I am trying to write a program in matlab, such that the sum of the reciprocals of the n first prime numbers exceeds a given value k. To clearify, I am trying to make a function
SumPrime(k)
And it is supposed to return an integer n such that
\sum_{i=1}^{n} 1/p_i > k
sum of primes and reciprocals of them and plot in matlab?
I tried looking here, but this does not quite answer my question. Neither did the command
sumInversePrimes = sum(1./primes(n));
Here is my attempt. First i define a function for finding the n`th prime number.
function Y = NthPrime(n)
if n==1
Y = 2;
return
end
if n < 1 || round(n)~=n
return
end
j = 2;
u = 0;
while u < n
T = primes(j);
u = numel(T);
j = 1 + j;
end
Y = T(numel(T));
After doing this (lengthy?) code for finding the n`th prime number, the rest is a cakewalk.
function Y = E(u)
sum = 0
n = 0
while sum < u
n = n + 1
sum = sum + 1/( NthPrime(n) )
end
Y = n;
Return the proper values. This somewhat works. Alas it is very slow, and I guess this is very bad code. I have merely started learning coding in matlab, Could someone please help me either write a better code or optimize mine ?
XOXOXOX
Nebby
Here's how to precompute the sums then find the first that exceeds a threshold:
>> p = primes(1000);
>> cs = cumsum(1./p);
>> find(cs > 1.8, 1)
ans = 25
Related
I want to compute the above summation, for a given 'x'. The summation is to be carried out over a block of lengths specified by an array , for example block_length = [5 4 3]. The summation is carried as follows: from -5 to 5 across one dimension, -4 to 4 in the second dimension and -3 to 3 in the last dimension.
The pseudo code will be something like this:
sum = 0;
for i = -5:5
for j = -4:4
for k = -3:3
vec = [i j k];
tv = vec * vec';
sum = sum + 1/(1+tv)*cos(2*pi*x*vec'));
end
end
end
The problem is that I want to find the sum when the number of dimensions are not known ahead of time, using some kind of variable nested loops hopefully. Matlab uses combvec, but it returns all possible combinations of vectors, which is not required as we only compute the sum. When there are many dimensions, combvec returning all combinations is not feasible memory wise.
Appreciate any ideas towards solutions.
PS: I want to do this at high number of dimensions, for example 650, as in machine learning.
Based on https://www.mathworks.com/matlabcentral/answers/345551-function-with-varying-number-of-for-loops I came up with the following code (I haven't tested it for very large number of indices!):
function sum = fun(x, block_length)
sum = 0;
n = numel(block_length); % Number of loops
vec = -ones(1, n) .* block_length; % Index vector
ready = false;
while ~ready
tv = vec * vec';
sum = sum + 1/(1+tv)*cos(2*pi*x*vec');
% Update the index vector:
ready = true; % Assume that the WHILE loop is ready
for k = 1:n
vec(k) = vec(k) + 1;
if vec(k) <= block_length(k)
ready = false;
break; % v(k) increased successfully, leave "for k" loop
end
vec(k) = -1 * block_length(k); % v(k) reached the limit, reset it
end
end
end
where x and block_length should be both 1-x-n vectors.
The idea is that, instead of using explicitly nested loops, we use a vector of indices.
How good/efficient is this when tackling the suggested use case where block_length can have 650 elements? Not much! Here's a "quick" test using merely 16 dimensions and a [-1, 1] range for the indices:
N = 16; tic; in = 0.1 * ones(1, N); sum = fun(in, ones(size(in))), toc;
which yields an elapsed time of 12.7 seconds on my laptop.
I want to find the smallest integer l that satisfies l^2 >= x, and mod(l,2)=0.
In the following example x=75, and hence l=10, since the previous even number doesn't fulfil the inequation: 8^2 <= 75 <= 10^2
I have tried this (ignoring the even-number requirement, which I can't to work):
syms l integer
eqn1 = l^2 >= 75;
% eqn2 = mod(l,2) == 0;
[sol_l, param, cond] = solve(eqn1, l, 'ReturnConditions', true);
But this does not give me anything helpful directly:
sol_l =
k
param =
k
cond =
(75^(1/2) <= k | k <= -75^(1/2)) & in(k, 'integer')
I would like to evaluate the conditions on the parameter and find the smallest value that satisfies the conditions.
Also, I would like to enforce the mod(l,2)=0 condition somehow, but I don't seem to get that work.
