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The code below currently plots the fourier series for a square wave for N terms. Is there any way I could change the range from [0;1] to [-1;1]?
% Assignment of variables
syms t
% Function variables
N = 5;
T0 = 1;
w0 = 2*pi/T0;
Imin = 0;
Imax = 0.5;
% Function
ft = 1;
% First term calculation
a0 = (1/T0)*int(ft, t, Imin, Imax);
y = a0;
% Calculation of n terms
for n = 1:N
an = (2/T0)*int(ft*cos(n*w0*t), t, Imin, Imax);
bn = (2/T0)*int(ft*sin(n*w0*t), t, Imin, Imax);
y = y + an*cos(n*w0*t) + bn*sin(n*w0*t);
end
fplot(y, [-4,4], "Black")
grid on
If you are talking about the figure scale, then ylim([-1 1])
1.- The following does what you asked for:
clear all;clc;close all
syms t
assume(t>0 & t<1)
% Function variables
N = 5;
T0 = 1;
w0 = 2*pi/T0;
Imin = 0;
Imax = 1;
% Function
h1=heaviside(t-.5)
h2=heaviside(t+.5)
ht=-2*((h1-h2)+.5)
% First term calculation
a0 = (1/T0)*int(ht, t, Imin, Imax);
y = a0;
% Calculation of n terms
for n = 1:N
an = (2/T0)*int(ht*cos(n*w0*t), t, Imin, Imax);
bn = (2/T0)*int(ht*sin(n*w0*t), t, Imin, Imax);
y = y + an*cos(n*w0*t) + bn*sin(n*w0*t);
end
fplot(y, [-4,4], "Black")
grid on
2.- You allocate a specific group of code lines headed with % Function to precisely define the function.
Yet you actually define the function with Imin and Imax.
It's good practice to constrain the function definition within the lines you intend for such purpose, not to scatter the function all over the place.
I have always used R, so I am quite new to Matlab and running into some troubleshooting issues. I am running some code for a tensor factorization method (available here: https://github.com/caobokai/tBNE). To start I tried to run the demo code, which generates simulated data to run the method with, which results in the following error(s):
Error using feval
Undefined function or variable 'Sfun'.
Error in OptStiefelGBB (line 199)
[F, G] = feval(fun, X , varargin{:}); out.nfe = 1;
Error in tbne_demo>tBNE_fun (line 124)
S, #Sfun, opts, B, P, X, L, D, W, Y, alpha, beta);
Here is the block of code I am running:
clear
clc
addpath(genpath('./tensor_toolbox'));
addpath(genpath('./FOptM'));
rng(5489, 'twister');
m = 10;
n = 10;
k = 10; % rank for tensor
[X, Z, Y] = tBNE_data(m, n, k); % generate the tensor, guidance and label
[T, W] = tBNE_fun(X, Z, Y, k);
[~, y1] = max(Y, [], 2);
[~, y2] = max(T{3} * W, [], 2);
fprintf('accuracy %3.2e\n', sum(y1 == y2) / n);
function [X, Z, Y] = tBNE_data(m, n, k)
B = randn(m, k);
S = randn(n, k);
A = {B, B, S};
X = ktensor(A);
Z = randn(n, 4);
Y = zeros(n, 2);
l = ceil(n / 2);
Y(1 : l, 1) = 1;
Y(l + 1 : end, 2) = 1;
X = tensor(X);
end
function [T, W] = tBNE_fun(X, Z, Y, k)
% t-BNE computes brain network embedding based on constrained tensor factorization
%
% INPUT
% X: brain networks stacked in a 3-way tensor
% Z: side information
% Y: label information
% k: rank of CP factorization
%
% OUTPUT
% T is the factor tensor containing
% vertex factor matrix B = T{1} and
% subject factor matrix S = T{3}
% W is the weight matrix
%
% Example: see tBNE_demo.m
%
% Reference:
% Bokai Cao, Lifang He, Xiaokai Wei, Mengqi Xing, Philip S. Yu,
% Heide Klumpp and Alex D. Leow. t-BNE: Tensor-based Brain Network Embedding.
% In SDM 2017.
