ode45 task.docx
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This document defines an ordinary differential equation (ODE) model of a cart-pendulum system. It sets the initial conditions and time span, calls the ODE solver ode45 to solve the system, and plots the results. The ODE function defines the equations of motion for the cart position and pendulum angle derivatives based on the system parameters.
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![clear all; close all; clc;
tspan=linspace(0,2.100) % time span ranges from 0 to 15 sec
% Defined the Inintial conditions
x0=[6;-1;1;1] % x1(0), x2(0), x1prime(0), x2prime(0);
%ode code
[t,x]= ode45(@odefun,tspan,x0) %odefun-name of function
%make plots
plot(t,x(:,1)) % first degree
hold on % add second plot
plot(t,x(:,2))% plot second degree
legend ('x1-cart','x2-pendulum') % created the legends on
the graph
% step 4: creating the derivaticce function
function dx2dt= odefun(t,x) % argument are t, x
% we defiend the parameters values
m1 = 0.5; % mass of cart
m2 = 0.2; % mass of pendulum
L=0.3;
Izz =0.006;
g = 9.81;
f=1
p1 = ((m1+m2)*(Izz+(m2*(L^2))))-((m2^2)*(L^2));
% initializing the column vector for the output derivative
dx2dt= zeros(4,1);
dx2dt(1)=x(3);
dx2dt(2)=x(4);
dx2dt(3)= (f/(m1+m2))-((((m2*L)^2)*g)/p1)*x(2); % x double
derivative equation
dx2dt(4)=m2*g*L*x(2) -((m2*L*f)/(m1+m2))/p1 ; % Phi double
derivative equation
% dx2dt is the output to our ode function
end](https://image.slidesharecdn.com/ode45task-220529191040-72c21f3e/85/ode45-task-docx-1-320.jpg)
Recommended
Use the same variable names and write the function F - Force(x-ks-kc-l.pdf
Use the same variable names and write the function F = Force(x,ks,kc,l0,S) ; function l0 =
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#!/usr/bin/env octave
% Only modify this file where you see a "% TODO"
% Functions that do not contain this should not be modified.
% If you modify them for debugging, please remove/comment those modifications.
% This line tells octave that this is not a function file.
1;
% Workaround fltk bug. You might need to comment this line out.
graphics_toolkit("gnuplot");
% Meanings of commonly encountered variables:
% n = number of particles
% x = 2*n dimensional column vector with the positions of the particles
% v = 2*n dimensional column vector with the velocities of the particles
% Note that the ordering is always x1 y1 x2 y2 x3 y3 ...
% ks = spring constant
% kc = penalty collision strength
% l0 = matrix of spring rest lengths.
% l0(i,j) = initial length of spring between particles i and j (i<j)
% Note that l0(i,j) should not be used if i>=j.
% S = matrix indicating which springs exist
% S(i,j) = 1 if there is a spring between particles i and j (i<j)
% Note that S(i,j) should not be used if i>=j.
% m = mass of the particles (all particles have the same mass)
% Computes the total potential energy for the system.
function E = Potential_Energy(x,ks,kc,l0,S)
n = rows(S);
E = 0;
% For each pair of particles (i<j)
for i = 1:n
xi = x(2*i-1:2*i,1); % Position of particle i
for j = i+1:n
% If there is a spring between particles i and j
if S(i,j)>0
xj = x(2*j-1:2*j,1); % Position of particle j
% Compute the potential energy of the spring.
E = E + .5*ks/l0(i,j)*(norm(xi-xj)-l0(i,j))^2;
end
end
% Compute the potential energy of the penalty collision force for particle i
% If the particle is inside the radius 2 circle centered at the origin,
% then there is no collision (no force, zero potential energy).
% Otherwise, the energy rises quadratically with distance from the circle.
E = E + .5*kc*max(norm(xi)-2,0)^2;
end
end
% Computes the total energy for the system.
function E = Total_Energy(x,v,m,ks,kc,l0,S)
n = rows(S);
KE = 0;
% Kinetic energy for particle = 1/2 m ||v||^2
for i = 1:n
vi = v(2*i-1:2*i,1);
KE = KE + .5*m*(vi'*vi);
end
% Total energy is kinetic + potential
E = KE + Potential_Energy(x,ks,kc,l0,S);
end
% Computes the total force for all of the particles in the system. x is a 2*n
% dimensional vector. On exit, F should be a 2*n dimensional vector containing
% the total force on each of the n particles. The force can be deduced from the
% potential energy. In particular, F(k) is the negative partial derivative of PE
% with respect to x(k). Here k=1..2*n and PE is the quantity computed by
% Potential_Energy. Treat the Potential_Energy function as a regular math
% function, which is a function of its input x.
function F = Force(x,ks,kc,l0,S)
n = rows(S);
F = zeros(2*n,1);
% TODO
end
% Computes the total momentum p and angular momentum L for t.
