Closed. This question does not meet Stack Overflow guidelines. It is not currently accepting answers.
This question does not appear to be about programming within the scope defined in the help center.
Closed 6 years ago.
Improve this question
I would like to know what h from the numerical differentiation formulas is and how I can calculate it when I have a function.
I am speaking about this formulas:
f'(x0) = (f(x0 + h) - f(x0)) / h
f'(x0) = (f(x0) - f(x0 - h)) / h
f'(x0) = (f(x0 + h) - f(x0 - h)) / 2*h
I would really appreciate any kind of help!
In such formulae h is usually a "very small number", similar to epsilon in Calculus.
For example, the derivative of f at a is defined as:
Note how h is defined as approaching 0.
When programming, e.g. doing numerical gradient computation, it usually works to set h to something very small - many programming environments have an "epsilon" quantity; lacking that, you can just use a very small floating-point number.
Using the usual 8 byte floats, sensible values for h are 1e-8 for the first and second formula and 1e-5 for the third central difference quotient. This is valid for medium values of x, for larger x one would have to include the scale of x in some way.
In general, for a kth order difference quotient with error order p, the balance between floating point noise and numerical error is reached for h about pow(2e-16, 1.0/(p+k)).
Related
Closed. This question does not meet Stack Overflow guidelines. It is not currently accepting answers.
Questions asking for code must demonstrate a minimal understanding of the problem being solved. Include attempted solutions, why they didn't work, and the expected results. See also: Stack Overflow question checklist
Closed 9 years ago.
Improve this question
I am trying to approximate an integral of a function f(x) from -inf to +inf using the Riemann sum.
integrand<-function(x){exp(x)/sum(exp(x))}
integrate(integrand,lower=-Inf,upper=Inf)
Error in integrate(integrand, lower = -Inf, upper = Inf) :
non-finite function value
The solution doesn't exist if that is your true function
You still haven't clarified the function that you want to integrate. Can you write the integral that you want to evaluate, in ordinary mathematical notation, forgetting about R for the moment?
Here's what you've written, and why it doesn't make sense:
x<- seq(-1000,1000, length=1000)
OK, now x is vector of length 1000: (-1000,-997.998,...,997.998,1000)
v <- sum(exp(x))
OK, now v is equal to the sum of exp evaluated for the values on the vector x. And, since exp(1000)=Inf, this means that, from the point of view of R, v is equal to infinity.
f <-(exp(x))/(v)
At this point, you have defined x to be a vector and v to be a scalar, so this expression will set f to be a vector of length 1000. However, since v is infinite, then every value on f is going to be either 0 or NaN.
This all means that
integrate(f, ???? )
is meaningless, because f is not a function.
If you can write, in ordinary mathematical notation, exactly what integral you want to evaluate, someone will probably be able to help you.
Closed. This question does not meet Stack Overflow guidelines. It is not currently accepting answers.
This question does not appear to be about programming within the scope defined in the help center.
Closed 9 years ago.
Improve this question
So I have an assignment question where I have to prove:
n^4 is in O(2^n)
Just by looking at the graphs of the functions I know that with c=1 and n[0] = 16 this is true.
While trying to prove it on paper I managed to reduce the inequality down to n <= 2^(n/4), however, I cannot figure out how to simplify this further or adequately prove from here that with n[0]=16 the big-O assertion holds.
Any help?
The title is incorrect, and the error is important.
You are not trying to prove that n ≤ 2n/4, you are trying to prove that n ∊ O(2n/4), which is a strictly weaker claim. It is impossible to prove that n ≤ 2n/4 because at n=2, the inequality is false.
By taking the logarithm of both sides, we can reduce the problem to that of showing that log n ∊ O(n), which is easy to show because d/dn log n ≤ 1 for n ≥ 1.
It is easy to prove that the inequality holds for n >= 16 using induction, no calculus required:
First, for n=16 you have 164=216.
If the inequality holds for n=k, for n=k+1 you have (k+1)4 = (####)·k4 < 2k4 ≤ 2·2k = 2k+1.
QED.
Since this is homework, I'll leave leave the crucial step, finding what goes in place of ####, to the reader.
Closed. This question does not meet Stack Overflow guidelines. It is not currently accepting answers.
This question does not appear to be about programming within the scope defined in the help center.
Closed 9 years ago.