%
% Dependency:
% [1] Matlab tensor toolbox v 2.6
% Brett W. Bader, Tamara G. Kolda and others
% http://www.sandia.gov/~tgkolda/TensorToolbox
% [2] A feasible method for optimization with orthogonality constraints
% Zaiwen Wen and Wotao Yin
% http://www.math.ucla.edu/~wotaoyin/papers/feasible_method_matrix_manifold.html
%% set algorithm parameters
printitn = 10;
maxiter = 200;
fitchangetol = 1e-4;
alpha = 0.1; % weight for guidance
beta = 0.1; % weight for classification loss
gamma = 0.1; % weight for regularization
u = 1e-6;
umax = 1e6;
rho = 1.15;
opts.record = 0;
opts.mxitr = 20;
opts.xtol = 1e-5;
opts.gtol = 1e-5;
opts.ftol = 1e-8;
%% compute statistics
dim = size(X);
normX = norm(X);
numClass = size(Y, 2);
m = dim(1);
n = dim(3);
l = size(Y, 1);
D = [eye(l), zeros(l, n - l)];
L = diag(sum(Z * Z')) - Z * Z';
%% initialization
B = randn(m, k);
P = B;
S = randn(n, k);
S = orth(S);
W = randn(k, numClass);
U = zeros(m, k); % Lagrange multipliers
%% main loop
fit = 0;
for iter = 1 : maxiter
fitold = fit;
% update B
ete = (S' * S) .* (P' * P); % compute E'E
b = 2 * ete + u * eye(k);
c = 2 * mttkrp(X, {B, P, S}, 1) + u * P + U;
B = c / b;
% update P
ftf = (S' * S) .* (B' * B); % compute F'F
b = 2 * ftf + u * eye(k);
c = 2 * mttkrp(X, {B, P, S}, 2) + u * B - U;
P = c / b;
% update U
U = U + u * (P - B);
% update u
u = min(rho * u, umax);
% update S
tic;
[S, out] = OptStiefelGBB(...
S, #Sfun, opts, B, P, X, L, D, W, Y, alpha, beta);
tsolve = toc;
fprintf(...
['[S]: obj val %7.6e, cpu %f, #func eval %d, ', ...
'itr %d, |ST*S-I| %3.2e\n'], ...
out.fval, tsolve, out.nfe, out.itr, norm(S' * S - eye(k), 'fro'));
% update W
H = D * S;
W = (H' * H + gamma * eye(k)) \ H' * Y;
% compute the fit
T = ktensor({B, P, S});
normresidual = sqrt(normX ^ 2 + norm(T) ^ 2 - 2 * innerprod(X, T));
fit = 1 - (normresidual / normX);
fitchange = abs(fitold - fit);
if mod(iter, printitn) == 0
fprintf(' Iter %2d: fitdelta = %7.1e\n', iter, fitchange);
end
% check for convergence
if (iter > 1) && (fitchange < fitchangetol)
break;
end
end
%% clean up final results
T = arrange(T); % columns are normalized
fprintf('factorization error %3.2e\n', fit);
end
I know that there is little context here, but my suspicion is that I need to have Simulink, as Sfun is a Simulink related function(?). The script requires two toolboxes: tensor_toolbox, and FOptM.
Available at:
https://www.sandia.gov/~tgkolda/TensorToolbox/index-2.6.html
https://github.com/andland/FOptM
Thank you so much for your help,
Paul
Although SFun is an often used abbreviation for a Simulink S-Function, in this case the error has nothing to do with Simulink, and the name is a coincidence. (There is no Simulink related function specifically called Sfun, it is just a general term.)
Your error message has #Sfun in it, which is a way in MATLAB of creating a function handle to an (m-code) function called Sfun. I'd summize from the code you've shown that this is a cost function used in the optimization.
If you look at the code that your code is based on (tBNE_fun.m) you'll see that there is a function at the end of the file called Sfun. It is this that you are missing.
I have a non-uniform rectangular grid along D dimensions, a matrix of logical values V on the grid, and a matrix of query data points X. The number of grid points differs across dimensions.
I run the interpolation multiple times for the same grid G and query X, but for different values V.
