V is a image matrix.D0 and D1 are right and left root of binary trees for level 1.
this is a binary tree and it has 8 level. this means a lots of code . I want to make it with recursive function.And as an Output I need all roots of means in array M. Please any idea to make it recursive?
clear all;clc;
V=imread('tire.tif');
[x y]=size(V);
U=V*0;
M=zeros(1,511);
% LEVEL 1
M(1,1)=mean(V(:));
% LEVEL 2
D0=V(V<=mean(V(:))); % right root for V
M(1,2)=mean(D0(:));
D1=V(V>mean(V(:))); %left root for V
M(1,3)=mean(D1(:));
% LEVEL 3
D00=D0(D0<=mean(D0(:))); %left root for D0
M(1,4)=mean(D00(:));
D01=D0(D0>mean(D0(:))); %left root for D0
M(1,5)=mean(D01(:));
D10=D1(D1<=mean(D1(:))); %right root for D1
M(1,6)=mean(D10(:));
D11=D1(D1>mean(D1(:))); %left root for D1
M(1,7)=mean(D11(:));
I believe this is the solution you are looking for. The tricky part is keeping track of the indices (as usual).
function M = myrecfun(V, M, n_max, n, i)
%n: current level (of recursions)
%i: an integer in [1, 2^(n-1)]
i_start = 2^n;
meanV = mean(V(:));
if n == 1
M(1) = meanV
end
DR = V(V <= meanV);
DL = V(V > meanV);
iR = i_start + 2*i - 2;
iL = i_start + 2*i - 1;
M(iR) = mean(DR);
M(iL) = mean(DL);
if n < n_max
M = myrecfun(DR, M, n_max, n+1, iR - i_start + 1);
M = myrecfun(DL, M, n_max, n+1, iL - i_start + 1);
else % else of if n < n_max
M;
end % end of if n < n_max
end % of myrecfun
Call the code:
n_max = 8;
V = 100*rand(100,100); %Just my example
M = zeros(1, 2^(n_max+1)-1);
Mout = myrecfun(V, M, n_max, 1, 1);
Test the output:
sum(Mout< 50)
ans =
256
sum(Mout > 50)
ans =
255
Related
Construct an experiment to study the performance of the Cramer rule (with two implementations
determinants) in relation to Gauss's algorithm.
In each iteration 10 random arrays A (NxN), and vectors b (Nx1) will be created.
The 10 linear systems will be solved using the Cramer rule ("cramer.m") using
of rec_det (A) and using det (A), and the Gaussian algorithm
(“GaussianElimination.m”), and the time for each technique will be the average of 10 values.
Repeat the above for N = 2 to 10 and make a graph of the average time
in relation to the dimension N.
This is my task. I dont know if the way that I calculate the average time is correct and the graphic is not displayed.
T1=0;
T2=0;
T3=0;
for N=2:10
for i=1:10
A=rand(N,N);
b=rand(N,1);
t1=[1,i];
t2=[1,i];
t3=[1,i];
tic;
crammer(A,b);
t1(i)=toc;
tic
crammer_rec(A,b);
t2(i)=toc;
tic
gaussianElimination(A,b);
t3(i)=toc;
T1=T1+t1(i);
T2=T2+t2(i);
T3=T3+t3(i);
end
avT1=T1/10;
avT2=T2/10;
avT3=T3/10;
end
plot(2:10 , avT1 , 2:10 , avT2 , 2:10 , avT3);
function x = cramer(A, b)
n = length(b);
d = det(A);
% d = rec_det(A);
x = zeros(n, 1);
for j = 1:n
x(j) = det([A(:,1:j-1) b A(:,j+1:end)]) / d;
% x(j) = rec_det([A(:,1:j-1) b A(:,j+1:end)]) / d;
end
end
function x = cramer(A, b)
n = length(b);
d = rec_det(A);
x = zeros(n, 1);
for j = 1:n
x(j) = rec_det([A(:,1:j-1) b A(:,j+1:end)]) / d;
end
end
function deta = rec_det(R)
if size(R,1)~=size(R,2)
error('Error.Matrix must be square.')
