I have to perform matrix updating by M = M + c*a*a' large number of times, where c is a constant and a is a column vector. If the size of matrix is larger than 1000, this simple updating will cost most of the time of my function, typically more than 1 min counted by profile.
Main codes are:
for i = 1:N
_do something..._
for k = 1:n
a(1:k) = M(1:k,1:k)*p(1:k);
M(1:k,1:k) = M(1:k,1:k)+c*a(1:k)*a(1:k)';
M(1:k, k+1) = b(1:k);
M(k+1, 1:k) = b(1:k)';
M(k+1, k+1) = x;
......
end
end
I have preallocated all variables, column vectors p and b are known, and x is another constant.
As I have large number of data to process by this function, does there exist more efficient alternative to this matrix updating?
You can concatenate a vectors to create a matrix A then apply multiplication just one time.
A =[a1 a2 a3];
M = c * A * A.';
consider the example
A = rand(5,5);
M = 0;
c=4;
for n = 1:5
M = M + c * A(:,n) * A(:,n).';
end
and this one
M1 = c * A * A.'
both M and M1 are equal
Have you tried using bsxfun?
In any case, bsxfun is much faster than regular multiplication, but the vectors/matrices have to be of equal length (which they are for you, aren't they?), and it's operating elementwise (i.e. a Nx1 vector bsx-multiplied with itself yields a Nx1 vector, multiplied with the transpose however yields a NxN matrix).
see https://mathworks.com/help/matlab/ref/bsxfun.html
use as
bsxfun(#times, a, a')
Related
I am given set of 50 data points with values {a^(i),b^(i)} for i=1,...,50
stored in the arrays a and b.
I know that the Vandermonde matrix A has size m x n, where n = 2 ... 11 and m is the size of the array a.
I want to to fit the data with a polynomial of degree (n ā 1), for n = 2,...,11. To do that for each n I have to set up the Vandermonde matrix A of size m Ć n.
The Vandermonde matrix A solves the following equation:
A^T*A*x = A^T*b
Where the A^T is the transpose matrix and I have b already given.
Also we know that Aij = (a^(i))^(jā1) for j = 1,...,n,
What confuses me is how to set the matrix for n = 2,..,11.
What my line of thought is:
I have m = length(a); this will set up m = 50;
n = 11;
Then A=ones(m,n); This creates a matrix A filled with ones that has the correct size.
However I am not sure how to populate the matrix.
I wrote the following for loop which I thought will populate the matrix:
for n = 2:11
j=n;
for i = 1:50
A(i,n) = (a^(i))^(j-1);
end
end
Could you help me please with setting up the matrix?
You should use the vander function. However, vander will return an m x m matrix, which is usually used to fit the data to a polynomial of degree (m-1). Since you want to fit to a polynomial of degree (n-1), you only need the last n columns of that matrix.
Here's the code:
A = vander(a);
A = A(:,end-n+1:end);
I have the variables
X = 1x20 vector of doubles
i = 0:M
j = 0:M
And the equation
sum n=1 to length(X) : (X(n)^(i+j))
Is there a way to obtain an MxM matrix (through the indices i,j) while summing out n in each cell? I tried this with symsum but it doesn't allow indexing with n.
Any help is appreciated!
By reshaping X to a vector of size [1 x 1 x 20] and using implicit expansion a 3D [M+1 x M+1 x 20] array is created then by summing along the third dimension the result can be obtained.
X = rand(1,20);
M = 30;
ii = 0:M;
jj = (0:M).';
Y = reshape(X,1,1,[]);
result = sum(Y.^(ii+jj), 3);
However as the expression Y.^(ii+jj) creates a 3D [M+1 x M+1 x 20] array it may need a large amount of memory that leads to decreased performance.
We know that x^(i+j) can be written as x^i * x^j So the expression can be written as:
result = sum(Y.^ii .* Y.^jj,3);
It has the same memory consumption as the previous method. But when we reach an expression that contains sum of products we should think about converting it to very fast matrix multiplication :
Z = X .^ jj; % A [M+1 x 20] matrix is created(implicit expansion)
result = Z * Z.' % multiply Z by its transpose
So the same result is obtained without the complexity of the other solutions.
I want to multiply two vectors to produce a matrix.
I have a vector 1*m and another 1*n which are in my case V (1*71) and I (1*315). The other vectors have same length as I.
I want to multiply every value of I with all values of V and have the answer in a matrix where every row or column of new matrix is I(t).*V
Ir and Temp are vectors with the size of 1*315 and all the variables have the same length and T is 315.
The other parameters that you see in the code are constant values.
This is the code :
function [I] = solar2diodedyn( Ir,time,Temp )
V = 0:0.01:0.7; %open circuit voltage of one cell in V.
for t=1:time;
T(t)= Temp(t)+273;
Vt(t)=(k*T(t))/q;
Iph(t) = Isc_cell*(Ir(t)/1000)*(1+(T_co*(Temp(t)-25)));
I0(t)=Is1*((T(t)/Tmeas)^(3/n1))*exp(Eg*((T(t)/Tmeas)-1)/(n1*Vt(t)));
I02(t)=Is2*((T(t)/Tmeas)^(3/n2))*exp(Eg*((T(t)/Tmeas)-1)/(n2*Vt(t)));
I(t) = zeros(size(t));
i=length(V);
for x=1:i
I(t) = Iph(t) - I0(t)*(exp((V(x)+I(t)*Rs)/(n1*Vt(t)))-1)-I02(t)*(exp((V(x)+I(t)*Rs)/(n2*Vt(t)))-1)-((V(x)+I(t)*Rs)/Rsh);
end
end
Thanks in advance
If you have two vectors x (of size 1-by-n) and y (of size 1-by-m) and you want a matrix M of size n-by-m such that M(i,j) = x(i) * y(j) then you are trying to compute the outer product of x and y.
