Search Results

"One-parameter family" simply means that one real parameter, $p$, appears in the definition. That method is simply a generalization of Newton's method exposed in the first equation; you can't derive i …

It is defined that way so that its maximum is exactly 1: $\|\phi_b(X)\|_{\infty}=1$.

To answer completely, one should know what is the application that you have in mind and if there is any specific convergence result for the Newton method. But, in general, I think that you are right: …

It can be unstable, and the condition number is not an adequate measure to tell when it fails. You may want to check the numerical experiments in http://www.jstor.org/stable/2153371 for specific examp …

I suspect that using the expm1 function would give you a better result. Computing $e^x -1$ with the trivial formula in machine precision gives you only limited accuracy for small inputs: the fundament …

I suggest you to take a look at Watkins' book "The Matrix Eigenvalue Problem - GR and Krylov subspace methods". It is very well written and will give you insight on the working of the QR method. I thi …

In addition to the convergence speed (and radius) mentioned in Pietro Majer's answer, there is another factor: if the problem is ill-conditioned, the solution is sensitive to perturbations. If you m …

One possibility is numerical rootfinding. A few iterations of Newton's method should get you home, and I imagine that for a third-degree polynomial one can construct (by sketching a plot of the functi …

If you're doing a full LU decomposition and ignoring sparsity, then you could switch to a Schur decomposition (costs $25n^3$ instead of $2/3n^3$, but allows you to solve any of the resulting systems w …

Newton's method? The derivative should be fairly straightforward to compute...

If you compute the (HO)SVD by solving the two eigenproblems, you always need to do some post-processing to sort the singular values and make them positive. I'd suggest moving the normalization issue …

Another good choice is the Aberth method http://en.wikipedia.org/wiki/Aberth_method. It has some advantages wrt the QR-based approaches, especially when the coefficients have very different magnitude …

Ill-conditioning isn't a concept that depends on the precision that you use to compute the solution. "A small change in the data turns into a large change of the solution" isn't a concept that involve …

There's a third option I'd consider: alternated iterations. Namely, call $N[p]$ the Newton iteration for a function $p(x)$, and solve $$ z_{k+1}=N[h](z_k) $$ $$ y_{k+1}=N[g(\cdot,z_{k+1})](y_k) $$ ` …

A variant of the argument in Carlo Beenakker's answer: if the $x_i$ are equispaced points with distance $h$ one from the next, then $$\frac{f(x_{i+1})-f(x_i)}{h} - f'(x_i) = O(h),$$ $$ \frac{f(x_{i+1} …