I am going to generate 4000 samples(x has 10 dimentions) from a pre-defined function in interval ([-10,10]^10) (f5 =#(x) 5*x(1)- 6*x(3)+x(4)^2+x(8)*x(10))
How can I do this in matlab? Actually, I thought that I should condition over all dimensions.
I would appreciate any help you can provide
you can generate some rand x in the range using the following method:
nums = rand(4000,10); % between zero to one
% If "x" in "[0,1]", "(b-a)*x + a" would be in "[a,b]"
nums = 20*nums - 10; % between -10 to 10
and then apply the function on the nums:
I = 1:size(nums,1);
F = #(i) f5(nums(i,:));
vals = arrayfun(F, I.');
Related
I'm trying to create a band matriz in Matlab, which should looks like this for matrix_size = 6 and band_width = 1:
[1,1,0,0,0,0]
[1,2,2,0,0,0]
[0,2,3,3,0,0]
[0,0,3,4,4,0]
[0,0,0,4,5,5]
[0,0,0,0,5,6]
The values should be like this.
I did function, which gives me result:
[1,1,0,0,0,0]
[1,1,1,0,0,0]
[0,1,1,1,0,0]
[0,0,1,1,1,0]
[0,0,0,1,1,1]
[0,0,0,0,1,1]
Code of my function:
function M=bandmatrix(n,r)
% n -- matriz size
% r -- band width
% n >= r + 2
M = sign(conv2(eye(n),ones(r+1),'same'));
end
How I can do this function? I also would be grateful for the function, in which walues are the same as I want, but function doesn't depends on band width. Thank you!
you can use diag as follows:
diag(1:6)+diag(1:5,1)+diag(1:5,-1)
Generally, for any order n:
diag(1:n)+diag(1:n-1,1)+diag(1:n-1,-1)
You could also do something like:
clear;
m = 6;
n = 6;
diags = repmat((1:6)', 1, 3);
diagidx = [-1:1];
for k = 1:length(diagidx)
if(diagidx(k) > 0)
diags(:, k) = circshift(diags(:, k), diagidx(k), 1);
end
end
sparseMat = spdiags(diags, [-1:1], m, n);
myMat = full(sparseMat)
Using sparse matrix allows us to specify the data and allocate only once.
But it also means the values used in the diagonal are needed to be shifted according to the matrix being fat or skinny.
Given a set of datapoints I'm trying to approximate the coefficients a,b in the function U(x)=8-ax^b using Newtons method in MATLAB.
x = [150 200 300 500 1000 2000]';
y = [2 3 4 5 6 7]';
a=170; b=-0.7; iter = 0;
for iter=1:5
f=8-a*x.^(b) -y;
J = [-x.^b -a*b*x.^(b-1)]; %Jacobis matrix
h=J\f;
a=a-h(1); b=b-h(2);
disp(norm(f))
iter = iter+1;
end
The results are incorrect and I've not been sucessful of finding the misstep. All help is appreciated.
The jacobi matrix is wrong. Using Newton's method, you're trying to find the values of a and bthat would solve the equations 8-ax^b - y = 0. So, your Jacobi should be the derivatives of f with respect to a and b. That is J = [df/da df/db], resulting in:
J = [-x.^b -a.*x.^b.*log(x)]
and you will get the following curve for the 5 iterations:
Note that you can easily linearize your model with
Log(8 - U(x)) = Log(a) + b Log(x)
I am trying to compute the value of this integral using Matlab
Here the other parameters have been defined or computed in the earlier part of the program as follows
N = 2;
sigma = [0.01 0.1];
l = [15];
meu = 4*pi*10^(-7);
f = logspace ( 1, 6, 500);
w=2*pi.*f;
for j = 1 : length(f)
q2(j)= sqrt(sqrt(-1)*2*pi*f(j)*meu*sigma(2));
q1(j)= sqrt(sqrt(-1)*2*pi*f(j)*meu*sigma(1));
C2(j)= 1/(q2(j));
C1(j)= (q1(j)*C2(j) + tanh(q1(j)*l))/(q1(j)*(1+q1(j)*C2(j)*tanh(q1(j)*l)));
Z(j) = sqrt(-1)*2*pi*f(j)*C1(j);
Apprho(j) = meu*(1/(2*pi*f(j))*(abs(Z(j))^2));
Phi(j) = atan(imag(Z(j))/real(Z(j)));
end
%integration part
c1=w./(2*pi);
rho0=1;
fun = #(x) log(Apprho(x)/rho0)/(x.^2-w^2);
c2= integral(fun,0,Inf);
phin=pi/4-c1.*c2;
I am getting an error like this
could anyone help and tell me where i am going wrong.thanks in advance
Define Apprho in a separate *.m function file, instead of storing it in an array:
function [ result ] = Apprho(x)
%
% Calculate f and Z based on input argument x
%
% ...
