I've been looking around but I can't seem to figure out how I should use fsolve to solve my system of nonlinear equations.
so I have a function k, and I know that I want .01 as the starting value (required input for fsolve I believe). I also know that k(1000) =12
Lastly, I have a formula for k,
k(N) = (1/(k(N).^.5 + .9*k(N) -k(N+1))) - ((.94 * .5 *k(N+1)^(1-.5) + .9)/(k(N+1)^.5 +
.9*k(N+1) - k(N+2)))
with N 1:1000. I figured I could just say fsolve(k(N), .01), but with the formula in place of k(N) (or is defining k(N) in a separate function necessary?), but I exceed the dimensions of N and also have issues with dimensions matching since there is N, N+1, and N+2 in the equation.
I think my issue may be stemming from the fact that I have defined vector valued functions where in reality I want them to be single valued, from 1:1000,but I don't know how to represent this in matlab (obviously I can't type in 1000 equations).
Any ideas, suggestions, or comments?
Related
I have the following matrix
R=(A-C)*inv(A+B-C-C')*(A-C');
where A and B are n by n matrices. I want to find n*n matrix C such that the determinant of R is minimized, SO:
C=arg min (det(R));
Is there any function in MATLAB that can handle this problem?
It seems like you are trying to find the minimum of an unconstrained multivariable function. This can probably be achieved with fminunc
fun = #(x)x(1)*exp(-(x(1)^2 + x(2)^2)) + (x(1)^2 + x(2)^2)/20;
x0 = [1,2];
[x,fval] = fminunc(fun,x0)
Note that there are no examples in the documentation where a matrix is used, this is probably because horrendous performance could be expected when trying to solve this problem for a matrix of any nontiny size. (This is not because of matlab, but because of the nature of the problem).
It is also good to realize that this method does not (cannot) guarantee an optimum, only a local optimum.
I'm using octave 3.8.1 which works like matlab.
I have an array of thousands of values I've only included three groupings as an example below:
(amp1=0.2; freq1=3; phase1=1; is an example of one grouping)
t=0;
amp1=0.2; freq1=3; phase1=1; %1st grouping
amp2=1.4; freq2=2; phase2=1.7; %2nd grouping
amp3=0.8; freq3=5; phase3=1.5; %3rd grouping
The Octave / Matlab code below solves for Y so I can plug it back into the equation to check values along with calculating values not located in the array.
clear all
t=0;
Y=0;
a1=[.2,3,1;1.4,2,1.7;.8,5,1.5]
for kk=1:1:length(a1)
Y=Y+a1(kk,1)*cos ((a1(kk,2))*t+a1(kk,3))
kk
end
Y
PS: I'm not trying to solve for Y since it's already solved for I'm trying to solve for Phase
The formulas located below are used to calculate Phase but I'm not sure how to put it into a for loop that will work in an array of n groupings:
How would I write the equation / for loop for finding the phase if I want to find freq=2.5 and amp=.23 and the phase is unknown I've looked online and it may require writing non linear equations which I'm not sure how to convert what I'm trying to do into such an equation.
phase1_test=acos(Y/amp1-amp3*cos(2*freq3*pi*t+phase3)/amp1-amp2*cos(2*freq2*pi*t+phase2)/amp1)-2*freq1*pi*t
phase2_test=acos(Y/amp2-amp3*cos(2*freq3*pi*t+phase3)/amp2-amp1*cos(2*freq1*pi*t+phase1)/amp2)-2*freq2*pi*t
phase3_test=acos(Y/amp3-amp2*cos(2*freq2*pi*t+phase2)/amp3-amp1*cos(2*freq1*pi*t+phase1)/amp3)-2*freq2*pi*t
Image of formula below:
I would like to do a check / calculate phases if given a freq and amp values.
I know I have to do a for loop but how do I convert the phase equation into a for loop so it will work on n groupings in an array and calculate different values not found in the array?
