# Questions tagged [numerical-algorithms]

Questions related to algorithms that use numerical approximations for the problems of mathematical analysis.

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### Implementing multidimensional integral for a custom function in C++ [closed]

I am not an expert with C++, but I am trying to implement a 4-dimensional integral using GSL numerical integration approach. The code below shows the whole algorithm. Although it seems correct what I ...
• 111
1 vote
41 views

### Algorithm to calculate lower incomplete gamma function

I'm trying to understand the implementation of the algoritm here. Please see GammaLowerRegularized(a, x) function. I understand 1st part of the function for ...
• 131
79 views

### Factor a number in the longest possible product of distinct numbers

I got stuck with quite a simple problem: Given a positive number $X$ find the largest number $k$, for which exists the positive distinct integers $Y_1,…,Y_k$ such that $(Y_1+1)(Y_2+1)⋯(Y_k+1)=X$ Any ...
• 121
1 vote
27 views

### What does it mean unambiguously that a number is value 0 up to numerical precision?

I was reading that a quantity $x$ is $0$ upt to numerical precision. What does this statement formally mean -- especially in the context of numerical methods or real computers. I looked up in google ...
• 2,910
56 views

### Bisecting Intervals of floating point numbers containing 0 and infinity fairly

It is seldom considered that floating points are not evenly distributed in the real number line. I've been working with interval arithmetic and noticed when bisecting $[a,b]$ on the real number line ...
34 views

### What is the most performant and reliable algorithm to find all roots of a nonlinear function within a range?

Following this question on MATLAB's Discord channel, I did a bit of search and found this interesting answer using circshift() function to detect zero-crossings. I ...
• 191
47 views

### Computing a series of matrix power - matrix products

Assuming we have two dense matrices $A \in \mathbb{R}^{m\times m}, B \in \mathbb{R}^{m\times n}$, is there a smart way to compute all entries of the series $A^1 B, A^2 B, A^3 B, \dots, A^k B$ up to ...
• 31
1 vote
52 views

### How many Integers can be represent in Double-Precision floating-point form

How to calculate the number of Integers that can be represent in Double-Precision floating-point form?
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1 vote
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### Given constants $c_i$ and $K$, find $b$ numerically such that $b^{c_1} + b^{c_2} + ... + b^{c_n} = K$

Given constants $c_i > 0$ and $K > 0$, find $b > 0$ numerically such that $b^{c_1} + b^{c_2} + \dots + b^{c_n} = K$. I'd like to solve this with a non-iterative method if possible. My attempt ...
69 views

### Efficient Algorithm to Find the Closest Integer Representation, in the Form $A\times\frac{N}{D}$ for a Value

The Problem I am working on a problem that boils down to finding the closest representation of an arbitrary number ($x$) in the form: $$x = A\times\frac{N}{D}$$ Where $A$ is a 32-bit integer, and $N$ ...
• 123
82 views

### Precise algorithm for finding higher order derivatives

I'm trying to make an algorithm that finds the first 10 or so terms of a function's Taylor series, which requires finding the nth derivative of the function for the nth term. It's easy to implement ...
• 35
1 vote
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### Finding points of local maximum error in Remez algorithm

So the Remez algorithm is an algorithm for finding optimal polynomial approximations or at the very least for converging towards them. To find an approximation of $f$ by an $N$th degree polynomial the ...
• 3,728
1 vote
47 views

### How to use Runge–Kutta methods in a second order ODE

Consider a second order equation $F=ma=m\ddot{x}$. In the language of Euler's method $\ddot{x}(t+dt)=F(t,x(t),\dot x(t))$ $\dot{x}(t+dt)=\dot x(t)+\ddot x(t)dt$ $x(t+dt)=x(t)+\dot x(t)dt$ Basically, ...
1 vote
15 views

### Numerically solving an ode with infinitely many variables of which only finitely many are significant in magnitude

Suppose I have an ode that involves infinitely many variables, with the property that at any given time, only finitely many of them are large enough to be of interest (say $>10^{-10}$). However, at ...
• 171
53 views

### fastest way to identify "singular row" of a matrix

Suppose I have a matrix that I know to be singular. This means that there is at least one row in the matrix which is a linear combination of the other rows. What is the fastest way to identify which ...
• 125
193 views

### Complexity of inverting a diagonal matrix

What is the complexity of inverting a $n \times n$ diagonal matrix? From what I learn in algebra, the inverse of a diagonal matrix is obtained by replacing each element in the diagonal with its ...
• 21
730 views

### What algorithm do computers use to compute the square root of a number?

What algorithm do computers use to compute the square root of a number ? EDIT It seems there is a similar question here: Finding square root without division and initial guess But I like the answers ...
• 119
54 views

### How to express a parabola as a Bezier curve

I need to create a parabolic shape with a given starting point, maximum point, and end point. I thought the most efficient way to do this is to use a bezier curve. However there is no major ...
1 vote
104 views

### Adaptive step size constrained to a limited number of iterations

I'm solving a differential equation on the form $\ddot x = f( \dot x, x)$ on a microchip within a limited (real world) time frame, hence I want to use an adaptive step size to get as good of a result ...
121 views

### How can I compute logarithm when comparison is undecidable?

In Haskell, I have the following datatypes that encodes arbitrary real numbers and arbitrary complex numbers, respectively: ...
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