5.n nmodels i
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Incorporate the SOR method in the multigridTest-m and apply the multig.pdf
Incorporate the SOR method in the multigridTest.m and apply the multigridTest.m, to
determine the effects of changes in
Pre-smoothing iterations
Post-smoothing iterations
Multigrid cycle type
Iterative method
Number of grids (or grid hierarchy)
% SOR (Successive Over-Relaxation)
n = input('Enter number of equations, n: ');
A = zeros(n,n+1);
x1 = zeros(1,n);
A=[5 -2 3 -1;-3 9 1 2;2 -1 -7 3];
x1 = [0 0 0];
tol = input('Enter tolerance, tol: ');
m = input('Enter maximum number of iterations, m: ');
w = input('Enter the parameter w (omega): ');
k = 1;
while k <= m
err = 0;
for i = 1 : n
s = 0;
for j = 1 : n
s = s-A(i,j)*x1(j);
end
s = w*(s+A(i,n+1))/A(i,i);
if abs(s) > err
err = abs(s);
end
x1(i) = x1(i)+s;
end
if err <= tol
break;
else
k = k+1;
end
end
fprintf('The solution vector after %d iterations is :\n', k);
for i = 1 : n
fprintf(' %11.8f \n', x1(i));
end
%% Multigrid test case with visulization
% Author: Ben Beaudry
clc; clear; close all
load A_b.mat;
A = full(A); % Unsparce matrix to show full power of Multigrid
pre = 2; % Number of presmoothing iterations
post = 2; % Number of postsmoothing iterations
% cycle = 1; % Type of multigrid cycle (1=V-cycle, 2=W-cycle, 3=F-cycle)
smooth = 1; % Smoother type (1=Jacobi, 2=Gauss-Seidel)
grids = 5; % Max grids in cycle
maxit = 10; % Max iterations of solver
tol = 1e-02; % Tolerance of solver
%% Solvers
% solve directly
t=cputime;
x_D = A\b;
fprintf('Direct Solve Time: %g Seconds\n',cputime-t)
% V-Cycle
t=cputime;
[x_V,res_V,it_V] = multigrid(A,b,pre,post,1,smooth,grids,maxit,tol);
fprintf('V-Cycle Solve Time: %g Seconds\n',cputime-t)
% W-Cycle
t=cputime;
[x_W,res_W,it_W] = multigrid(A,b,pre,post,2,smooth,grids,maxit,tol);
fprintf('W-Cycle Solve Time: %g Seconds\n',cputime-t)
% F-Cycle
t=cputime;
[x_F,res_F,it_F] = multigrid(A,b,pre,post,3,smooth,grids,maxit,tol);
fprintf('F-Cycle Solve Time: %g Seconds\n',cputime-t)
% max it for iterative methods
it = 100;
% Gauss-Siedel
t=cputime;
L = tril(A);
U = triu(A,1);
x_G = zeros(length(b),1);
res_G= zeros(it,1);
for g = 1:it
x_G = L\(b-U*x_G);
r = b - A * x_G;
res_G(g) = norm(r);
if res_G(g) < tol
break;
end
end
fprintf('Gauss-Seidel Solve Time: %g Seconds\n',cputime-t)
% Jacobi
t=cputime;
d = diag(A);
dinv = 1./d;
LU = tril(A,-1)+triu(A,1);
U = triu(A,1);
x_J = zeros(length(b),1);
res_J= zeros(it,1);
for j = 1:it
x_J = dinv.*(b-(LU*x_J));
r = b - A * x_J;
res_J(j) = norm(r);
if res_J(j) < tol
break;
end
end
fprintf('Jacobi Solve Time: %g Seconds\n',cputime-t)
%% Plotting
figure;
hold on
plot(1:g,res_G(1:g),'o-','DisplayName','Guass-Seidel')
plot(1:j,res_J(1:j),'o-','DisplayName','Jacobi')
plot(1:it_V,res_V(1:it_V),'o-','DisplayName','V-Cycle Multigrid')
plot(1:it_F,res_F(1:it_F),'o-','DisplayName','F-Cycle Multigrid')
plot(1:it_W,res_W(1:it_W),'o-','DisplayName','W-Cycle Multigrid')
set(gca,'XLim',[0 100]);
grid on
legend('location','best')
ylabel('Relative Convergance')
xlabel('Iterations')
title('Convergance of Numerical System')
hold off
MULTIGRID FUNC.
assignment_2
The document discusses using Monte Carlo simulation to solve a partial differential equation (PDE) and using explicit time stepping to solve a different PDE. For the Monte Carlo method, as the number of random samples (M) increases, the approximation converges to the exact solution. For explicit time stepping, increasing the number of time and space steps (M and N) causes the error to diverge instead of converge due to exceeding memory cache capacity.
Please use the same variables and only write the TODO part #!-usr-bi.pdf
Please use the same variables and only write the TODO part
#!/usr/bin/env octave
% Only modify this file where you see a "% TODO"
% Functions that do not contain this should not be modified.
