I want to generate array or 1 x M matrix with variables such as (please note that sum goes from i=1 to m-1):
Because I want to use Obj later as a general array depends on given M for some optimization purposes. I tried following code in MATLAB but that symbolic L does not support:
function testSymbolic
M=3;
Obj = ones(1,M);
L = sym('L', [1 M]);
tempL = log(1-L);
for m=1:M
Obj(1,m) = log((L(m))/(1+L(m))) + sum(tempL(1,1:m-1),2);
end
Obj
However, when I see that following when I run separately:
L = sym('L', [1 3])
L =
[ L1, L2, L3]
L(1)
ans =
L1
can some one please help me to fix this issue?
Try
M = 3;
for i = 1:M
L(i) = sym(['L(' num2str(i) ')'])
end
tempL = log(1-L);
for m=1:M
Obj(1,m) = log((L(m))/(1+L(m))) + sum(tempL(1:m-1));
end
This yields e.g.
Obj(3)
ans =
log(L(3)/(L(3) + 1)) + log(1 - L(1)) + log(1 - L(2))
EDIT:
Considering you want to calculate the maximum of the Obj-vector I would suggest this:
clear;
M = 3;
L = sym('L', [1 M]);
tempL = log(1-L);
for m=1:M
Obj(1,m) = real(log((L(m))/(1+L(m))) + sum(tempL(1:m-1)));
end
F = matlabFunction(Obj,'vars',{L})
ft = #(v) (max (-F(v)));
Now you can optimize ft.
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 write a Matlab code to calculate the following:
\sum_{k=0}^{N-1} \frac{1}{k!} \sum_{i=0}^{k} {k \choose i}(a-1)^{k-i} a^k
and my code is:
N = 3;
a = [3 4];
for k = 0:N-1
f = 0;
for i = 0:k
f = f + nchoosek(k,i).* a.^k .* (a-1).^(k-i);
end
sumoff = sum(f);
all = (( 1./ (factorial(k))).*sumoff);
end
overall= sum(all);
'all' variable gives different value when it is inside the for loop rather than outside. But I want it to calculate when k = 0:N-1. What am I doing wrong?
Thank you.
The issue is your current code overwrites all on every iteration. Moving it outside the loop also doesn't work because you'll only save the result of the last iteration.
To save the all of every iteration, define all as a vector and then assign each intermediate result into that vector:
N = 3;
a = [3 4];
% preallocate a vector for `all`
all = nan(N-1, 1);
for k = 0:N-1
f = 0;
for i = 0:k
f = f + nchoosek(k,i) .* a.^k .* (a-1).^(k-i);
end
sumoff = sum(f);
% assign your intermediate result into the `all` vector
all(k+1) = ((1./(factorial(k))) .* sumoff);
end
overall = sum(all);
I am trying to use minimization to calculate coefficients of the polynomial p(x) = 1 + c(1)x + c(2)x^2 to approximate e^x. I need to use points xi = 1 + i/n for natural numbers i on [1,n], first for n=5, then n=10, etc. The approach is to minimize the 1, 2, and inf norm(p(x) - e^x) using fminsearch. So the output should be the 2 coefficients for the 3 p(x)'s. Any suggestions is appreciated.
Well, if anyone was wondering, I did figure the question out finally.
for l = [1 2 inf]
fprintf('For norm %d\n', l)
fprintf('Coefficients c1 c2\n')
for n = [5 10 100]
i = 1:n ;
x = 1 + i/n ;
c = [1 1] ;
%Difference function that we want to minimize
g = #(c) x.*c(1) + x.^2.*c(2) + 1 - exp(x);
f_norm = #(c) norm(g(c), l) ;
C = fminsearch(f_norm, c);
fprintf('n = %d ', n)
fprintf('%f %f\n', C(1), C(2))
% Compare plot of e^x and p(x).
p = #(x) C(1)*x + C(2)*x.^2 + 1;
xx = linspace(1,2,1e5);
figure;
plot(xx, p(xx), '--r', xx, exp(xx));
str = sprintf('Plot with n = %d, for norm %d', n,l);
title(str,'FontSize',24)
xlabel('x','FontSize',20)
ylabel('y','FontSize',20)
legend('p2 approximation','exponential');
end
end
This worked in the end to answer the question.
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 need to evaluate following expression (in pseudo-math notation):
∑ipi⋅n
where p is a matrix of three-element vectors and n is a three-element vector. I can do this with for loops as follows but I can't figure out
how to vectorize this:
p = [1 1 1; 2 2 2];
n = [3 3 3];
s = 0;
for i = 1:size(p, 1)
s = s + dot(p(i, :), n)
end
Why complicate things? How about simple matrix multiplication:
s = sum(p * n(:))
where p is assumed to be an M-by-3 matrix.
I think you can do it with bsxfun:
sum(sum(bsxfun(#times,p,n)))
----------
% Is it the same for this case?
----------
n = 200; % depending on the computer it might be
m = 1000*n; % that n needs to be chosen differently
A = randn(n,m);
x = randn(n,1);
p = zeros(m,1);
q = zeros(1,m);
tic;
for i = 1:m
p(i) = sum(x.*A(:,i));
q(i) = sum(x.*A(:,i));
end
time = toc; disp(['time = ',num2str(time)]);