simply after speed, the fastest approach will be a mex routine to do the calculations inline (i.e., explicitly without calling a separate generic matrix multiply package ala BLAS). This avoids all of the forming of intermediate variables. How many is "... lots of times"? Which variable(s) are changing in each iteration of the loop?
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Julián Francisco on 14 May 2012
Yes, you are right. I use functions of angles for each matrix element. For the matrix rotation I have written the following code:
function [R] = rot(axis,theta)
% axis is refering to the axis around the rotation is made (use 1 for x-axis, 2 for y-axis and 3 for z-axis)
% theta is the rotation angle
S = sin(theta);
C = cos(theta);
R = eye(3);
switch axis
case 1
R(2,2) = C;
R(3,2) = -S;
R(2,3) = S;
R(3,3) = C;
case 2
R(1,1) = C;
R(3,1) = S;
R(1,3) = -S;
R(3,3) = C;
otherwise
R(1,1) = C;
R(2,1) = -S;
R(1,2) = S;
R(2,2) = C;
end

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