% If you modify them for debugging, please remove/comment those modifications.
% This line tells octave that this is not a function file.
1;
% Workaround fltk bug. You might need to comment this line out .
graphics_toolkit("gnuplot");
% Meanings of commonly encountered variables:
% n = number of particles
% x = 2*n dimensional column vector with the positions of the particles
% v = 2*n dimensional column vector with the velocities of the particles
% Note that the ordering is always x1 y1 x2 y2 x3 y3 ...
% ks = spring constant
% kc = penalty collision strength
% l0 = matrix of spring rest lengths.
% l0(i,j) = initial length of spring between particles i and j (i<j)
% Note that l0(i,j) should not be used if i>=j.
% S = matrix indicating which springs exist
% S(i,j) = 1 if there is a spring between particles i and j (i<j)
% Note that S(i,j) should not be used if i>=j.
% m = mass of the particles (all particles have the same mass)
% Computes the total potential energy for the system.
function E = Potential_Energy(x,ks,kc,l0,S)
n = rows(S);
E = 0;
% For each pair of particles (i<j)
for i = 1:n
xi = x(2*i-1:2*i,1); % Position of particle i
for j = i+1:n
% If there is a spring between particles i and j
if S(i,j)>0
xj = x(2*j-1:2*j,1); % Position of particle j
% Compute the potential energy of the spring.
E = E + .5*ks/l0(i,j)*(norm(xi-xj)-l0(i,j))^2;
end
end
% Compute the potential energy of the penalty collision force for particle i
% If the particle is inside the radius 2 circle centered at the origin,
% then there is no collision (no force, zero potential energy).
% Otherwise, the energy rises quadratically with distance from the circle.
E = E + .5*kc*max(norm(xi)-2,0)^2;
end
end
% Computes the total energy for the system.
function E = Total_Energy(x,v,m,ks,kc,l0,S)
n = rows(S);
KE = 0;
% Kinetic energy for particle = 1/2 m ||v||^2
for i = 1:n
vi = v(2*i-1:2*i,1);
KE = KE + .5*m*(vi'*vi);
end
% Total energy is kinetic + potential
E = KE + Potential_Energy(x,ks,kc,l0,S);
end
% Computes the total force for all of the particles in the system. x is a 2*n
% dimensional vector. On exit, F should be a 2*n dimensional vector containing
% the total force on each of the n particles. The force can be deduced from the
% potential energy. In particular, F(k) is the negative partial derivative of PE
% with respect to x(k). Here k=1..2*n and PE is the quantity computed by
% Potential_Energy. Treat the Potential_Energy function as a regular math
% function, which is a function of its input x.
function F = Force(x,ks,kc,l0,S)
n = rows(S);
F = zeros(2*n,1);
% TODO
end
% Computes the total momentum p and angular momentum L for the system
function [p L] = Momentum(x,v,m)
n = rows(x)/2;
p = zeros(2,1); % 2D vector; total momentum
L = 0; % Scalar; total angular momentum
for i = 1:.
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Use the same variable names and write the function F - Force(x-ks-kc-l.pdf
Use the same variable names and write the function F = Force(x,ks,kc,l0,S) ; function l0 =
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answer the 6 questions below
#!/usr/bin/env octave
% Only modify this file where you see a "% TODO"
% Functions that do not contain this should not be modified.
% If you modify them for debugging, please remove/comment those modifications.
% This line tells octave that this is not a function file.
1;
% Workaround fltk bug. You might need to comment this line out.
graphics_toolkit("gnuplot");
% Meanings of commonly encountered variables:
% n = number of particles
% x = 2*n dimensional column vector with the positions of the particles
% v = 2*n dimensional column vector with the velocities of the particles
% Note that the ordering is always x1 y1 x2 y2 x3 y3 ...
% ks = spring constant
% kc = penalty collision strength
% l0 = matrix of spring rest lengths.
% l0(i,j) = initial length of spring between particles i and j (i<j)
% Note that l0(i,j) should not be used if i>=j.
% S = matrix indicating which springs exist
% S(i,j) = 1 if there is a spring between particles i and j (i<j)
% Note that S(i,j) should not be used if i>=j.
% m = mass of the particles (all particles have the same mass)
% Computes the total potential energy for the system.
function E = Potential_Energy(x,ks,kc,l0,S)
n = rows(S);
E = 0;
% For each pair of particles (i<j)
for i = 1:n
xi = x(2*i-1:2*i,1); % Position of particle i
for j = i+1:n
% If there is a spring between particles i and j
if S(i,j)>0
xj = x(2*j-1:2*j,1); % Position of particle j
% Compute the potential energy of the spring.
E = E + .5*ks/l0(i,j)*(norm(xi-xj)-l0(i,j))^2;
end
end
% Compute the potential energy of the penalty collision force for particle i
% If the particle is inside the radius 2 circle centered at the origin,
% then there is no collision (no force, zero potential energy).
% Otherwise, the energy rises quadratically with distance from the circle.