else
n = size(R,1);
if ( n == 2 )
deta=(R(1,1)*R(2,2))-(R(1,2)*R(2,1));
else
for i=1:n
deta_temp=R;
deta_temp(1,:)=[ ];
deta_temp(:,i)=[ ];
if i==1
deta=(R(1,i)*((-1)^(i+1))*rec_det(deta_temp));
else
deta=deta+(R(1,i)*((-1)^(i+1))*rec_det(deta_temp));
end
end
end
end
end
function x = gaussianElimination(A, b)
[m, n] = size(A);
if m ~= n
error('Matrix A must be square!');
end
n1 = length(b);
if n1 ~= n
error('Vector b should be equal to the number of rows and columns of A!');
end
Aug = [A b]; % build the augmented matrix
C = zeros(1, n + 1);
% elimination phase
for k = 1:n - 1
% ensure that the pivoting point is the largest in its column
[pivot, j] = max(abs(Aug(k:n, k)));
C = Aug(k, :);
Aug(k, :) = Aug(j + k - 1, :);
Aug(j + k - 1, :) = C;
if Aug(k, k) == 0
error('Matrix A is singular');
end
for i = k + 1:n
r = Aug(i, k) / Aug(k, k);
Aug(i, k:n + 1) = Aug(i, k:n + 1) - r * Aug(k, k: n + 1);
end
end
% back substitution phase
x = zeros(n, 1);
x(n) = Aug(n, n + 1) / Aug(n, n);
for k = n - 1:-1:1
x(k) = (Aug(k, n + 1) - Aug(k, k + 1:n) * x(k + 1:n)) / Aug(k, k);
end
end
I think the easiest way to do this is by creating a 9 * 3 dimensional matrix to contain all the total times, and then take the average at the end.
allTimes = zeros(9, 3);
for N=2:10
for ii=1:10
A=rand(N,N);
b=rand(N,1);
tic;
crammer(A,b);
temp = toc;
allTimes(N-1,1) = allTimes(N-1,1) + temp;
tic
crammer_rec(A,b);
temp = toc;
allTimes(N-1,2) = allTimes(N-1,2) + temp;
tic
gaussianElimination(A,b);
temp = toc;
allTimes(N-1,3) = allTimes(N-1,3) + temp;
end
end
allTimes = allTimes/10;
figure; plot(2:10, allTimes);
You can use this approach because the numbers are quite straightforward and simple. If you had a more complicated setup, the way to store the times/calculate the averages would have to be tweaked.
If you had more functions you could also use function handles and create a third inner loop, but this is a little more advanced.
I would like to apply loop over a function. I have the following "mother" code:
v = 1;
fun = #root;
x0 = [0,0]
options = optimset('MaxFunEvals',100000,'MaxIter', 10000 );
x = fsolve(fun,x0, options)
In addition, I have the following function in a separate file:
function D = root(x)
v = 1;
D(1) = x(1) + x(2) + v - 2;
D(2) = x(1) - x(2) + v - 1.8;
end
Now, I would like to find roots when v = sort(rand(1,1000)). In other words, I would like to iterate over function for each values of v.
You will need to modify root to accept an additional variable (v) and then change the function handle to root to an anonymous function which feeds in the v that you want
function D = root(x, v)
D(1) = x(1) + x(2) + v - 2;
D(2) = x(1) - x(2) + v - 1.8;
end
% Creates a function handle to root using a specific value of v
fun = #(x)root(x, v(k))
Just in case that equation is your actual equation (and not a dummy example): that equation is linear, meaning, you can solve it for all v with a simple mldivide:
v = sort(rand(1,1000));
x = [1 1; 1 -1] \ bsxfun(#plus, -v, [2; 1.8])
And, in case those are not your actual equations, you don't need to loop, you can vectorize the whole thing:
function x = solver()
options = optimset('Display' , 'off',...
'MaxFunEvals', 1e5,...
'MaxIter' , 1e4);
v = sort(rand(1, 1000));
x0 = repmat([0 0], numel(v), 1);
x = fsolve(#(x)root(x,v'), x0, options);
end
function D = root(x,v)
D = [x(:,1) + x(:,2) + v - 2
x(:,1) - x(:,2) + v - 1.8];
end
This may or may not be faster than looping, it depends on your actual equations.
It may be slower because fsolve will need to compute a Jacobian of 2000×2000 (4M elements), instead of 2×2, 1000 times (4k elements).
But, it may be faster because the startup cost of fsolve can be large, meaning, the overhead of many calls may in fact outweigh the cost of computing the larger Jacobian.
In any case, providing the Jacobian as a second output will speed everything up rather enormously:
function solver()
options = optimset('Display' , 'off',...
'MaxFunEvals', 1e5,...
'MaxIter' , 1e4,...
