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If your final result is of the order of magnitude of exp(-1000) $\approx 5 \cdot 10^{-435}$, then you are out of luck; no matter how you compute it, it will always underflow. There is simply no repres …

This can be done optimally, mapping bijectively each combination of $k$ numbers among $n$ into a distinct integer in $\{1,...,N\}$, with $N = \binom{n}{k}$ is as small as possible. This problem is kno …

When implementing this is there some care I have to take besides checking that the denominator is not zero in order to achieve the best numerical results? No. The naive algorithm (compute the numera …

, and it looks very fast: sage: %timeit lcm(range(1,1000)) 100 loops, best of 3: 820 µs per loop If you are doing number theoretical computations, I'd recommend you to move to Sage instead of pure Python …

Why don't you use regular Newton? Your system is simple enough that you can find its closed-form Jacobian and write your own Newton solver. If you just need one solution which is close to a given star …

As the comments notice, you may have some confusion in your head between covariance and sample covariance. However, that's not what causes your error. First of all, forget about getting the covarianc …

Those proposed tolerances look fine, but in my (opinionated) view this is really a problem with no satisfying solution, as the comments also argue. For most algorithms, the error bounds one gets look …

They are commonly called block matrices. You can create them with hstack, vstack, and block.

Are you sure the implementations of the function that you wish to differentiate return the same results both in Julia and Python? That seems the first place to look for bugs. …

Algorithms for the SVD, as more or less every classical linear algebra algorithm based on orthogonal transformations, are normwise backward stable, i.e., it should be guaranteed that $\frac{\|USV^* - …

If you are familiar with einsum, maybe this explanation does it: axes[0] and axes[1] specify the locations of the repeated letters in the parameters of einsum. For instance, np.tensordot(a, b, axes= …

This will get technical, though: you will need to call Lapack functions by hand, I am afraid (*potrf and *potrs), since Python doesn't help you here, so to use the exact same algorithm you may want to …

scipy.linalg.solve, in its newer versions, has a parameter assume_a that can be used to specify that the matrix $A$ is symmetric or positive definite; in these cases, LDL or Cholesky are used rather t …

(in Python notation). There are orthogonal transformations thrown in between these sums that make it difficult to use different strategies. …

output[i*n + j][k*n + l] = com[k][l] That's your mistake I think -- reversed indices. To compute the matrix $M$ associated to a linear operator $f$ (the way it's usually taught in a linear algebra c …