Improve this question
Let's say I want to predict a dependent variable D, where:
D<-rnorm(100)
I cannot observe D, but I know the values of three predictor variables:
I1<-D+rnorm(100,0,10)
I2<-D+rnorm(100,0,30)
I3<-D+rnorm(100,0,50)
I want to predict D by using the following regression equation:
I1 * w1 + I2 * w2 + I3 * w3 = ~D
however, I do not know the correct values of the weights (w), but I would like to fine-tune them by repeating my estimate:
in the first step I use equal weights:
w1= .33, w2=.33, w3=.33
and I estimate D using these weights:
EST= I1 * .33 + I2 * .33 + I3 *. 33
I receive feedback, which is a difference score between D and my estimate (diff=D-EST)
I use this feedback to modify my original weights and fine-tune them to eventually minimize the difference between D and EST.
My question is:
Is the difference score sufficient for being able to fine-tune the weights?
What are some ways of manually fine-tuning the weights? (e.g. can I look at the correlation between diff and I1,I2,I3 and use that as a weight?
The following command,
coefficients(lm(D ~ I1 + I2 + I3))
will give you the ideal weights to minimize diff.
Your defined diff will not tell you enough to manually manipulate the weights correctly as there is no way to isolate the error component of each I.
The correlation between D and the I's is not sufficient either as it only tells you the strength of the predictor, not the weight. If your I's are truly independent (both from each other, all together and w.r.t. D - a strong assumption, but true when using rnorm for each), you could try manipulating one at a time and notice how it affects diff, but using a linear regression model is the simplest way to do it.
Closed. This question is off-topic. It is not currently accepting answers.
Want to improve this question? Update the question so it's on-topic for Stack Overflow.
Closed 10 years ago.
Improve this question
I have run into the following problem that I need to solve in a project that I'm working on:
Given some number of vectors v_i (in the math sense), and a target vector H, compute a linear combination of the vectors v_i that most closely matches the target vector H, with the constraint that the coefficients must be in [0, 1].
I do not know much about what kind of algorithms / math should be used to approach such a problem. Any prods in the right general direction would be much appreciated!
It's a constrained least square problem. Basically you want to solve the optimization problem:
argmin ||Ax-H||
x
s.t. 0<=x_j<=1
where x=(x_1, ..., x_j, ..., x_n) consists the coefficients you are seeking, and a column of A corresponds to a vector v_i.
Assuming that you want to solve in the least squares sense, then you have a quadratic programming problem. For example, say that your set of vectors is
x1 = 1 2 3]' x2 = [3 2 1]'
and your target vector is
H = [1 -1 1]'
Then you can create the matrix whose columns are your vectors:
A = [1 3;
2 2;
3 1]
and the thing you are trying to minimize is
norm(A*x - H) = (A*x - H)' * (A*x - H) = x' * (A'*A) * x - (2*H'*A) * x + const
If you define
B = A' * A
C = -2 * H' * A
then you have a problem that can be solved optimally my Matlab's quadprog function
quadprog(B,C,[],[],[],[],0,1)
ans =
0.16667
0.16667
so the optimal solution in this case is
1/6 * x1 + 1/6 * x2 = [2/3, 2/3, 2/3]
This is a combinatorial optimization problem. This kind of problems are NP-hard. But I guess for the binary one, there should be polynomial algorithms that can solve, or there may be some relaxation to get an approximate solution. Some googling on "integer programming" may help.
Closed. This question is off-topic. It is not currently accepting answers.
Want to improve this question? Update the question so it's on-topic for Stack Overflow.
Closed 10 years ago.
Improve this question
I am trying to find unknown matrix multiply matrix with knowing matrix
A*c=b
where b is defined vector, A is defined matrix 8x8, c is unknown vector.
I know, I can not divide matrix but what is the solution for this situation ??
This is basically a system of simultaneous linear equations. You can solve it using Gaussian elimination.
As for matrix "division", what you really have in mind is an inverse matrix, i.e. a matrix A-1 such that
AA-1=A-1A=I
where I is the identity matrix. If A is invertible then A*c=b is equivalent to c=A-1b.
The answer by Adam is certainly correct, but you should know that calculating the inverse of the matrix might not be the best solution.
Another to look into is LU decomposition and forward-back substitution. It will be more computationally stable that full Gaussian elimination and calculating the inverse.
You solve the problem in steps like this:
Decompose A = LU; now you'll have LUc = b. L is lower triangular; U is upper triangular.
Let y = Uc; solve Ly = b for y.
Now that you have y, solve for the c vector you want: y = Uc.