# Questions tagged [machine-precision]

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1answer
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### Avoiding catastrophic cancellation in $1-\operatorname{sinc}x$

When I try to calculate the function $f(x)=1-\operatorname{sinc}x$ for small values of $x$ I get large relative errors due to catastrophic cancellation. I want an accurate way to calculate $f(x)$ ...
0answers
11 views

### Relative error of a number in machine epsilon units

I came across an estimation of the relative error between two representations of the same number, one implemented in C++ and another one via a computer algebra program, that was in units of machine ...
1answer
73 views

1answer
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### Floating point numbers, real numbers and machine precision

My math-book states that when a real number $x$ is replaced by a floating-point number $F(x)$ then the error between the two is: $|error|=|x-F(x)|\le\epsilon |x|$ Now the book asks: Consider two ...
1answer
173 views

### Solving strict inequalities with integer maths accounting for rounding errors

I have an array of floating point numbers $X_i$ of type T, where T could be either float or double. These numbers are strictly ...
1answer
1k views

### How to Calculate Mean Cross Entropy for Accuracy

I have 2 arrays of size N. X= array of predicted probabilities Y= observed probabilities This can be interpreted as X[i] = the predicted probability of the event occurring on trial i....
1answer
503 views

### Question on benign and catastrophic cancellations

I am studying theorem 4 in this article (its proof is under theorem 13). From what I have learned, the theorem is helpful when $x$ is small but $1 \oplus x \neq 1$. At the end of theorem 4, it ...
1answer
344 views

### Rounding error of $\ln(1 + x)$

I am struggling to understand the theorem 4 in this article. From what I understood, the theorem indicates that since we lost precision in calculating $\ln(x+1)$ for small $x$, a more accurate way to ...
1answer
485 views

### Relative error of machine summation

Let $\mathbb{F}(b,t,L,U)$ (briefly $\mathbb{F}$) the set of all machine numbers. The definition is the usual, i.e. $\mathbb{F}$ is defined as \begin{equation*} \mathbb{F} := \left\lbrace (-1)^{s} m ...
1answer
248 views

### calculating exponential function with high precision.

How can I calculate $e^{-x}$ with 128 bit precision. I am using c on a machine of 32 bit wordsize. Thank you.
0answers
100 views

### What is the best method to compute the eigenvalues of a large non-symmetric matrix?

I have a square matrix with the dimension of $1600\times1600$. I need to compute the eigenvalues of this matrix and investigate its observability. The usual way is to compute the eigenvalues of this ...