%
meu = 4*pi*10^(-7);
result = meu*(1/(2*pi*f)*(abs(Z)^2));
end
How you calculate f and Z is up to you.
MATLAB's integral works by calling the function (in this case, Apprho) repeatedly at many different x values. The x values called by integral don't necessarily correspond to the 1: length(f) values used in your original code, which is why you received errors.
I have a function that transforms R^N to R^M. For simplicity, lets just let it be the identity function #(z) z where z may be a vector. I want to apply a function to a list of parameters of size K x N and have it map to K x M output.
Here is my attempt:
function out_matrix = array_fun_matrix(f, vals)
for i=1:size(vals,1)
f_val = f(vals(i,:));
if(size(f_val,1) > 1) %Need to stack up rows, so convert as required.
f_val = f_val';
end
out_matrix(i,:) = f_val;
end
end
You can try it with
array_fun_matrix(#(z) z(1)^2 + z(2)^2 + z(3), [0 1 0; 1 1 1; 1 2 1; 2 2 2])
The question: Is there a better and more efficient way to do this with vectorization, etc.? Did I miss a built-in function?
Examples of non-vectorizable functions: There are many, usually involving elaborate sub-steps and numerical solutions. A trivial example is something like looking for the numerical solution to an equation, which in term is using numerical quadrature. i.e. let params = [b c] and solve for the a such that int_0^a ((z + b)^2) dz = c
(I know here you could do some calculus, but the integral here is stripped down). Implementing this example,
find_equilibrium = #(param) fzero(#(a) integral(#(x) (x + param(1)).^2 - param(2), 0, a), 1)
array_fun_matrix(find_equilibrium, [0 1; 0 .8])
You can use the cellfun function, but you'll need to manipulate your data a bit:
function out_matrix = array_fun_matrix(f, vals)
% Convert your data to a cell array:
cellVals = mat2cell(vals, ones(1,size(vals,1)));
% apply the function:
out_cellArray = cellfun(f, cellVals, 'UniformOutput', false);
% Convert back to matrix:
out_matrix = cell2mat(out_cellArray);
end
If you don't like this implementation, you can improve the performance of yours by preallocating the out_matrix:
function out_matrix = array_fun_matrix(f, vals)
firstOutput = f(vals(1,:));
out_matrix = zeros(size(vals,1), length(firstOutput)); % preallocate for speed.
for i=1:size(vals,1)
f_val = f(vals(i,:));
if(size(f_val,1) > 1) %Need to stack up rows, so convert as required.
f_val = f_val';
end
out_matrix(i,:) = f_val;
end
end
I want to vectorize the following MATLAB code. I think it must be simple but I'm finding it confusing nevertheless.
r = some constant less than m or n
[m,n] = size(C);
S = zeros(m-r,n-r);
for i=1:m-r+1
for j=1:n-r+1
S(i,j) = sum(diag(C(i:i+r-1,j:j+r-1)));
end
end
The code calculates a table of scores, S, for a dynamic programming algorithm, from another score table, C.
The diagonal summing is to generate scores for individual pieces of the data used to generate C, for all possible pieces (of size r).
Thanks in advance for any answers! Sorry if this one should be obvious...
Note
The built-in conv2 turned out to be faster than convnfft, because my eye(r) is quite small ( 5 <= r <= 20 ). convnfft.m states that r should be > 20 for any benefit to manifest.
If I understand correctly, you're trying to calculate the diagonal sum of every subarray of C, where you have removed the last row and column of C (if you should not remove the row/col, you need to loop to m-r+1, and you need to pass the entire array C to the function in my solution below).