Using the solve for this task is like using a cannon to kill a mosquito. Actually, the answer of Lidia Parrilla is good and fast, although it can be simplified as follows:
l = ceil(sqrt(x));
if (mod(x,2) ~= 0)
l = l + 1;
end
% if x = 75, then l = 10
But I would like to point out something that no one else noticed. The condition provided by the solve function for l^2 >= 75 is:
75^(1/2) <= k | k <= -75^(1/2)
and it's absolutely correct. Since l is being raised to the power of 2, and since a negative number raised to the power of 2 produces a positive number, the equation will always have two distinct solutions: a negative one and a positive one.
For x = 75, the solutions will be l = 10 and l = -10. So, if you want to find the smallest number (and a negative number is always smaller than a positive one), the right solution will be:
l = ceil(sqrt(x));
if (mod(x,2) ~= 0)
l = l + 1;
end
l = l * -1;
If you want to return both solutions, the result will be:
l_pos = ceil(sqrt(x));
if (mod(x,2) ~= 0)
l_pos = l_pos + 1;
end
l_neg = l_pos * -1;
l = [l_neg l_pos];
I guess the easiest solution without employing the inequality and solve function would be to find the exact solution to your equation l^2 >= x, and then finding the next even integer. The code would look like this:
x = 75;
y = ceil(sqrt(x)); %Ceil finds the next bigger integer
if(~mod(y,2)) %If it's even, we got the solution
sol = y;
else %If not, get the next integer
sol = y+1;
end
The previous code gives the correct solution to the provided example (x = 75; sol = 10)
I am trying to sum a function and then attempting to find the root of said function. That is, for example, take:
Consider that I have a matrix,X, and vector,t, of values: X(2*n+1,n+1), t(n+1)
for j = 1:n+1
sum = 0;
for i = 1:2*j+1
f = #(g)exp[-exp[X(i,j)+g]*(t(j+1)-t(j))];
sum = sum + f;
end
fzero(sum,0)
end
That is,
I want to evaluate at
j = 1
f = #(g)exp[-exp[X(1,1)+g]*(t(j+1)-t(j))]
fzero(f,0)
j = 2
f = #(g)exp[-exp[X(1,2)+g]*(t(j+1)-t(j))] + exp[-exp[X(2,2)+g]*(t(j+1)-t(j))] + exp[-exp[X(3,2)+g]*(t(j+1)-t(j))]
fzero(f,0)
j = 3
etc...
However, I have no idea how to actually implement this in practice.
Any help is appreciated!
PS - I do not have the symbolic toolbox in Matlab.
I suggest making use of matlab's array operations:
zerovec = zeros(1,n+1); %preallocate
for k = 1:n+1
f = #(y) sum(exp(-exp(X(1:2*k+1,k)+y)*(t(k+1)-t(k))));
zerovec(k) = fzero(f,0);
end
However, note that the sum of exponentials will never be zero, unless the exponent is complex. Which fzero will never find, so the question is a bit of a moot point.
Another solution is to write a function:
function [ sum ] = func(j,g,t,X)
sum = 0;
for i = 0:2*j
f = exp(-exp(X(i+1,j+1)+g)*(t(j+3)-t(j+2)));
sum = sum + f;
end
end
Then loop your solver
for j=0:n
fun = #(g)func(j,g,t,X);
fzero(fun,0)
end
I'm trying to multiply (element wise) a vector V of length N by a randomly generated number in the range (a,b), while keeping the sum of the vector equal to a total amount, E. I want to do this in MATLAB, but I'm not sure how. Getting random numbers between a certain range I know how to do:
minrand = 0;
maxrand = 1;
randfac = (maxrand-minrand).*rand(1,N) + minrand;
But yeah, beyond that I'm pretty clueless. I guess the random numbers can't really be generated like this, because if we call the random numbers the vector R, then I want that
R_1*V1 + R_2*V2 .... + R_N*V_N = E. So I guess it's a big equation. Is there any way to solve it, while putting constraints on the max and min values of R?
You can pick pairs of two elements (in all combinations) and add and subtract an equal random number.
% Make up a random vector
N=10;
randfac = 10*rand(1,N);
%OP Answer here: Given randfac with sum E re-randomize it
E = sum(randfac);
minrand = 0;
maxrand = 2;
disp(randfac)
% v = [6.4685 2.9652 6.6567 1.6153 7.3581 0.0237 7.1025
% 3.2381 1.9176 1.3561]
disp(sum(randfac))
% E = 38.7019
r = minrand + (maxrand-minrand)*rand(N*N,1);
k = 1;
for i=1:N
for j=1:N
randfac(i) = randfac(i)-r(k);
randfac(j) = randfac(j)+r(k);
k = k + 1;
end
end
disp(randfac)
% v = [5.4905 0.7051 4.7646 1.3479 9.3722 -1.4222 7.9275
% 7.5777 1.7549 1.1836]
disp(sum(randfac))
% E = 38.7019
Just divide the vector with the sum and multiply with the target E.
randfac = (maxrand-minrand).*rand(1,N) + minrand;
randfac = E*randfac/sum(randfac);
as long as the operator is linear, the result is going to retain it's randomness. Below is some sample code:
minrand = 0;
maxrand = 1;
N = 1000; %size
v = (maxrand-minrand).*rand(1,N) + minrand;
E = 100; %Target sum
A = sum(v);
randfac = (E/A)*v;
disp(sum(randfac))
% 100.0000
First of all with random numbers in the interval of [a b] you can't guarantee that you will have the same summation (same E). For example if [a b]=[1 2] of course the E will increase.
