Numerical algorithms for problems in analysis and algebra, scientific computation

**106**

votes

**2**answers

10k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**7**

votes

**4**answers

1k views

### Is there a name for the matrix equation A X B + B X A + C X C = D?

I happen to be working on a problem that reduces to solving the following equation:
$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$
where A through D are known matrices ( A, B, D ...

**6**

votes

**2**answers

3k views

### Computational complexity of calculating the nth root of a real number

Several sources state that the computational or time complexity of square rooting is the same as that of multiplication (or division). See for example:
Jean-Michel Muller, "Elementary Functions: ...

**12**

votes

**7**answers

4k views

### Numerical integration over 2D disk

I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...

**6**

votes

**1**answer

856 views

### Approximately Invert x^x

What is the best asymptotic approximation of the inverse $x=g(y)$ of $y = x^x$ for large $x$? [Clearly, if $x>e$, then $f(x) > e^x$ implies $g(x) < \log x$.]

**3**

votes

**1**answer

170 views

### Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?

**2**

votes

**0**answers

146 views

### Are there workable numerical approaches for the pentagon equation?

Warning: this post is the "numerical" analog of
Are there workable algebraic geometry approaches for the pentagon equation?
I've replaced "algebraic geometry" by "numerical" in its content,
...

**9**

votes

**3**answers

507 views

### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

**7**

votes

**1**answer

166 views

### Compute only selected components of an eigenvector

I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...

**6**

votes

**3**answers

898 views

### Dependence of error on mesh for Riemann sums

Suppose $f$ is continuous on $[a,b]$ with $I = \int_a^b f(x)\: dx$,
and for every $\epsilon > 0$ let $\delta(\epsilon)$ be the largest
$\delta > 0$ such that every Riemann sum arising from a ...

**5**

votes

**0**answers

104 views

### Error of midpoint method for differentiable functions

Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$?
...

**5**

votes

**2**answers

798 views

### error estimates for multi-dimensional Riemann sums

Suppose that $f$ is a continuous function of bounded variation from $R^2$ to $R$ that's negative outside of some bounded set, and let $F=\max(f,0)$. Let $S_n$ be the Riemann sum for the integral of ...

**3**

votes

**0**answers

883 views

### Eigenvalue Problems with Linear Constraints

The motivation for this problem comes from trying to develop a simple way to decompose domains into non-overlapping subdomains to solve for the eigenvalues of some differential operator. The idea is ...

**0**

votes

**1**answer

164 views

### Regular Perturbation Series soln to eqn

I want to find the a 3 term perturbation soln of
(i) $(1+x)^3 = ex$ where $e\ll1$
Direct substitution of the regular perturbation series $x = x_0 + ex_1 + e^2x_2$
into (i) does not work
I ...

**4**

votes

**1**answer

83 views

### The link and equivalence between variant definition of computation model and computational complexity over reals

To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational ...

**3**

votes

**1**answer

151 views

### Approximate the square root of (1-X) efficiently through (nested) products

Currently, I encountered a problem of approximating the following
series:
$$
(I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots
$$
where ...

**3**

votes

**1**answer

211 views

### Error of midpoint method for functions that are not twice-differentiable

All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not ...

**2**

votes

**1**answer

208 views

### Literature on root finding of convex Functions

I am interested in using a result about Newton's method, which basically states that if f is convex on $[a,b]$ and it holds $f(a)<0$ and $f(b)>0$, then the Newton iteration converges to ...

**2**

votes

**2**answers

2k views

### Finding the formula for Bezier curve ratios (hull/point : point/baseline)

Given a cubic Bezier curve defined by points p₁, p₂, p₃, and p₄, a point B on that curve at some t value (where 0 ≤ t ≤ 1), a point A on the line (p₂ — p₃) at distance ratio t from p₂, and a point C ...

**2**

votes

**1**answer

88 views

### Is is preferable to use a difference formula of higher order of accuracy for spatial derivatives to solve this IVP problem ?

I want to numerically integrate the equation $\partial_t u= a(t) \partial_xu+b\partial_{xxx}u+c$ to get $u(t)$. Is is preferable to use a difference formula of higher order of accuracy for spatial ...

**0**

votes

**0**answers

166 views

### Solving a parabolic PDE with boundary conditions given over ranges

How can one solve a Parabolic PDE (like the wave or diffusion equations) if the boundary conditions were given over ranges?
Here is an example: How to solve the equation ...