You can do this operation via a convolution, like so:
S = conv2(C(1:end-1,1:end-1),eye(r),'valid');
If C and r are large, you may want to have a look at CONVNFFT from the Matlab File Exchange to speed up calculations.
Based on the idea of JS, and as Jonas pointed out in the comments, this can be done in two lines using IM2COL with some array manipulation:
B = im2col(C, [r r], 'sliding');
S = reshape( sum(B(1:r+1:end,:)), size(C)-r+1 );
Basically B contains the elements of all sliding blocks of size r-by-r over the matrix C. Then we take the elements on the diagonal of each of these blocks B(1:r+1:end,:), compute their sum, and reshape the result to the expected size.
Comparing this to the convolution-based solution by Jonas, this does not perform any matrix multiplication, only indexing...
I would think you might need to rearrange C into a 3D matrix before summing it along one of the dimensions. I'll post with an answer shortly.
EDIT
I didn't manage to find a way to vectorise it cleanly, but I did find the function accumarray, which might be of some help. I'll look at it in more detail when I am home.
EDIT#2
Found a simpler solution by using linear indexing, but this could be memory-intensive.
At C(1,1), the indexes we want to sum are 1+[0, m+1, 2*m+2, 3*m+3, 4*m+4, ... ], or (0:r-1)+(0:m:(r-1)*m)
sum_ind = (0:r-1)+(0:m:(r-1)*m);
create S_offset, an (m-r) by (n-r) by r matrix, such that S_offset(:,:,1) = 0, S_offset(:,:,2) = m+1, S_offset(:,:,3) = 2*m+2, and so on.
S_offset = permute(repmat( sum_ind, [m-r, 1, n-r] ), [1, 3, 2]);
create S_base, a matrix of base array addresses from which the offset will be calculated.
S_base = reshape(1:m*n,[m n]);
S_base = repmat(S_base(1:m-r,1:n-r), [1, 1, r]);
Finally, use S_base+S_offset to address the values of C.
S = sum(C(S_base+S_offset), 3);
You can, of course, use bsxfun and other methods to make it more efficient; here I chose to lay it out for clarity. I have yet to benchmark this to see how it compares with the double-loop method though; I need to head home for dinner first!
Is this what you're looking for? This function adds the diagonals and puts them into a vector similar to how the function 'sum' adds up all of the columns in a matrix and puts them into a vector.
function [diagSum] = diagSumCalc(squareMatrix, LLUR0_ULLR1)
%
% Input: squareMatrix: A square matrix.
% LLUR0_ULLR1: LowerLeft to UpperRight addition = 0
% UpperLeft to LowerRight addition = 1
%
% Output: diagSum: A vector of the sum of the diagnols of the matrix.
%
% Example:
%
% >> squareMatrix = [1 2 3;
% 4 5 6;
% 7 8 9];
%
% >> diagSum = diagSumCalc(squareMatrix, 0);
%
% diagSum =
%
% 1 6 15 14 9
%
% >> diagSum = diagSumCalc(squareMatrix, 1);
%
% diagSum =
%
% 7 12 15 8 3
%
% Written by M. Phillips
% Oct. 16th, 2013
% MIT Open Source Copywrite
% Contact mphillips#hmc.edu fmi.
%
if (nargin < 2)
disp('Error on input. Needs two inputs.');
return;
end
if (LLUR0_ULLR1 ~= 0 && LLUR0_ULLR1~= 1)
disp('Error on input. Only accepts 0 or 1 as input for second condition.');
return;
end
[M, N] = size(squareMatrix);
if (M ~= N)
disp('Error on input. Only accepts a square matrix as input.');
return;
end
diagSum = zeros(1, M+N-1);
if LLUR0_ULLR1 == 1
squareMatrix = rot90(squareMatrix, -1);
end
for i = 1:length(diagSum)
if i <= M
countUp = 1;
countDown = i;
while countDown ~= 0
diagSum(i) = squareMatrix(countUp, countDown) + diagSum(i);
countUp = countUp+1;
countDown = countDown-1;
end
end
if i > M
countUp = i-M+1;
countDown = M;
while countUp ~= M+1
diagSum(i) = squareMatrix(countUp, countDown) + diagSum(i);
countUp = countUp+1;
countDown = countDown-1;
end
end
end
Cheers