The goal is to precompute the indexes and weights for the interpolation and to reuse them, because they are always the same.
Here is an example in 2 dimensions, in which I have to compute indexes and values every time within the loop, but I want to compute them only once before the loop. I keep the data types from my application (mostly single and logical gpuArrays).
% Define grid
G{1} = single([0; 1; 3; 5; 10]);
G{2} = single([15; 17; 18; 20]);
% Steps and edges are reduntant but help make interpolation a bit faster
S{1} = G{1}(2:end)-G{1}(1:end-1);
S{2} = G{2}(2:end)-G{2}(1:end-1);
gpuInf = 1e10;
% It's my workaround for a bug in GPU version of discretize in Matlab R2017a.
% It throws an error if edges contain Inf, realmin, or realmax. Seems fixed in R2017b prerelease.
E{1} = [-gpuInf; G{1}(2:end-1); gpuInf];
E{2} = [-gpuInf; G{2}(2:end-1); gpuInf];
% Generate query points
n = 50; X = gpuArray(single([rand(n,1)*14-2, 14+rand(n,1)*7]));
[G1, G2] = ndgrid(G{1},G{2});
for i = 1 : 4
% Generate values on grid
foo = #(x1,x2) (sin(x1+rand) + cos(x2*rand))>0;
V = gpuArray(foo(G1,G2));
% Interpolate
V_interp = interpV(X, V, G, E, S);
% Plot results
subplot(2,2,i);
contourf(G1, G2, V); hold on;
scatter(X(:,1), X(:,2),50,[ones(n,1), 1-V_interp, 1-V_interp],'filled', 'MarkerEdgeColor','black'); hold off;
end
function y = interpV(X, V, G, E, S)
y = min(1, max(0, interpV_helper(X, 1, 1, 0, [], V, G, E, S) ));
end
function y = interpV_helper(X, dim, weight, curr_y, index, V, G, E, S)
if dim == ndims(V)+1
M = [1,cumprod(size(V),2)];
idx = 1 + (index-1)*M(1:end-1)';
y = curr_y + weight .* single(V(idx));
else
x = X(:,dim); grid = G{dim}; edges = E{dim}; steps = S{dim};
iL = single(discretize(x, edges));
weightL = weight .* (grid(iL+1) - x) ./ steps(iL);
weightH = weight .* (x - grid(iL)) ./ steps(iL);
y = interpV_helper(X, dim+1, weightL, curr_y, [index, iL ], V, G, E, S) +...
interpV_helper(X, dim+1, weightH, curr_y, [index, iL+1], V, G, E, S);
end
end
I found a way to do this and posting it here because (as of now) two more people are interested. It takes only a slight modification to my original code (see below).
% Define grid
G{1} = single([0; 1; 3; 5; 10]);
G{2} = single([15; 17; 18; 20]);
% Steps and edges are reduntant but help make interpolation a bit faster
S{1} = G{1}(2:end)-G{1}(1:end-1);
S{2} = G{2}(2:end)-G{2}(1:end-1);
gpuInf = 1e10;
% It's my workaround for a bug in GPU version of discretize in Matlab R2017a.
% It throws an error if edges contain Inf, realmin, or realmax. Seems fixed in R2017b prerelease.
