I am trying to solve a system of nonlinear equations using fsolve. The system is given in the matrix form (image), u_i being the unknowns.
plz suggest how can I create the function to be given input to fsolve.
thanks
Check out the documentation for fsolve. You can create a function handle, either for a function in its own file or anonymously, and call it using fsolve:
h = #(U) P'*U.^2; % function handle
U_next = fsolve(h, U);
If you are doing an inner product, you also need to make sure that your left-hand side has the same number of columns as your right-hand side has rows. For instance, if you have some arbitrary 4x1 matrix P and initial 1x4 vector U, you have to take some transposes to make sure that your inner product works out:
P = rand(4, 1); % random column vector
U = rand(1, 4); % random row vector
h = #(U) P'*U'.^2; % (1x4) * (4x1) = scalar result
Or if you are trying to do element-wise multiplication:
h = #(U) P'.*U.^2; % (1x4) .* (1x4) = (1x4) result with element-wise multiplication
Related
I have a function:
Where ||x|| is an Euclidean distance.
For given n (number of variables) I want Matlab to differentiate this function and then substitute real numbers into it.
What I truly don't understand how to do is:
1) How let Matlab create all these new n variables for later differentiation?
2) Each variable is a vector of dimension = d, i.e x=[x1, x2, ... xd], so later I want to differentiate function with respect to certain vector elements, for example with respect to x1, how can I do it?
EDIT 1: function should be differentiated at each x_i, where i=1:n
The derivative of a sum, is the sum of the derivatives of each element... so, we need to find the derivative only once (you can do this manually, if it is a simple function like in your toy-example, but we do this the general way, using the Symbolic Math Toolbox):
syms x y z % declaring 3 symolic variables
F = 1/(norm([x,y,z])); % declaring a function
f = diff(F,x) % calculate the derivative with regard to the symbolic variable x
f = -(abs(x)*sign(x))/(abs(x)^2 + abs(y)^2 + abs(z)^2)^(3/2)
you now have different options. You can use subs to evaluate the function f (simply assign numeric values to x,y, and z and call subs(f). Or, you create a (numeric) function handle by using matlabFunction (this is the way, I prefer)
fnc = matlabFunction(f); % convert to matlab function
Then, you just need to sum up the vector that you created (well, you need to sum up the sum of each of the two vector elements...)
% create arbitrary vector
n = 10;
x = rand(n+1,3);
% initialize total sum
SumFnc = 0;
% loop through elements
for i = 1:n
% calculate local sum
s = x(i,:)+x(i+1,:);
% call derivative-function + sum up
SumFnc = SumFnc + fnc(s(1),s(2),s(3));
end
I have been using the following custom function to perform the multiplication of a vector by a matrix, in which each element of the vector multiplies a 3x3 block within a (3xN)x(3) matrix:
function [B] = BlockScalar(v,A)
N=size(v,2);
B=zeros(3*N,3);
for i=1:N
B(3*i-2:3*i,:) = v(i).*A(3*i-2:3*i,:);
end
end
Similarly, when I want to multiply a collection of 3x3 matrices by a collection of 3x3 vectors, I use the following
function [B] = BlockMatrix(A,u)
N=size(u,2);
B=zeros(N,3);
for i=1:N
B(i,:) = A(3*i-2:3*i,:)*u(:,i);
end
end
Since I call them very often, these unfortunately, slow down the running of my code significantly. I was wondering if there was a more efficient (perhaps vectorised) version of the above operations.
In both instance, you are able to do away with the for-loops (although without testing, I cannot confirm if this will necessarily speed up your computation).
For the first function, you can do it as follows:
function [B] = BlockScalar(v,A)
% We create a vector N = [1,1,1,2,2,2,3,3,3,...,N,N,N]
N=ceil((1:size(A,1))/3);
% Use N to index v, and let matlab do the expansion
B = v(N).*A;
end
For the second function, we can make a block-diagonal matrix.