'Jacobian' , 'on');
v = sort(rand(1, 1000));
x0 = repmat([1 1], numel(v), 1);
x = fsolve(#(x)root(x,v'), x0, options);
end
function [D, J] = root(x,v)
% Jacobian is constant:
persistent J_out
if isempty(J_out)
one = speye(numel(v));
J_out = [+one +one
+one -one];
end
% Function values at x
D = [x(:,1) + x(:,2) + v - 2
x(:,1) - x(:,2) + v - 1.8];
% Jacobian at x:
J = J_out;
end
vvec = sort(rand(1,2));
x0 = [0,0];
for v = vvec,
fun = #(x) root(v, x);
options = optimset('MaxFunEvals',100000,'MaxIter', 10000 );
x = fsolve(fun, x0, options);
end
with function definition:
function D = root(v, x)
D(1) = x(1) + x(2) + v - 2;
D(2) = x(1) - x(2) + v - 1.8;
end
this is my first time here so I hope that someone can help me.
I'm trying to implementing the Gauss-Seidel method and the power method using a matrix with the storage CSR or called Morse storage. Unfortunately I can't manage to do better then the following codes:
GS-MORSE:
function [y] = gs_morse(aa, diag, col, row, nmax, tol)
[n, n] = size(A);
y = [1, 1, 1, 1];
m = 1;
while m < nmax,
for i = 1: n,
k1 = row(i);
k2 = row(i + 1) - 1;
for k = k1: k2,
y(i) = y(i) + aa(k) * x(col(k));
y(col(k)) = y(col(k)) + aa(k) * diag(i);
end
k2 = k2 + 1;
y(i) = y(i) + aa(k) * diag(i);
end
if (norm(y - x)) < tol
disp(y);
end
m = m + 1;
for i = 1: n,
x(i) = y(i);
end
end
POWER-MORSE:
I was able only to implement the power method but I don't understand how to use the former matrix... so my code for power method is:
function [y, l] = potencia_iterada(A, v)
numiter=100;
eps=1e-10;
x = v(:);
y = x/norm(x);
l = 0;
for k = 1: numiter,
x = A * y;
y = x / norm(x);
l0 = x.' * y;
if abs(l0) < eps
return
end
l = l0;
end
Please anyone can help me for completing these codes or can explain me how can I do that? I really don't understand how to do. Thank you very much
I was wondering if there was a way to vectorize the nested for loop in this function which is filling up the entries of the 2D dynamic programming table DP. I believe that at the very least the inner loop could be vectorized as each row only depends on the previous row. I'm not sure how to do it though. Note this function is called on large 2D arrays (images) so the nested for loop really doesn't cut it.
function [cols] = compute_seam(energy)
[r, c, ~] = size(energy);
cols = zeros(r);
DP = padarray(energy, [0, 1], Inf);
BP = zeros(r, c);
for i = 2 : r
for j = 1 : c
[x, l] = min([DP(i - 1, j), DP(i - 1, j + 1), DP(i - 1, j + 2)]);
DP(i, j + 1) = DP(i, j + 1) + x;
BP(i, j) = j + (l - 2);
end
end
[~, j] = min(DP(r, :));
j = j - 1;
for i = r : -1 : 1
cols(i) = j;
j = BP(i, j);
end
end
Vectorization of the innermost nested loop
You were right in postulating that at least the inner loop is vectorizable. Here's the modified code for the nested loops part -
rows_DP = size(DP,1); %// rows in DP
%// Get first row linear indices for a group of neighboring three columns,
%// which would be incremented as we move between rows with the row iterator
start_ind1 = bsxfun(#plus,[1:rows_DP:2*rows_DP+1]',[0:c-1]*rows_DP); %//'
for i = 2 : r
ind1 = start_ind1 + i-2; %// setup linear indices for the row of this iteration
[x,l] = min(DP(ind1),[],1); %// get x and l values in one go
DP(i,2:c+1) = DP(i,2:c+1) + x; %// set DP values of a row in one go
BP(i,1:c) = [1:c] + l-2; %// set BP values of a row in one go
end
Benchmarking
Benchmarking Code -
N = 3000; %// Datasize
energy = rand(N);
[r, c, ~] = size(energy);
disp('------------------------------------- With Original Code')
DP = padarray(energy, [0, 1], Inf);
BP = zeros(r, c);
tic
for i = 2 : r
for j = 1 : c
[x, l] = min([DP(i - 1, j), DP(i - 1, j + 1), DP(i - 1, j + 2)]);
DP(i, j + 1) = DP(i, j + 1) + x;
BP(i, j) = j + (l - 2);
end
end
toc,clear DP BP x l
disp('------------------------------------- With Vectorized Code')
DP = padarray(energy, [0, 1], Inf);
BP = zeros(r, c);
tic
rows_DP = size(DP,1); %// rows in DP
start_ind1 = bsxfun(#plus,[1:rows_DP:2*rows_DP+1]',[0:c-1]*rows_DP); %//'
for i = 2 : r
ind1 = start_ind1 + i-2; %// setup linear indices for the row of this iteration
[x,l] = min(DP(ind1),[],1); %// get x and l values in one go
DP(i,2:c+1) = DP(i,2:c+1) + x; %// set DP values of a row in one go
BP(i,1:c) = [1:c] + l-2; %// set BP values of a row in one go
end
toc
Results -
------------------------------------- With Original Code
Elapsed time is 44.200746 seconds.
