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0
votes
0answers
29 views

Most stable algorithm to solve a system of linear equations?

I am doing some image processing involving solving a system of linear equations. I am getting some errors and bits of the image looks corrupted. I would like to know what is the most stable way to ...
4
votes
2answers
91 views

“Compressing” rationals given error bounds

I'm working on implementing some exact real arithmetic operations for fun. I've got the rough outline of how I want to do things as well and have figured out how to write most of the important ...
3
votes
0answers
42 views

Testing whether an analytic function vanishes identically

I have an application that basically reduces to testing whether a given function vanishes identically. The function is given symbolically, using unary and binary operators on complex numbers. For ...
4
votes
1answer
79 views

A good-enough version of good-enough? When should I stop iterating?

I am studying Abelson and Sussman's Structure and Interpretation of Computer Programs, 2/e. I solved the exercise 1.7, which asks for a better stopping criteria for an iterative function to calculate ...
1
vote
0answers
23 views

Conservation Law in Finite difference Scheme [closed]

I am using a Finite Difference scheme to solve a simple PDE in conserved form: $$\partial_t u = \partial_x (\partial_x u +au\partial_x u) = (1+a)\partial_x^2u +a(\partial_x u)^2 $$ $$\frac{u_n+1,j - ...
-3
votes
1answer
83 views

Determine machine epsilon

Consider a base 2 computer that stores floating point numbers using a 6 bit normalized mantissa (x.xxxxx), a 4 digit exponent and a sign for each. a) For this machine, what is machine epsilon? b) ...
1
vote
2answers
90 views

Gap between numbers in fixed-point vs. floating point arithmetic

If $r$ is a machine-representable number and $f(r)$ is the next larger machine representable number, are the following true or false? In fixed-point arithmetic, the distance between $r$ and $f(r)$ ...
10
votes
5answers
2k views

Is 2**x faster to compute than exp(x)?

Forgive the naïveté that will be obvious in the way I ask this question as well as the fact that I'm asking it. Mathematicians typically use $\exp$ as it's the simplest/nicest base in ...
2
votes
1answer
322 views

Numerical stability of C++: will a C++ program using a float or double library return the same arithmetical results on two different computers?

Say I am using boost or the built-in float or double mathematical libraries of my C++ compiler. I distribute the program. Will the execution of my C++ program on different machines given different ...
0
votes
1answer
62 views

How to logarithmic interpolation? [closed]

I'm trying to interpolate a logarithmic function but it always reaches a singularity due to $\log(0)$ being $-\infty$ is there a correct way to interpolate logarithmic functions? (as in correct ...
2
votes
0answers
58 views

0 error interpolation for discrete finite value points

I am working on an algorithm that requires me to interpolate a couple trillion positive discrete points with f(x) having low finite value (for example 0 - 5). It there a specialized algorithm specific ...
0
votes
1answer
44 views

Can someone interpret what this is asking for

I have this programming problem, but I really cant figure out what it wants me to do. Heres what it is: The cube root of a number can be found based on the observation that, if $t$ is an ...
4
votes
1answer
263 views

Is “ternary search” an appropriate term for the algorithm that optimizes a unimodal function on a real interval?

Suppose that I want to optimize a unimodal function defined on some real interval. I can use the well-known algorithm as described in Wikipedia under the name of ternary search. In case of the ...
0
votes
0answers
54 views

Numerically approximating an inverse integral transform?

My question is about numerical methods for inverting integral transforms; I'm trying to numerically invert the following integral transform: $$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + ...
1
vote
0answers
79 views

Error accumulation in a numerical integration

I have this problem: Consider the problem of calculating the integral $$y_n =\int_{0}^{1} \dfrac{x^n}{x+10} \mathrm{d}x $$ for a positive integer $n$. Observe that $$y_n + 10y_{n-1} = ...
2
votes
0answers
116 views

Does Automatic Differentiation handle conditional branches, if yes how?

I'm trying to understand how Automatic Differentiation (AD) works. For simple algebraic operation, I get the chain rule thing. But, when the code contains conditional statement like ...
2
votes
1answer
105 views

What are the drawbacks of using an algorithm that is not backwards stable?

(This question might be legitimately crossposted to stackoverflow or mathoverflow or programming StackExchanges.) Preface I'm reading this paper on solving linear systems of equations ...
7
votes
2answers
89 views

numerical integral vs counting roots

I have a problem that can be viewed in two different ways: Compute an $n$-dimensional integral, numerical context. The domain of integration is an $n$-dimensional hyper-cube of side $L$. Count (just ...
13
votes
2answers
560 views

Computing inverse matrix when an element changes

Given an $n \times n$ matrix $\mathbf{A}$. Let the inverse matrix of $\mathbf{A}$ be $\mathbf{A}^{-1}$ (that is, $\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}$). Assume that one element in $\mathbf{A}$ is ...
0
votes
2answers
230 views

Implications of truncation of numbers when converted into binary

I have been posed with a question whereby a computer truncates numbers to x number of digits. Due to this, if this computer is trying to store a decimal number which has a binary equivalent greater ...
0
votes
1answer
431 views

In a 32-bit floating number with normalized mantissa and excess-64 exponent base 16, the number $16^{-65}$ denotes

In a 32-bit floating number with normalized mantissa and excess-64 exponent base 16, the number $16^{-65}$ denotes Floating point overflow. Negative floating point overflow. All 0's in the exponent ...
7
votes
0answers
342 views

Alternatives to SVD for rank factorization

I have rank-deficient matrix $M \in \mathbb{R}^{n\times m}$ with $\text{rank}(M) = k$ and I want to find a rank factorization $M = PQ$ with $P \in \mathbb{R}^{n \times k}$ and $Q \in \mathbb{R}^{k ...
5
votes
0answers
407 views

Arc-Length parameterization of a cubic bezier curve

I like to implement an arc-length Parameterization of a cubic bezier curve. So far I have implemented the method of calculating the arc length of the curve and now I'm stuck at calculating the times ...
11
votes
1answer
262 views

Floating point rounding

Can an IEEE-754 floating point number < 1 (i.e. generated with a random number generator which generates a number >= 0.0 and < 1.0) ever be multiplied by some integer (in floating point form) to ...
1
vote
1answer
164 views

Which method for ODE instead of Euler's?

I need a super-fast method for ordinary differential equations. Should I use the midpoint method? I need this for a reaction-diffusion system (Gray-Scott).
3
votes
1answer
114 views

An argument for error accumulation during complex DFT

I am doing FFT-based multiplication of polynomials with integer coefficients (long integers, in fact). The coefficients have a maximum value of $BASE-1, \quad BASE \in \mathbb{n},\quad BASE > 1$. ...
4
votes
1answer
353 views

Fast Poisson quantile computation

I am seeking a fast algorithm to compute the following function, a quantile of the Poisson distribution: $$f(n, \lambda) = e^{-\lambda} \sum_{k=0}^{n} \frac{\lambda^k}{k!} $$ I can think of an ...
3
votes
0answers
288 views

Cyclic coordinate method: how does it differ from Hook & Jeeves and Rosenbrock?

I have trouble understanding the cyclic coordinate method. How does it differ with the Hook and Jeeves method and the Rosenbrock method? From a past exam text: Describe the cyclic coordinate ...
7
votes
2answers
287 views

Detecting overflow in summation

Suppose I am given an array of $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$. I want to compute the sum $S = a_1 + \ldots + a_n$ on a machine with 2's ...
10
votes
1answer
193 views

Overflow safe summation

Suppose I am given $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$ such that their sum $a_1 + a_2 + \dots + a_n = S$ also fits in a register of width $w$. ...