% Runs Newton\'s method to find roots of f(z) . It col
function [a,n,TOL,N]=P3(a,n,TOL,N)
A=zeros(n,n,\'uint8\');
f = inline(\'z^3-1\'); % Define f and f\'.
fprime = inline(\'3*z^2\');
rts=roots([1 0 0 -1])
h=2*a/n;
for j=1:n+1, % Try initial values with imaginary parts between
y = -a + (j-1)*h; % -a and a
for i=1:n+1, % and with real parts between
x = -a + (i-1)*h; % -a and a.
z = x + 1i*y;
zk = z;
kount = 0; % kount is the total number of iterations.
conv1 = 0; % conv1,2,3 count iterations when approx soln is within
conv2 = 0; % TOL of root1,2,3.
conv3 = 0;
while kount < 40 & conv1 < 5 & conv2 < 5 & conv3 < 5,
kount = kount + 1;
zk = zk - f(zk)/fprime(zk); % This is the Newton step.
if abs(zk-root1) < TOL, % Check for convergence to root1.
conv1 = conv1 + 1;
else
conv1 = 0;
end;
if abs(zk-root2) < TOL, % Check for convergence to root2.
conv2 = conv2 + 1;
else
conv2 = 0;
end;
if abs(zk-root3) < TOL, % Check for convergence to root3.
conv3 = conv3 + 1;
else
conv3 = 0;
end;
end;
if conv1 >=3, M(j,i,1) = 255; end; % Converged to root 1. Color point green.
if conv2 >=3, M(j,i,2) = 255; end; % Converged to root 2. Color point red.
if conv3 >=3, M(j,i,3) = 255; end; % Converged to root 3. Color point blue.
end;
end;
imshow(A)
Solution
% Runs Newton\'s method to find roots of f(z) . It col
function [a,n,TOL,N]=P3(a,n,TOL,N)
A=zeros(n,n,\'uint8\');
f = inline(\'z^3-1\'); % Define f and f\'.
fprime = inline(\'3*z^2\');
rts=roots([1 0 0 -1])
h=2*a/n;
for j=1:n+1, % Try initial values with imaginary parts between
y = -a + (j-1)*h; % -a and a
for i=1:n+1, % and with real parts between
x = -a + (i-1)*h; % -a and a.
z = x + 1i*y;
zk = z;
kount = 0; % kount is the total number of iterations.
conv1 = 0; % conv1,2,3 count iterations when approx soln is within
conv2 = 0; % TOL of root1,2,3.
conv3 = 0;
while kount < 40 & conv1 < 5 & conv2 < 5 & conv3 < 5,
kount = kount + 1;
zk = zk - f(zk)/fprime(zk); % This is the Newton step.
if abs(zk-root1) < TOL, % Check for convergence to root1.
conv1 = conv1 + 1;
else
conv1 = 0;
end;
if abs(zk-root2) < TOL, % Check for convergence to root2.
conv2 = conv2 + 1;
else
conv2 = 0;
end;
if abs(zk-root3) < TOL, % Check for convergence to root3.
conv3 = conv3 + 1;
else
conv3 = 0;
end;
end;
if conv1 >=3, M(j,i,1) = 255; end; % Converged to root 1. Color point green.
if conv2 >=3, M(j,i,2) = 255; end; % Converged to root 2. Color point red.
if conv3 >=3, M(j,i,3) = 255; end; % Converged to root 3. Color point blue.
end;
end;
imshow(A).
PSYPACT- Practicing Over State Lines May 2024.pptx
Runs Newtons method to find roots of f(z) . It col function [a.pdf
1. % Runs Newton's method to find roots of f(z) . It col
function [a,n,TOL,N]=P3(a,n,TOL,N)
A=zeros(n,n,'uint8');
f = inline('z^3-1'); % Define f and f'.
fprime = inline('3*z^2');
rts=roots([1 0 0 -1])
h=2*a/n;
for j=1:n+1, % Try initial values with imaginary parts between
y = -a + (j-1)*h; % -a and a
for i=1:n+1, % and with real parts between
x = -a + (i-1)*h; % -a and a.
z = x + 1i*y;
zk = z;
kount = 0; % kount is the total number of iterations.
conv1 = 0; % conv1,2,3 count iterations when approx soln is within
conv2 = 0; % TOL of root1,2,3.
conv3 = 0;
while kount < 40 & conv1 < 5 & conv2 < 5 & conv3 < 5,
kount = kount + 1;
zk = zk - f(zk)/fprime(zk); % This is the Newton step.
if abs(zk-root1) < TOL, % Check for convergence to root1.
conv1 = conv1 + 1;
else
conv1 = 0;
end;
if abs(zk-root2) < TOL, % Check for convergence to root2.
conv2 = conv2 + 1;
else
conv2 = 0;
end;
if abs(zk-root3) < TOL, % Check for convergence to root3.
conv3 = conv3 + 1;
else
conv3 = 0;
end;
2. end;
if conv1 >=3, M(j,i,1) = 255; end; % Converged to root 1. Color point green.