E = E + .5*kc*max(norm(xi)-2,0)^2;
end
end
% Computes the total energy for the system.
function E = Total_Energy(x,v,m,ks,kc,l0,S)
n = rows(S);
KE = 0;
% Kinetic energy for particle = 1/2 m ||v||^2
for i = 1:n
vi = v(2*i-1:2*i,1);
KE = KE + .5*m*(vi'*vi);
end
% Total energy is kinetic + potential
E = KE + Potential_Energy(x,ks,kc,l0,S);
end
% Computes the total force for all of the particles in the system. x is a 2*n
% dimensional vector. On exit, F should be a 2*n dimensional vector containing
% the total force on each of the n particles. The force can be deduced from the
% potential energy. In particular, F(k) is the negative partial derivative of PE
% with respect to x(k). Here k=1..2*n and PE is the quantity computed by
% Potential_Energy. Treat the Potential_Energy function as a regular math
% function, which is a function of its input x.
function F = Force(x,ks,kc,l0,S)
n = rows(S);
F = zeros(2*n,1);
% TODO
end
% Computes the total momentum p and angular momentum L for t.
5.n nmodels i
The document describes the Delta rule for training artificial neural networks. It begins by introducing a simple linear neural network model and then adds a non-linear activation function. The Delta rule is then derived as a method for minimizing the mean squared error between the network's output and the target values by adjusting the network's weights. Specifically, the weight updates are proportional to the derivative of the error function with respect to each weight. Regularization is also discussed as a method for improving generalization.
Incorporate the SOR method in the multigridTest-m and apply the multig.pdf
Incorporate the SOR method in the multigridTest.m and apply the multigridTest.m, to
determine the effects of changes in
Pre-smoothing iterations
Post-smoothing iterations
Multigrid cycle type
Iterative method
Number of grids (or grid hierarchy)
% SOR (Successive Over-Relaxation)
n = input('Enter number of equations, n: ');
A = zeros(n,n+1);
x1 = zeros(1,n);
A=[5 -2 3 -1;-3 9 1 2;2 -1 -7 3];
x1 = [0 0 0];
tol = input('Enter tolerance, tol: ');
m = input('Enter maximum number of iterations, m: ');
w = input('Enter the parameter w (omega): ');
k = 1;
while k <= m
err = 0;
for i = 1 : n
s = 0;
for j = 1 : n
s = s-A(i,j)*x1(j);
end
s = w*(s+A(i,n+1))/A(i,i);
if abs(s) > err
err = abs(s);
end
x1(i) = x1(i)+s;
end
if err <= tol
break;
else
k = k+1;
end
end
fprintf('The solution vector after %d iterations is :\n', k);
for i = 1 : n
fprintf(' %11.8f \n', x1(i));
end
%% Multigrid test case with visulization
% Author: Ben Beaudry
clc; clear; close all
load A_b.mat;
A = full(A); % Unsparce matrix to show full power of Multigrid
pre = 2; % Number of presmoothing iterations
post = 2; % Number of postsmoothing iterations
% cycle = 1; % Type of multigrid cycle (1=V-cycle, 2=W-cycle, 3=F-cycle)
smooth = 1; % Smoother type (1=Jacobi, 2=Gauss-Seidel)
grids = 5; % Max grids in cycle
maxit = 10; % Max iterations of solver
tol = 1e-02; % Tolerance of solver
%% Solvers
% solve directly
t=cputime;
x_D = A\b;
fprintf('Direct Solve Time: %g Seconds\n',cputime-t)
% V-Cycle
t=cputime;
[x_V,res_V,it_V] = multigrid(A,b,pre,post,1,smooth,grids,maxit,tol);
fprintf('V-Cycle Solve Time: %g Seconds\n',cputime-t)
% W-Cycle
t=cputime;
[x_W,res_W,it_W] = multigrid(A,b,pre,post,2,smooth,grids,maxit,tol);
fprintf('W-Cycle Solve Time: %g Seconds\n',cputime-t)
% F-Cycle
t=cputime;
[x_F,res_F,it_F] = multigrid(A,b,pre,post,3,smooth,grids,maxit,tol);
fprintf('F-Cycle Solve Time: %g Seconds\n',cputime-t)
% max it for iterative methods
it = 100;
% Gauss-Siedel
t=cputime;
L = tril(A);
U = triu(A,1);
x_G = zeros(length(b),1);
res_G= zeros(it,1);
for g = 1:it
x_G = L\(b-U*x_G);
r = b - A * x_G;
res_G(g) = norm(r);
if res_G(g) < tol
break;
end
end
fprintf('Gauss-Seidel Solve Time: %g Seconds\n',cputime-t)
% Jacobi
t=cputime;
d = diag(A);
dinv = 1./d;
LU = tril(A,-1)+triu(A,1);
U = triu(A,1);
x_J = zeros(length(b),1);
res_J= zeros(it,1);
for j = 1:it
x_J = dinv.*(b-(LU*x_J));
r = b - A * x_J;
res_J(j) = norm(r);
if res_J(j) < tol
break;
end
end
fprintf('Jacobi Solve Time: %g Seconds\n',cputime-t)
%% Plotting
figure;
hold on
plot(1:g,res_G(1:g),'o-','DisplayName','Guass-Seidel')
plot(1:j,res_J(1:j),'o-','DisplayName','Jacobi')
plot(1:it_V,res_V(1:it_V),'o-','DisplayName','V-Cycle Multigrid')
plot(1:it_F,res_F(1:it_F),'o-','DisplayName','F-Cycle Multigrid')
plot(1:it_W,res_W(1:it_W),'o-','DisplayName','W-Cycle Multigrid')
set(gca,'XLim',[0 100]);
grid on
legend('location','best')
ylabel('Relative Convergance')
xlabel('Iterations')
title('Convergance of Numerical System')
hold off
MULTIGRID FUNC.