Basically I would be given an array of n groupings and freq=2.5 and amp=.23 and use the formula to calculate phase. Note: freq will not always be in the array hence why I'm trying to calculate the phase using a formula.
Ok, I think I finally understand your question:
you are trying to find a set of phase1, phase2,..., phaseN, such that equations like the ones you describe are satisfied
You know how to find y, and you supply values for freq and amp.
In Matlab, such a problem would be solved using, for example fsolve, but let's look at your problem step by step.
For simplicity, let me re-write your equations for phase1, phase2, and phase3. For example, your first equation, the one for phase1, would read
amp1*cos(phase1 + 2 freq1 pi t) + amp2*cos(2 freq2 pi t + phase2) + amp3*cos(2 freq3 pi t + phase3) - y = 0
Note that ampX (X is a placeholder for 1, 2, 3) are given, pi is a constant, t is given via Y (I think), freqX are given.
Hence, you are, in fact, dealing with a non-linear vector equation of the form
F(phase) = 0
where F is a multi-dimensional (vector) function taking a multi-dimensional (vector) input variable phase (comprised of phase1, phase2,..., phaseN). And you are looking for the set of phaseX, where all of the components of your vector function F are zero. N.B. F is a shorthand for your functions. Therefore, the first component of F, called f1, for example, is
f1 = amp1*cos(phase1+...) + amp2*cos(phase2+...) + amp3*cos(phase3+...) - y = 0.
Hence, f1 is a one-dimensional function of phase1, phase2, and phase3.
The technical term for what you are trying to do is find a zero of a non-linear vector function, or find a solution of a non-linear vector function. In Matlab, there are different approaches.
For a one-dimensional function, you can use fzero, which is explained at http://www.mathworks.com/help/matlab/ref/fzero.html?refresh=true
For a multi-dimensional (vector) function as yours, I would look into using fsolve, which is part of Matlab's optimization toolbox (which means I don't know how to do this in Octave). The function fsolve is explained at http://www.mathworks.com/help/optim/ug/fsolve.html
If you know an approximate solution for your phases, you may also look into iterative, local methods.
In particular, I would recommend you look into the Newton's Method, which allows you to find a solution to your system of equations F. Wikipedia has a good explanation of Newton's Method at https://en.wikipedia.org/wiki/Newton%27s_method . Newton iterations are very simple to implement and you should find a lot of resources online. You will have to compute the derivative of your function F with respect to each of your variables phaseX, which is very simple to compute since you're only dealing with cos() functions. For starters, have a look at the one-dimensional Newton iteration method in Matlab at http://www.math.colostate.edu/~gerhard/classes/331/lab/newton.html .
Finally, if you want to dig deeper, I found a textbook on this topic from the society for industrial and applied math: https://www.siam.org/books/textbooks/fr16_book.pdf .
As you can see, this is a very large field; Newton's method should be able to help you out, though.
Good luck!
I have the following differential equation which I'm not able to solve.
We know the following about the equation:
D(r) is a third grade polynom
D'(1)=D'(2)=0
D(2)=2D(1)
u(1)=450
u'(2)=-K * (u(2)-Te)
Where K and Te are constants.
I want to approximate the problem using a matrix and I managed to solve
the similiar equation: with the same limit conditions for u(1) and u'(2).
On this equation I approximated u' and u'' with central differences and used a finite difference method between r=1 to r=2. I then placed the results in a matrix A in matlab and the limit conditions in the vector Y in matlab and ran u=A\Y to get how the u value changes. Heres my matlab code for the equation I managed to solve:
clear
a=1;
b=2;
N=100;
h = (b-a)/N;
K=3.20;
Ti=450;
Te=20;
A = zeros(N+2);
A(1,1)=1;
A(end,end)=1/(2*h*K);
A(end,end-1)=1;
A(end,end-2)=-1/(2*h*K);
r=a+h:h:b;
%y(i)
for i=1:1:length(r)
yi(i)=-r(i)*(2/(h^2));
end
A(2:end-1,2:end-1)=A(2:end-1,2:end-1)+diag(yi);
%y(i-1)
for i=1:1:length(r)-1
ymin(i)=r(i+1)*(1/(h^2))-1/(2*h);
end
A(3:end-1,2:end-2) = A(3:end-1,2:end-2)+diag(ymin);
%y(i+1)
for i=1:1:length(r)
ymax(i)=r(i)*(1/(h^2))+1/(2*h);
end
A(2:end-1,3:end)=A(2:end-1,3:end)+diag(ymax);
Y=zeros(N+2,1);
Y(1) =Ti;
Y(2)=-(Ti*(r(1)/(h^2)-(1/(2*h))));
Y(end) = Te;
r=[1,r];
u=A\Y;
plot(r,u(1:end-1));
My question is, how do I solve the first differential equation?