E{1} = [-gpuInf; G{1}(2:end-1); gpuInf];
E{2} = [-gpuInf; G{2}(2:end-1); gpuInf];
% Generate query points
n = 50; X = gpuArray(single([rand(n,1)*14-2, 14+rand(n,1)*7]));
[G1, G2] = ndgrid(G{1},G{2});
[W, I] = interpIW(X, G, E, S); % Precompute weights W and indexes I
for i = 1 : 4
% Generate values on grid
foo = #(x1,x2) (sin(x1+rand) + cos(x2*rand))>0;
V = gpuArray(foo(G1,G2));
% Interpolate
V_interp = sum(W .* single(V(I)), 2);
% Plot results
subplot(2,2,i);
contourf(G1, G2, V); hold on;
scatter(X(:,1), X(:,2), 50,[ones(n,1), 1-V_interp, 1-V_interp],'filled', 'MarkerEdgeColor','black'); hold off;
end
function [W, I] = interpIW(X, G, E, S)
global Weights Indexes
Weights=[]; Indexes=[];
interpIW_helper(X, 1, 1, [], G, E, S, []);
W = Weights; I = Indexes;
end
function [] = interpIW_helper(X, dim, weight, index, G, E, S, sizeV)
global Weights Indexes
if dim == size(X,2)+1
M = [1,cumprod(sizeV,2)];
Weights = [Weights, weight];
Indexes = [Indexes, 1 + (index-1)*M(1:end-1)'];
else
x = X(:,dim); grid = G{dim}; edges = E{dim}; steps = S{dim};
iL = single(discretize(x, edges));
weightL = weight .* (grid(iL+1) - x) ./ steps(iL);
weightH = weight .* (x - grid(iL)) ./ steps(iL);
interpIW_helper(X, dim+1, weightL, [index, iL ], G, E, S, [sizeV, size(grid,1)]);
interpIW_helper(X, dim+1, weightH, [index, iL+1], G, E, S, [sizeV, size(grid,1)]);
end
end
To do the task the whole process of interpolation ,except computing the interpolated values, should be done. Here is a solution translated from the Octave c++ source. Format of the input is the same as the frst signature of the interpn function except that there is no need to the v array. Also Xs should be vectors and should not be of the ndgrid format. Both the outputs W (weights) and I (positions) have the size (a ,b) that a is the number of neighbors of a points on the grid and b is the number of requested points to be interpolated.
function [W , I] = lininterpnw(varargin)
% [W I] = lininterpnw(X1,X2,...,Xn,Xq1,Xq2,...,Xqn)
n = numel(varargin)/2;
x = varargin(1:n);
y = varargin(n+1:end);
sz = cellfun(#numel,x);
scale = [1 cumprod(sz(1:end-1))];
Ni = numel(y{1});
index = zeros(n,Ni);
x_before = zeros(n,Ni);
x_after = zeros(n,Ni);
for ii = 1:n
jj = interp1(x{ii},1:sz(ii),y{ii},'previous');
index(ii,:) = jj-1;
x_before(ii,:) = x{ii}(jj);
x_after(ii,:) = x{ii}(jj+1);
end
coef(2:2:2*n,1:Ni) = (vertcat(y{:}) - x_before) ./ (x_after - x_before);
coef(1:2:end,:) = 1 - coef(2:2:2*n,:);
bit = permute(dec2bin(0:2^n-1)=='1', [2,3,1]);
%I = reshape(1+scale*bsxfun(#plus,index,bit), Ni, []).'; %Octave
I = reshape(1+sum(bsxfun(#times,scale(:),bsxfun(#plus,index,bit))), Ni, []).';
W = squeeze(prod(reshape(coef(bsxfun(#plus,(1:2:2*n).',bit),:).',Ni,n,[]),2)).';
end
Testing:
x={[1 3 8 9],[2 12 13 17 25]};
v = rand(4,5);
y={[1.5 1.6 1.3 3.5,8.1,8.3],[8.4,13.5,14.4,23,23.9,24.2]};
[W I]=lininterpnw(x{:},y{:});
sum(W.*v(I))
interpn(x{:},v,y{:})
Thanks to #SardarUsama for testing and his useful comments.
Introduction
I am using Matlab to simulate some dynamic systems through numerically solving systems of Second Order Ordinary Differential Equations using ODE45. I found a great tutorial from Mathworks (link for tutorial at end) on how to do this.
In the tutorial the system of equations is explicit in x and y as shown below:
x''=-D(y) * x' * sqrt(x'^2 + y'^2)
y''=-D(y) * y' * sqrt(x'^2 + y'^2) + g(y)
Both equations above have form y'' = f(x, x', y, y')
Question
However, I am coming across systems of equations where the variables can not be solved for explicitly as shown in the example. For example one of the systems has the following set of 3 second order ordinary differential equations:
y double prime equation
y'' - .5*L*(x''*sin(x) + x'^2*cos(x) + (k/m)*y - g = 0
x double prime equation
.33*L^2*x'' - .5*L*y''sin(x) - .33*L^2*C*cos(x) + .5*g*L*sin(x) = 0
A single prime is first derivative
A double prime is second derivative
L, g, m, k, and C are given parameters.