assignment_2
The document discusses using Monte Carlo simulation to solve a partial differential equation (PDE) and using explicit time stepping to solve a different PDE. For the Monte Carlo method, as the number of random samples (M) increases, the approximation converges to the exact solution. For explicit time stepping, increasing the number of time and space steps (M and N) causes the error to diverge instead of converge due to exceeding memory cache capacity.
Please use the same variables and only write the TODO part #!-usr-bi.pdf
Please use the same variables and only write the TODO part
#!/usr/bin/env octave
% Only modify this file where you see a "% TODO"
% Functions that do not contain this should not be modified.
% If you modify them for debugging, please remove/comment those modifications.
% This line tells octave that this is not a function file.
1;
% Workaround fltk bug. You might need to comment this line out .
graphics_toolkit("gnuplot");
% Meanings of commonly encountered variables:
% n = number of particles
% x = 2*n dimensional column vector with the positions of the particles
% v = 2*n dimensional column vector with the velocities of the particles
% Note that the ordering is always x1 y1 x2 y2 x3 y3 ...
% ks = spring constant
% kc = penalty collision strength
% l0 = matrix of spring rest lengths.
% l0(i,j) = initial length of spring between particles i and j (i<j)
% Note that l0(i,j) should not be used if i>=j.
% S = matrix indicating which springs exist
% S(i,j) = 1 if there is a spring between particles i and j (i<j)
% Note that S(i,j) should not be used if i>=j.
% m = mass of the particles (all particles have the same mass)
% Computes the total potential energy for the system.
function E = Potential_Energy(x,ks,kc,l0,S)
n = rows(S);
E = 0;
% For each pair of particles (i<j)
for i = 1:n
xi = x(2*i-1:2*i,1); % Position of particle i
for j = i+1:n
% If there is a spring between particles i and j
if S(i,j)>0
xj = x(2*j-1:2*j,1); % Position of particle j
% Compute the potential energy of the spring.
E = E + .5*ks/l0(i,j)*(norm(xi-xj)-l0(i,j))^2;
end
end
% Compute the potential energy of the penalty collision force for particle i
% If the particle is inside the radius 2 circle centered at the origin,
% then there is no collision (no force, zero potential energy).
% Otherwise, the energy rises quadratically with distance from the circle.
E = E + .5*kc*max(norm(xi)-2,0)^2;
end
end
% Computes the total energy for the system.
function E = Total_Energy(x,v,m,ks,kc,l0,S)
n = rows(S);
KE = 0;
% Kinetic energy for particle = 1/2 m ||v||^2
for i = 1:n
vi = v(2*i-1:2*i,1);
KE = KE + .5*m*(vi'*vi);
end
% Total energy is kinetic + potential
E = KE + Potential_Energy(x,ks,kc,l0,S);
end
% Computes the total force for all of the particles in the system. x is a 2*n
% dimensional vector. On exit, F should be a 2*n dimensional vector containing
% the total force on each of the n particles. The force can be deduced from the
% potential energy. In particular, F(k) is the negative partial derivative of PE
% with respect to x(k). Here k=1..2*n and PE is the quantity computed by
% Potential_Energy. Treat the Potential_Energy function as a regular math
% function, which is a function of its input x.
function F = Force(x,ks,kc,l0,S)
n = rows(S);
F = zeros(2*n,1);
% TODO
end
% Computes the total momentum p and angular momentum L for the system
function [p L] = Momentum(x,v,m)
n = rows(x)/2;
p = zeros(2,1); % 2D vector; total momentum
L = 0; % Scalar; total angular momentum
for i = 1:.
Fourier series example
This MATLAB code provides an example of plotting a truncated Fourier series representation of a square wave signal. It computes the Fourier series in both complex exponential form (yce) and trigonometric form (yt) up to the Nth term, where N is an odd integer. It plots the original square wave, the truncated Fourier series approximations yce and yt, and their amplitude and phase spectra. The code demonstrates how to calculate and visualize truncated Fourier series representations of a periodic signal.