Here is an idea, I don't know how random is this!
For even N I randomize V then divide it in two rows and multiply one of them with random numbers in [a b] but the second column will be multiplied to a vector to hold the summation fixed.
N = 10;
V = randi(100,[1 N]);
E = sum(V);
idx = randperm(N);
Vr = V(idx);
[~,ridx] = sort(idx);
Vr = reshape(Vr,[2 N/2]);
a = 1;
b = 3;
r1 = (b - a).*rand(1,N/2) + a;
r2 = (sum(Vr) - r1.*Vr(1,:))./Vr(2,:);
r = reshape([r1;r2],1,[]);
r = r(ridx);
Enew = sum(V.*r);
The example results are,
V = [12 82 25 51 81 51 31 87 6 74];
r = [2.8018 0.7363 1.9281 0.5451 1.9387 -0.4909 1.3076 0.8904 2.9236 0.8440];
with E = 500 as well as Enew.
I'm simply assigning one random number to a pair (It can be considered as half random!).
Okay, I have found a way to somewhat do this, but it is not elegant and there are probably better solutions. Starting with an initial vector e, for which sum(e) = E, I can randomize its values and end up with an e for which sum(e) is in the range [(1-threshold)E,(1+thresholdE)]. It is computationally expensive, and not pretty.
The idea is to first multiply e by random numbers in a certain range. Then, I will check what the sum is. If it is too big, I will decrease the value of the random numbers smaller than half of the range until the sum is no longer too big. If it is too small, I do the converse, and iterate until the sum is within the desired range.
e = somepredefinedvector
minrand = 0;
maxrand = 2;
randfac = (maxrand-minrand).*rand(1,N) + minrand;
e = randfac.*e;
threshold = 0.001;
while sum(e) < (1-threshold)*E || sum(e) > (1+threshold)*E
if sum(e) > (1+threshold)*E
for j = 1:N
if randfac(j) > (maxrand-minrand)/2
e(j) = e(j)/randfac(j);
randfac(j) = ((maxrand-minrand)/2-minrand).*rand(1,1) + minrand;
e(j) = randfac(j)*e(j);
end
if sum(e) > (1-threshold)*E && sum(e) < (1+threshold)*E
break
end
end
elseif sum(e) < (1-threshold)*E
for j = 1:N
if randfac(j) < (maxrand-minrand)/2
e(j) = e(j)/randfac(j);
randfac(j) = (maxrand-(maxrand-minrand)/2).*rand(1,1) + (maxrand-minrand)/2;
e(j) = randfac(j)*e(j);
end
if sum(e) > (1-threshold)*E && sum(e) < (1+threshold)*E
break
end
end
end
end
I have 2 matrices = X in R^(n*m) and W in R^(k*m) where k<<n.
Let x_i be the i-th row of X and w_j be the j-th row of W.
I need to find, for each x_i what is the j that maximizes <w_j,x_i>
I can't see a way around iterating over all the rows in X, but it there a way to find the maximum dot product without iterating every time over all of W?
A naive implementation would be:
n = 100;
m = 50;
k = 10;
X = rand(n,m);
W = rand(k,m);
Y = zeros(n, 1);
for i = 1 : n
max_ind = 1;
max_val = dot(W(1,:), X(i,:));
for j = 2 : k
cur_val = dot(W(j,:),X(i,:));
if cur_val > max_val
max_val = cur_val;
max_ind = j;
end
end
Y(i,:) = max_ind;
end
Dot product is essentially matrix multiplication:
[~, Y] = max(W*X');
bsxfun based approach to speed-up things for you -
[~,Y] = max(sum(bsxfun(#times,X,permute(W,[3 2 1])),2),[],3)
On my system, using your dataset I am getting a 100x+ speedup with this.
One can think of two more "closeby" approaches, but they don't seem to give any huge improvement over the earlier one -
[~,Y] = max(squeeze(sum(bsxfun(#times,X,permute(W,[3 2 1])),2)),[],2)
and
[~,Y] = max(squeeze(sum(bsxfun(#times,X',permute(W,[2 3 1]))))')