As TroyHaskin pointed out in comments, one can determine D up to a constant factor, and that constant factor cancels out in D'/D anyway. Put another way: we can assume that D(1)=1 (a convenient number), since D can be multiplied by any constant. Now it's easy to find the coefficients (done with Wolfram Alpha), and the polynomial turns out to be
D(r) = -2r^3+9r^2-12r+6
with derivative D'(r) = -6r^2+18r-12. (There is also a smarter way to find the polynomial by starting with D', which is quadratic with known roots.)
I would probably use this information right away, computing the coefficient k of the first derivative:
r = a+h:h:b;
k = 1+r.*(-6*r.^2+18*r-12)./(-2*r.^3+9*r.^2-12*r+6);
It seems that k is always positive on the interval [1,2], so if you want to minimize the changes to existing code, just replace r(i) by r(i)/k(i) in it.
By the way, instead of loops like
for i=1:1:length(r)
yi(i)=-r(i)*(2/(h^2));
end
one usually does simply
yi=-r*(2/(h^2));
This vectorization makes the code more compact and can benefit the performance too (not so much in your example, where solving the linear system is the bottleneck). Another benefit is that yi is properly initialized, while with your loop construction, if yi happened to already exist and have length greater than length(r), the resulting array would have extraneous entries. (This is a potential source of hard-to-track bugs.)
I'm new to matlab and try to do some energy minimization work with it. The energy function takes a 3-channel image as input. For every channel, there's a energy term looks like this:
E = x'Ax + ||Bx||^2 + w*||x-c||^2,
where x,c are vectors of length N, A is a matrix of size N*N. A is sparse and positive semi-definite and has 25 non-zero elements per row, giving constraints to all elements of x. B is of size M*N. B is sparse too and has 2 non-zero elements per row. N is about 850,000. M is about 1,000,000. Although B gives more than N constraints, some elements of x have nothing to do with ||Bx||^2 term. The weight w of term ||x-c||^ is quite small, say 1e-3.
I've searched matlab documentation. It looks like I should use lsqnonlin for this problem. Is there a special designed function or option for quadratic form minimization in matlab?
For those who are familiar with computer vision literature, I'm actually trying to implement the algorithm in "Coherent Intrinsic Images from Photo Collections". The authors said they use matlab backslash operator to minimize the energy, but I can't see how a backslash operator can be used in quadratic form problem.
Yes, there is a function specifically for optimizing quadratic cost functions: quadprog. However, if you don't have any linear constraints, then you should be able to write your cost function as
E = x'Mx/2 + vx + k
Finding the point of zero gradient (hopefully a minimum) can then be achieved by taking first derivatives:
dE/dx = Mx + v
setting them to zero giving the solution:
x = -M\v
I need to solve the following SOCP in Matlab:
argmin_x ||R*x||_2 s.t. s^H * x = 1 and ||x||_2 < d,
where x is an Nx1 vector and R is an MxN matrix.
CVX can solve this type of problem. However, CVX requires me to give R and does not allow me to instead give a function handle that will return R*x. This is a problem for me since once R becomes large, computing R*x directly takes too long. There exists an efficient algorithm for computing R*x that I would like to take advantage of, so I am hoping that there is another SOCP solver that I could use.