Applied Digital Signal Processing 1st Edition Manolakis Solutions Manual
Full download http://alibabadownload.com/product/applied-digital-signal-processing-1st-edition-manolakis-solutions-manual/
Applied Digital Signal Processing 1st Edition Manolakis Solutions Manual
Explanation on Tensorflow example -Deep mnist for expert
you can find the exact and detailed network architecture of 'Deep mnist for expert' example of tensorflow's tutorial. I also added descriptions on the program for your better understanding.
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2. Define custom functions to generate basic sequences like unit sample, unit step, exponential rise and decay.
3. Plot signals together and manipulate signals using operations like convolution.
4. Quantize signals and calculate signal-to-quantization noise ratio (SQNR).
5. Perform operations like filtering using transfer functions and z-transforms.
6. Implement the discrete Fourier transform and inverse discrete Fourier transform using a custom function.
residue
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2) An ideal sampler can be modeled as a carrier signal modulated by the input signal. The output of the sampler is then sent to a zero-order hold.
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Application of Differential Equation
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2. A differential equation representing current i in an electric circuit is solved given initial conditions when t=0 of i=0 and di/dt=34.
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Numerical solution of ordinary differential equations by using Taylor method and Euler method
How to write the C++ codes?
This problem illustrates one of the possible pitfalls of blindly appl.pdf
This problem illustrates one of the possible pitfalls of blindly applying numerical methods
without paying attention to the theoretical aspects of the differential equation itself. Consider the
equation ty\' + 3y - 9t^2 = 0. Use the MATLAB program myeuler.m from Chapter 8 to compute
the Euler Method approximation to the solution with initial condition y(-0.5) = 3.15, using step
size h = 0.2 and n = 10 steps The program will generate a lot of ordered pairs (x_i, .y_i). Use plot
to graph the piecewise linear function connecting the points (x_i, y_i). Now modify the program
to implement the Improved Euler Method. Can you make sense of your answers? Next, use
ode45 to find an approximate solution on the interval (-0.5, 0.5), and plot it with plot. Print out
the values of the solution at the points -0.06: 0.02: 0.06. What is the interval on which the
approximate solution is defined? Solve the equation explicitly and graph the solutions for the
initial conditions y(0) = 0, y(-0.5) = 3.15, y(0, 5) = 3.15, y(-0.5) = -3.45, and y(0.5) = -3.45. Now
explain your results in (a)-(c). Could we have known, without solving the equation, whether to
expect meaningful results in parts (a) and (b)? Why? Can you explain how ode45 avoids making
the same mistake?
Solution
clc
clear all
close all
x=input(\'Enter the first sequence: \');
l1=input(\'Enter the lower limit: \');
u1=input(\'Enter the upper limit: \');
x1=l1:1:u1;
h=input(\'Enter the second sequence: \');
l2=input(\'Enter the lower limit: \');
u2=input(\'Enter the upper limit: \');
h1=l2:1:u2;
l=l1+l2;
u=u1+u2;
n=l:1:u;
s=numel(n);
i=1;
for i=1:s
y(i)=0;
for k=1:numel(x)
if (i+1-k)<=0
y(i)=y(i)+(x(k)*0);
else if (i+1-k)>numel(h)
y(i)=y(i)+(x(k)*0);
else
y(i)=y(i)+(x(k)*h(i+1-k));
k=k+1;
end
end
end
i=i+1;
end
disp(y);
subplot(2,2,1);stem(x1,x);
title(\'First sequence\');xlabel(\'n\');ylabel(\'x(n)\');
subplot(2,2,2);stem(h1,h);
title(\'Second Sequence\');xlabel(\'n\');ylabel(\'h(n)\');
subplot(2,2,[3 4]);stem(n,y);
title(\'Convoluted sequence\');xlabel(\'n\');ylabel(\'y(n)\');
Comment only
X = input(\'Enter x: \'); %input vector X and H
H = input(\'Enter h: \') ;
LenX = length(X); %defining their lenghts
LenH = length(H);
y = zeros(1,LenX+LenH); %defing vector y of zeroes and of size
% lenth of X + length of H
t = zeros(1,LenH); % definign a vector t of same length as H
for i = 1:LenH+LenX-1 % Running a for loop from 1 to length of Y -1
if i<=LenX % till I IS Lesser then length of X i.e overlap about to begin
t(1)= X(i); % put x(i) on t(1) later it is shifted forwards in the vector t i.e. later t(2)=t(1)
for j = 1:LenH % in the if condition a for loop from 1 to length of H
y(i) = y(i) + H(j)*t(j); % summing for all H(j)*t(j) and putting it at y(1) or y(2) or Y(i) in first
iteration
% i.e. for i=1 only firt multiplication would
% be non zero rest all zeroes.
end
for k = LenH:-1:2 % shifting old value of t(i) to t(i+1) now there would me 1+ non zeroes values
in t
% this cycle would continue until i is lesse.