if conv2 >=3, M(j,i,2) = 255; end; % Converged to root 2. Color point red.
if conv3 >=3, M(j,i,3) = 255; end; % Converged to root 3. Color point blue.
end;
end;
imshow(A)
Solution
% Runs Newton's method to find roots of f(z) . It col
function [a,n,TOL,N]=P3(a,n,TOL,N)
A=zeros(n,n,'uint8');
f = inline('z^3-1'); % Define f and f'.
fprime = inline('3*z^2');
rts=roots([1 0 0 -1])
h=2*a/n;
for j=1:n+1, % Try initial values with imaginary parts between
y = -a + (j-1)*h; % -a and a
for i=1:n+1, % and with real parts between
x = -a + (i-1)*h; % -a and a.
z = x + 1i*y;
zk = z;
kount = 0; % kount is the total number of iterations.
conv1 = 0; % conv1,2,3 count iterations when approx soln is within
conv2 = 0; % TOL of root1,2,3.
conv3 = 0;
while kount < 40 & conv1 < 5 & conv2 < 5 & conv3 < 5,
kount = kount + 1;
zk = zk - f(zk)/fprime(zk); % This is the Newton step.
if abs(zk-root1) < TOL, % Check for convergence to root1.
conv1 = conv1 + 1;
else
conv1 = 0;
end;
if abs(zk-root2) < TOL, % Check for convergence to root2.
3. conv2 = conv2 + 1;
else
conv2 = 0;
end;
if abs(zk-root3) < TOL, % Check for convergence to root3.
conv3 = conv3 + 1;
else
conv3 = 0;
end;
end;
if conv1 >=3, M(j,i,1) = 255; end; % Converged to root 1. Color point green.
if conv2 >=3, M(j,i,2) = 255; end; % Converged to root 2. Color point red.
if conv3 >=3, M(j,i,3) = 255; end; % Converged to root 3. Color point blue.
end;
end;
imshow(A)
![% Runs Newton's method to find roots of f(z) . It col
function [a,n,TOL,N]=P3(a,n,TOL,N)
A=zeros(n,n,'uint8');
f = inline('z^3-1'); % Define f and f'.
fprime = inline('3*z^2');
rts=roots([1 0 0 -1])
h=2*a/n;
for j=1:n+1, % Try initial values with imaginary parts between
y = -a + (j-1)*h; % -a and a
for i=1:n+1, % and with real parts between
x = -a + (i-1)*h; % -a and a.
z = x + 1i*y;
zk = z;
kount = 0; % kount is the total number of iterations.
conv1 = 0; % conv1,2,3 count iterations when approx soln is within
conv2 = 0; % TOL of root1,2,3.
conv3 = 0;
while kount < 40 & conv1 < 5 & conv2 < 5 & conv3 < 5,
kount = kount + 1;
zk = zk - f(zk)/fprime(zk); % This is the Newton step.
if abs(zk-root1) < TOL, % Check for convergence to root1.
conv1 = conv1 + 1;
else
conv1 = 0;
end;
if abs(zk-root2) < TOL, % Check for convergence to root2.
conv2 = conv2 + 1;
else
conv2 = 0;
end;
if abs(zk-root3) < TOL, % Check for convergence to root3.
conv3 = conv3 + 1;
else
conv3 = 0;
end;](https://image.slidesharecdn.com/runsnewtonsmethodtofindrootsoffz-230409124500-f2079460/85/Runs-Newtons-method-to-find-roots-of-f-z-It-col-function-a-pdf-1-320.jpg)
![end;
if conv1 >=3, M(j,i,1) = 255; end; % Converged to root 1. Color point green.
if conv2 >=3, M(j,i,2) = 255; end; % Converged to root 2. Color point red.
if conv3 >=3, M(j,i,3) = 255; end; % Converged to root 3. Color point blue.
end;
end;
imshow(A)
Solution
% Runs Newton's method to find roots of f(z) . It col
function [a,n,TOL,N]=P3(a,n,TOL,N)
A=zeros(n,n,'uint8');
f = inline('z^3-1'); % Define f and f'.
fprime = inline('3*z^2');
rts=roots([1 0 0 -1])
h=2*a/n;
for j=1:n+1, % Try initial values with imaginary parts between
y = -a + (j-1)*h; % -a and a
for i=1:n+1, % and with real parts between
x = -a + (i-1)*h; % -a and a.
z = x + 1i*y;
zk = z;
kount = 0; % kount is the total number of iterations.
conv1 = 0; % conv1,2,3 count iterations when approx soln is within
conv2 = 0; % TOL of root1,2,3.
conv3 = 0;
while kount < 40 & conv1 < 5 & conv2 < 5 & conv3 < 5,
kount = kount + 1;
zk = zk - f(zk)/fprime(zk); % This is the Newton step.
if abs(zk-root1) < TOL, % Check for convergence to root1.
conv1 = conv1 + 1;
else
conv1 = 0;
end;
if abs(zk-root2) < TOL, % Check for convergence to root2.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)