function [B] = BlockMatrix(A,u)
N=size(u,2)*3;
% We use a little meshgrid+sparse magic to convert A to a block matrix
[X,Y] = meshgrid(1:N,1:3);
% Use sparse matrices to speed up multiplication and save space
B = reshape(sparse(Y+(ceil((1:N)/3)-1)*3,X,A) * (u(:)),3,size(u,2))';
end
Note that if you are able to access the individual 3x3 matrices, you can potentially make this faster/simpler by using the native blkdiag:
function [B] = BlockMatrix(a,b,c,d,...,u)
% Where A = [a;b;c;d;...];
% We make one of the input matrices sparse to make the whole block matrix sparse
% This saves memory and potentially speeds up multiplication by a lot
% For small enough values of N, however, using sparse may slow things down.
reshape(blkdiag(sparse(a),b,c,d,...) * (u(:)),3,size(u,2))';
end
Here are vectorized solutions:
function [B] = BlockScalar(v,A)
N = size(v,2);
B = reshape(reshape(A,3,N,3) .* v, 3*N, 3);
end
function [B] = BlockMatrix(A,u)
N = size(u,2);
A_r = reshape(A,3,N,3);
B = (A_r(:,:,1) .* u(1,:) + A_r(:,:,2) .* u(2,:) + A_r(:,:,3) .* u(3,:)).';
end
function [B] = BlockMatrix(A,u)
N = size(u,2);
B = sum(reshape(A,3,N,3) .* permute(u, [3 2 1]) ,3).';
end
I have two lists of functions, for instance: log(n*x), n=1:2017 and cos(m*x), m=1:6. I want/need to construct the matrix product of these vectors and then integrating each element of the matrix between 10 and 20.
I have read this post:
Matrix of symbolic functions
but I think that it is not useful for this problem.
I'm trying to do this by using a loop but I can not get it.
Thanks in advance for reading it.
You can solve this problem by assigning the appropriate vectors to n and m as follows:
n = (1:2017)'; % column vector
m = 1:6; % row vector
syms x;
l = log(n*x); % column vector of logs
c = cos(m*x); % row vector of cos
product = l*c; % matrix product
i = int(product, x, 10, 20); % integral from 10 to 20
iDouble = double(i); % convert the result to double
I have a matrix 2NxN.
And I want get some parametrs by this matrix. For example it:
How, I can do it?
You may want to break your 12x6 matrix, into two 6x6 matrix; let's say: Z and Zb (last one for z bar). Odd rows are Z and evens are Zb.
Considering M to be the combined matrices:
Z = M(1:2:end,:)
Zb = M(2:2:end,:)
read about the colon(:) operator and end to see what 1:2:end means.
Hope it helps.
From what I understand here are the first three:
% Random Matrix
% Needs to be defined before the functions since the functions look for
% the m variable
m = rand(12,6);
% Function 1
p = #(i,j) sign(m(i,j)+m(i+1,j)) * max(abs(m(i,j)),abs(m(i+1,j)));
p(2,2)
% Function 2 - Avg of row
pavg = #(i) mean(m(i,:));
pavg(2)
% Function 3
c = #(i,j) abs(m(i,j)+m(i+1,j)) / (abs(m(i,j)) + abs(m(i+1,j)));
c(2,2)
Suppose I have a function y(t,x) = exp(-t)*sin(x)
In Matlab, I define
t = [0: 0.5: 5];
x = [0: 0.1: 10*2*pi];
y = zeros(length(t), length(x)); % empty matrix init
Now, how do I define matrix y without using any loop, such that each element y(i,j) contains the value of desired function y at (t(i), x(j))? Below is how I did it using a for loop.
for i = 1:length(t)
y(i,:) = exp(-t(i)) .* sin(x);
end
Your input vectors x is 1xN and t is 1xM, output matrix y is MxN. To vectorize the code both x and t must have the same dimension as y.
[x_,t_] = meshgrid(x,t);
y_ = exp(-t_) .* sin(x_);
Your example is a simple 2D case. Function meshgrid() works also 3D. Sometimes you can not avoid the loop, in such cases, when your loop can go either 1:N or 1:M, choose the shortest one. Another function I use to prepare vector for vectorized equation (vector x matrix multiplication) is diag().
there is no need for meshgrid; simply use:
y = exp(-t(:)) * sin(x(:)'); %multiplies a column vector times a row vector.
Those might be helpful:
http://www.mathworks.com/access/helpdesk/help/techdoc/ref/meshgrid.html
http://www.mathworks.com/company/newsletters/digest/sept00/meshgrid.html
Good Luck.