This can be done easily with matlab
>> M = x.' * y;
I have an equation of the type c = Ax + By where c, x and y are vectors of dimensions say 50,000 X 1, and A and B are matrices with dimensions 50,000 X 50,000.
Is there any way in Matlab to find matrices A and B when c, x and y are known?
I have about 100,000 samples of c, x, and y. A and B remain the same for all.
Let X be the collection of all 100,000 xs you got (such that the i-th column of X equals the x_i-th vector).
In the same manner we can define Y and C as 2D collections of ys and cs respectively.
What you wish to solve is for A and B such that
C = AX + BY
You have 2 * 50,000^2 unknowns (all entries of A and B) and numel(C) equations.
So, if the number of data vectors you have is 100,000 you have a single solution (up to linearly dependent samples). If you have more than 100,000 samples you may seek for a least-squares solution.
Re-writing:
C = [A B] * [X ; Y] ==> [X' Y'] * [A';B'] = C'
So, I suppose
[A' ; B'] = pinv( [X' Y'] ) * C'
In matlab:
ABt = pinv( [X' Y'] ) * C';
A = ABt(1:50000,:)';
B = ABt(50001:end,:)';
Correct me if I'm wrong...
EDIT:
It seems like there is quite a fuss around dimensionality here. So, I'll try and make it as clear as possible.
Model: There are two (unknown) matrices A and B, each of size 50,000x50,000 (total 5e9 unknowns).
An observation is a triplet of vectors: (x,y,c) each such vector has 50,000 elements (total of 150,000 observed points at each sample). The underlying model assumption is that an observation is generated by c = Ax + By in this model.
The task: given n observations (that is n triplets of vectors { (x_i, y_i, c_i) }_i=1..n) the task is to uncover A and B.
Now, each sample (x_i,y_i,c_i) induces 50,000 equations of the form c_i = Ax_i + By_i in the unknown A and B. If the number of samples n is greater than 100,000, then there are more than 50,000 * 100,000 ( > 5e9 ) equations and the system is over constraint.
To write the system in a matrix form I proposed to stack all observations into matrices:
A matrix X of size 50,000 x n with its i-th column equals to observed x_i
A matrix Y of size 50,000 x n with its i-th column equals to observed y_i
A matrix C of size 50,000 x n with its i-th column equals to observed c_i
With these matrices we can write the model as:
C = A*X + B*Y
I hope this clears things up a bit.
Thank you #Dan and #woodchips for your interest and enlightening comments.
EDIT (2):
Submitting the following code to octave. In this example instead of 50,000 dimension I work with only 2, instead of n=100,000 observations I settled for n=100:
n = 100;
A = rand(2,2);
B = rand(2,2);
X = rand(2,n);
Y = rand(2,n);
C = A*X + B*Y + .001*randn(size(X)); % adding noise to observations
ABt = pinv( [ X' Y'] ) * C';
Checking the difference between ground truth model (A and B) and recovered ABt:
ABt - [A' ; B']
Yields
ans =
5.8457e-05 3.0483e-04
1.1023e-04 6.1842e-05
-1.2277e-04 -3.2866e-04
-3.1930e-05 -5.2149e-05
Which is close enough to zero. (remember, the observations were noisy and solution is a least-square one).
my current code is below.
What I have is two sets of data phi and theta both 18x30 and I have a for loop running from 1 to 30 which are the 30 columns of my data. Each of these individual columns will produce a matrix 'B' for me. The matrix 'B' is produced after going through the 18 rows of each column.
The problem I have is that I need to multiply all the resulting 'B' matrices for each of the 18 rows with each other in order to get a final matrix for each of the 30 columns, that is why I have set A(:,:,i) = eye(2) so that my first matrix will multiply by A. But I don't want to store A or B for each loop, instead what I want is that on the first loop of i my matrix B will multiply the identity matrix A. Then that matrix A will multiply the next matrix B...with the result of each multiplication being carried forward to the next matrix B that will be calculated, so the multiplications will get done as the program loops. That's why I have the line:
A(:,:,i) = A.*B;
but it doesn't seem to work. I get an error message saying the dimensions must match.
At the end of the program I want to be able to access each of the 30 matrices using a command like:
A(:,:,3), for example, to get my 2x2 matrix for the 3rd column.
Hope that was clear enough!
theta = dlmread('theta.dat');
phi = dlmread('phi.dat');
ne = 1.7;
no = 1.5;
d = 0.000001;
w = 0.000000555;
for i = 1:30
A(:,:,i) = eye(2);
for j = 1:18
nx =((cos(theta(j,i)).^2)/(no^2) + ((sin(theta(j,i)).^2)/(ne^2))).^(-1/2);
D = (2*pi*(nx-no)*d)/w;
x = ((cos(phi(j,i))).^2).*exp((-1i.*D)/2) + ((sin(phi(j,i))).^2).*exp((1i.*D)/2);
y = 1i*(sin(D/2)).*sin(2*phi(j,i));
z = ((cos(phi(j,i))).^2).*exp((1i.*D/2) + ((sin(phi(j,i))).^2).*exp((-1i.*D)/2));
B = [x y;y z];
A(:,:,i) = A.*B;
end
end
B is a 2x2 matrix. For A.*B to work, A must also be 2x2. But A in your program is three-dimensional.
From your problem description, I think you want
A(:,:,i) = A(:,:,i)*B; % Edited now that I see this happens 18 times on the same i
(Please note, I also replaced element-wise multiply .* with matrix multiply *, because that's what it sounds like you want.)
But I suggest
A = eye(2);
and
A = A*B;
and store it at the end like
results(:,:,i) = A;