------------------------------------- With Vectorized Code
Elapsed time is 1.694288 seconds.
Thus, you might enjoy a good 26x speedup improvement in performance with that little vectorization tweak.
More tweaks
Few more optimization tweaks could be tried into your code for performance -
cols = zeros(r) could be replaced with col(r,r) = 0.
DP = padarray(energy, [0, 1], Inf) could be replaced with
DP(1:size(energy,1),1:size(energy,2)+2)=Inf;
DP(:,2:end-1) = energy;
BP = zeros(r, c) could be replaced with BP(r, c) = 0.
The pre-allocation tweaks used here are inspired by this blog post.
I am unable to get converging values using a Gauss-Seidel algorithm
Here is the code:
A = [12 3 -5 2
1 6 3 1
3 7 13 -1
-1 2 -1 7];
b = [2
-3
10
-11];
ep = 1e-8;
[m, n] = size(A);
[n, p] = size(b);
x = zeros(n, 1001);
x(:, 1) = []
for k=0:1000
ka = k + 1;
if ka == 1001
break;
end
xnew = zeros(n,1);
for i=1:n
sum = 0;
j = 1;
while j < i
s1 = s1 + A(i,j) * x(j, ka + 1);
j = j + 1;
end
j = i + 1;
while j <= n
sum = sum + A(i,j) * x(j, ka);
j = j + 1;
end
xnew(i) = (b(i) - sum) / A(i, i);
% if result is within error bounds exit loop
if norm(b - A * xnew, 2) < ep * norm(b, 2)
'ending'
break
end
end
x(:,ka + 1) = xnew;
end
I cannot get the A * xnew to converge on b what am I doing wrong?
I have tried running this changing the syntax several times, but I keep getting values that are way off.
Thanks!
Gabe
You have basically two problems with your code:
(1) You are using two different variables "sum" and "s1". I replaced it by mySum. By the way, dont use "sum", since there is a matlab function with this name.
(2) I think there is also a problem in the update of x;
I solved this problem and I also tried to improve your code:
(1) You dont need to save all "x"s;
(2) It is better to use a "while" than a for when you dont know how many iterations you need.
(3) It is good to use "clear all" and "close all" in general in order to keep your workspace. Sometimes old computations may generate errors. For instance, when you use matrices with different sizes and the same name.
(4) It is better to use dot/comma to separate the lines of the matrices
You still can improve this code:
(1) You can test if A is square and if it satisfies the conditions necessary to use this numerical method: to be positive definite or to be diagonally dominant.
clear all
close all
A = [12 3 -5 2;
1 6 3 1;
3 7 13 -1;
-1 2 -1 7];
b = [2;
-3;
10;
-11];
ep = 1e-8;
n = length(b); % Note this method only works for A(n,n)
xNew=zeros(n,1);
xOld=zeros(n,1);
leave=false;
while(~leave)
xOld=xNew;
for i=1:n
mySum = 0;
j = i + 1;
while j <= n
mySum = mySum + A(i,j) * xOld(j,1);
j = j + 1;
end
j = 1;
while j < i
mySum = mySum + A(i,j) * xNew(j,1);
j = j + 1;
end
mySum=b(i,1)-mySum;
xNew(i,1) = mySum / A(i, i);
end
if (norm(b - A * xNew, 2) < ep * norm(b, 2))
disp('ending');
leave=true;
end
xOld = xNew;
end
xNew