Computational Method to Solve the Partial Differential Equations (PDEs)
This document discusses various computational methods for solving partial differential equations (PDEs) using MATLAB. It begins by introducing three types of PDEs - elliptic, parabolic, and hyperbolic - and provides examples of each. It then describes explicit methods like the Forward Time Centered Space (FTCS) method, Lax method, and Crank-Nicolson (CTCS) method for solving the advection equation. The document provides MATLAB code implementing these methods for a test case of solving the advection equation modeling a square wave.
Numerical Method Assignment
1) The document is a numerical methods assignment submitted by a student to their professor. It contains 4 problems solved using various numerical methods like Gauss-Seidel iteration, Lagrange interpolation, and Newton divided differences.
2) For problem 1, the solutions to a system of equations using Gauss-Seidel iteration are found to be x1= 1, x2= -2, x3= -3.
3) For problem 4, the value of f(5) is estimated using Newton divided differences to be 92.
Numerical methods generating polynomial
Numerical methods generating polynomialDepartment of Telecommunications, Ministry of Communication & IT (INDIA)
The smallest positive guess for which the Newton method diverges for the function f(x)=atan(x) is 1.4. The Newton method was able to find the root for guesses up to 1.39 but produced a divide by zero error at 1.4, indicating that 1.4 is the smallest positive value where the method diverges.Amth250 octave matlab some solutions (1)
This document contains the solutions to 5 questions regarding numerical integration and differential equations. Question 1 involves numerically evaluating several integrals. Question 2 computes the Fresnel integrals. Question 3 uses Monte Carlo integration to estimate volumes. Question 4 examines the convergence and stability of the Euler method. Question 5 simulates the Lorenz system and demonstrates its sensitivity to initial conditions.
matlab codes.pdf
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Write Python for Speed
Being a slow interpreter, Python may drive a system to deliver utmost speed if some guidelines are followed. The key is to treat programming languages as syntactic sugar to the machine code. It expedites the workflow of timing, iterative design, automatic testing, optimization, and realize an HPC system balancing the time to market and quality of code.
Speed is the king. 10x productive developers change business. So does 10x faster code. Python is 100x slower than C++ but it only matters when you really use Python to implement number-crunching algorithms. We should not do that, and instead go directly with C++ for speed. It calls for strict disciplines of software engineering and code quality, but it should be noted that here the quality is defined by the runtime and the time to market.
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2) Using character strings to print output and combine text.
3) Common mathematical operations like exponents, logarithms, trigonometric functions, rounding, and converting between numeric and string formats.
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Adaptive dynamic programming for control
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Recursion in C
C program to find factorial of number using recursion as well as iteration ,
Calculate power of a number program in c using Recursion and Iteration, Write a C program to count digits of a number using Recursion and Iteration, Write a C program to find sum of first n natural numbers using Recursion, C program to print sum of digits of a given number using recursion ,Write a C program to find nth term in Fibonacci Series using Recursion, C program to find out the GCD (Greatest Common Divisor )of the two numbers using recursion,
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2. Define custom functions to generate basic sequences like unit sample, unit step, exponential rise and decay.
3. Plot signals together and manipulate signals using operations like convolution.
4. Quantize signals and calculate signal-to-quantization noise ratio (SQNR).
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without paying attention to the theoretical aspects of the differential equation itself. Consider the
equation ty\' + 3y - 9t^2 = 0. Use the MATLAB program myeuler.m from Chapter 8 to compute
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to implement the Improved Euler Method. Can you make sense of your answers? Next, use
ode45 to find an approximate solution on the interval (-0.5, 0.5), and plot it with plot. Print out
the values of the solution at the points -0.06: 0.02: 0.06. What is the interval on which the
approximate solution is defined? Solve the equation explicitly and graph the solutions for the
initial conditions y(0) = 0, y(-0.5) = 3.15, y(0, 5) = 3.15, y(-0.5) = -3.45, and y(0.5) = -3.45. Now
explain your results in (a)-(c). Could we have known, without solving the equation, whether to
expect meaningful results in parts (a) and (b)? Why? Can you explain how ode45 avoids making
the same mistake?