How can Matlab be used to numerically solve a set of second order ordinary differential equations where second order can not be explicitly solved for?
Thanks!
Your second system has the form
a11*x'' + a12*y'' = f1(x,y,x',y')
a21*x'' + a22*y'' = f2(x,y,x',y')
which you can solve as a linear system
[x'', y''] = A\f
or in this case explicitly using Cramer's rule
x'' = ( a22*f1 - a12*f2 ) / (a11*a22 - a12*a21)
y'' accordingly.
I would strongly recommend leaving the intermediate variables in the code to reduce chances for typing errors and avoid multiple computation of the same expressions.
Code could look like this (untested)
function dz = odefunc(t,z)
x=z(1); dx=z(2); y=z(3); dy=z(4);
A = [ [-.5*L*sin(x), 1] ; [.33*L^2, -0.5*L*sin(x)] ]
b = [ [dx^2*cos(x) + (k/m)*y-g]; [-.33*L^2*C*cos(x) + .5*g*L*sin(x)] ]
d2 = A\b
dz = [ dx, d2(1), dy, d2(2) ]
end
Yes your method is correct!
I post the following code below:
%Rotating Pendulum Sym Main
clc
clear all;
%Define parameters
global M K L g C;
M = 1;
K = 25.6;
L = 1;
C = 1;
g = 9.8;
% define initial values for theta, thetad, del, deld
e_0 = 1;
ed_0 = 0;
theta_0 = 0;
thetad_0 = .5;
initialValues = [e_0, ed_0, theta_0, thetad_0];
% Set a timespan
t_initial = 0;
t_final = 36;
dt = .01;
N = (t_final - t_initial)/dt;
timeSpan = linspace(t_final, t_initial, N);
% Run ode45 to get z (theta, thetad, del, deld)
[t, z] = ode45(#RotSpngHndl, timeSpan, initialValues);
%initialize variables
e = zeros(N,1);
ed = zeros(N,1);
theta = zeros(N,1);
thetad = zeros(N,1);
T = zeros(N,1);
V = zeros(N,1);
x = zeros(N,1);
y = zeros(N,1);
for i = 1:N
e(i) = z(i, 1);
ed(i) = z(i, 2);
theta(i) = z(i, 3);
thetad(i) = z(i, 4);
T(i) = .5*M*(ed(i)^2 + (1/3)*L^2*C*sin(theta(i)) + (1/3)*L^2*thetad(i)^2 - L*ed(i)*thetad(i)*sin(theta(i)));
V(i) = -M*g*(e(i) + .5*L*cos(theta(i)));
E(i) = T(i) + V(i);
end
figure(1)
plot(t, T,'r');
hold on;
plot(t, V,'b');
plot(t,E,'y');
title('Energy');
xlabel('time(sec)');
legend('Kinetic Energy', 'Potential Energy', 'Total Energy');
Here is function handle file for ode45:
function dz = RotSpngHndl(~, z)
% Define Global Parameters
global M K L g C
A = [1, -.5*L*sin(z(3));
-.5*L*sin(z(3)), (1/3)*L^2];
b = [.5*L*z(4)^2*cos(z(3)) - (K/M)*z(1) + g;
(1/3)*L^2*C*cos(z(3)) + .5*g*L*sin(z(3))];
X = A\b;
% return column vector [ed; edd; ed; edd]
dz = [z(2);
X(1);
z(4);
X(2)];
I am trying to compute x1^i * x2^j * x3^k * ......
This is my code so far:
for l = 1:N
f = 1;
for i = 0:2
for j = 0:2-i
for k = 0:2-j
for m = 0:2-k
g(l,f) = x1(l)^i*x2(l)^j*x3(l)^k*x4(l)^m;
f = f+1;
end
end
end
end
end
How can I do this easier or without a loop?
I do not have MATLAB on hand here, but what I'd do is make a vector X = [x1, x2, ..., xn] of bases and a vector P = [i, j, k, ..., z] of powers, and then compute prod(power(X, P)).
power() does an element-wise power function, and prod takes the product of every element in the vector.