Solution
clc
clear all
close all
x=input(\'Enter the first sequence: \');
l1=input(\'Enter the lower limit: \');
u1=input(\'Enter the upper limit: \');
x1=l1:1:u1;
h=input(\'Enter the second sequence: \');
l2=input(\'Enter the lower limit: \');
u2=input(\'Enter the upper limit: \');
h1=l2:1:u2;
l=l1+l2;
u=u1+u2;
n=l:1:u;
s=numel(n);
i=1;
for i=1:s
y(i)=0;
for k=1:numel(x)
if (i+1-k)<=0
y(i)=y(i)+(x(k)*0);
else if (i+1-k)>numel(h)
y(i)=y(i)+(x(k)*0);
else
y(i)=y(i)+(x(k)*h(i+1-k));
k=k+1;
end
end
end
i=i+1;
end
disp(y);
subplot(2,2,1);stem(x1,x);
title(\'First sequence\');xlabel(\'n\');ylabel(\'x(n)\');
subplot(2,2,2);stem(h1,h);
title(\'Second Sequence\');xlabel(\'n\');ylabel(\'h(n)\');
subplot(2,2,[3 4]);stem(n,y);
title(\'Convoluted sequence\');xlabel(\'n\');ylabel(\'y(n)\');
Comment only
X = input(\'Enter x: \'); %input vector X and H
H = input(\'Enter h: \') ;
LenX = length(X); %defining their lenghts
LenH = length(H);
y = zeros(1,LenX+LenH); %defing vector y of zeroes and of size
% lenth of X + length of H
t = zeros(1,LenH); % definign a vector t of same length as H
for i = 1:LenH+LenX-1 % Running a for loop from 1 to length of Y -1
if i<=LenX % till I IS Lesser then length of X i.e overlap about to begin
t(1)= X(i); % put x(i) on t(1) later it is shifted forwards in the vector t i.e. later t(2)=t(1)
for j = 1:LenH % in the if condition a for loop from 1 to length of H
y(i) = y(i) + H(j)*t(j); % summing for all H(j)*t(j) and putting it at y(1) or y(2) or Y(i) in first
iteration
% i.e. for i=1 only firt multiplication would
% be non zero rest all zeroes.
end
for k = LenH:-1:2 % shifting old value of t(i) to t(i+1) now there would me 1+ non zeroes values
in t
% this cycle would continue until i is lesse.
Computational Method to Solve the Partial Differential Equations (PDEs)
This document discusses various computational methods for solving partial differential equations (PDEs) using MATLAB. It begins by introducing three types of PDEs - elliptic, parabolic, and hyperbolic - and provides examples of each. It then describes explicit methods like the Forward Time Centered Space (FTCS) method, Lax method, and Crank-Nicolson (CTCS) method for solving the advection equation. The document provides MATLAB code implementing these methods for a test case of solving the advection equation modeling a square wave.
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Speed is the king. 10x productive developers change business. So does 10x faster code. Python is 100x slower than C++ but it only matters when you really use Python to implement number-crunching algorithms. We should not do that, and instead go directly with C++ for speed. It calls for strict disciplines of software engineering and code quality, but it should be noted that here the quality is defined by the runtime and the time to market.
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This document provides a summary of basic MATLAB commands organized into sections:
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Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
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TIME DIVISION MULTIPLEXING TECHNIQUE FOR COMMUNICATION SYSTEM
Time Division Multiplexing (TDM) is a method of transmitting multiple signals over a single communication channel by dividing the signal into many segments, each having a very short duration of time. These time slots are then allocated to different data streams, allowing multiple signals to share the same transmission medium efficiently. TDM is widely used in telecommunications and data communication systems.
### How TDM Works
1. **Time Slots Allocation**: The core principle of TDM is to assign distinct time slots to each signal. During each time slot, the respective signal is transmitted, and then the process repeats cyclically. For example, if there are four signals to be transmitted, the TDM cycle will divide time into four slots, each assigned to one signal.
2. **Synchronization**: Synchronization is crucial in TDM systems to ensure that the signals are correctly aligned with their respective time slots. Both the transmitter and receiver must be synchronized to avoid any overlap or loss of data. This synchronization is typically maintained by a clock signal that ensures time slots are accurately aligned.
3. **Frame Structure**: TDM data is organized into frames, where each frame consists of a set of time slots. Each frame is repeated at regular intervals, ensuring continuous transmission of data streams. The frame structure helps in managing the data streams and maintaining the synchronization between the transmitter and receiver.
4. **Multiplexer and Demultiplexer**: At the transmitting end, a multiplexer combines multiple input signals into a single composite signal by assigning each signal to a specific time slot. At the receiving end, a demultiplexer separates the composite signal back into individual signals based on their respective time slots.
### Types of TDM
1. **Synchronous TDM**: In synchronous TDM, time slots are pre-assigned to each signal, regardless of whether the signal has data to transmit or not. This can lead to inefficiencies if some time slots remain empty due to the absence of data.
2. **Asynchronous TDM (or Statistical TDM)**: Asynchronous TDM addresses the inefficiencies of synchronous TDM by allocating time slots dynamically based on the presence of data. Time slots are assigned only when there is data to transmit, which optimizes the use of the communication channel.
### Applications of TDM
- **Telecommunications**: TDM is extensively used in telecommunication systems, such as in T1 and E1 lines, where multiple telephone calls are transmitted over a single line by assigning each call to a specific time slot.
- **Digital Audio and Video Broadcasting**: TDM is used in broadcasting systems to transmit multiple audio or video streams over a single channel, ensuring efficient use of bandwidth.
- **Computer Networks**: TDM is used in network protocols and systems to manage the transmission of data from multiple sources over a single network medium.
### Advantages of TDM
- **Efficient Use of Bandwidth**: TDM all
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scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
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