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Questions tagged [numerics]

Questions on the numerical functions of Mathematica, implementing numerical methods and numerical computing with Mathematica.

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110
votes
1answer
2k views

Why are numeric division and subtraction not handled better in Mathematica?

There is something that has been troubling me for a while. At least through version 10.0 the performance of a / b and a - b is ...
81
votes
4answers
37k views

Why round to even integers?

According to the Mathematica help: Round rounds numbers of the form x.5 toward the nearest even integer. For example: Round[{0.5, 1.5, 2.5, 3.5, 4.5}] gives ...
76
votes
9answers
5k views

Updating Wagon's FindAllCrossings2D[] function

Stan Wagon's Mathematica in Action (second edition; I haven't read the third edition and I'm hoping to eventually see it), demonstrates a nifty function called ...
72
votes
3answers
7k views

Numerically solving Helmholtz equation in 2D for arbitrary shapes

I would like to solve the Helmholtz equation with Dirichlet boundary conditions in two dimensions for an arbitrary shape (for a qualitative comparison of the eigenstates to periodic orbits in the ...
64
votes
3answers
3k views

When can I assume that all decimal digits returned by Mathematica are provably correct?

How to Control the Precision and Accuracy of Numerical Results Arbitrary-Precision Numbers Mathematica works with exact numbers and with two different types of approximate numbers: machine-...
59
votes
11answers
5k views

Can Mathematica propose an exact value based on an approximate one?

Sometimes, I use Mathematica to do some hypothesis on homework to make the question easier. For instance, when I have to compute big sums when $n\to\infty$ and Mathematica can't give the exact answer, ...
51
votes
1answer
3k views

Adaptive sampling for slow to compute functions in 2D

EDIT: Although I have posted an answer based on my current progress, this in incomplete. Please see the "open issues" section in the answer. Most plotting functions in Mathematica adjust the ...
47
votes
6answers
10k views

Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)

Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$. When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
46
votes
3answers
2k views

Is manual adjustment of AccuracyGoal and PrecisionGoal useless?

This is a problem confusing me for years. AccuracyGoal and PrecisionGoal are two options that I never truly understand and, to ...
44
votes
8answers
10k views

About multi-root search in Mathematica for transcendental equations

I have some questions for multiroot search for transcendental equations. Is there any clever solution to find all the roots for a transcendental equation in a specific range? Perhaps ...
42
votes
3answers
12k views

Understanding differences between Maple and Mathematica in examples picked by Maplesoft

I am reading the document How Maple Compares to Mathematica. On page 15 there is an example where Mathematica produces wrong results. Does anybody know why? MAPLE: MATHEMATICA: Also on page 17 the ...
41
votes
3answers
2k views

Funny behaviour when plotting a polynomial of high degree and large coefficients

I am trying to plot a polynomial of degree $29$ on the domain $[0,1]$, with fairly large coefficients: ...
41
votes
1answer
2k views

How to compare power towers in Mathematica?

First I tried it directly, but it overflowed: ...
40
votes
1answer
1k views

Is there a difference between Divide[a,b] and a/b?

In this comment it was asserted that Divide[a,b] and a/b are different, though the documentation indicates that they are the ...
40
votes
3answers
4k views

Identifying critical points/lines of 2/3D image/cubes

Upshot I am interested in identifying critical points of a 3D field/cubes (maxima, minima, tube-like and wall-like saddle points) and 2D field/image (maxima, minima, saddle points). I.e. the ...
40
votes
0answers
2k views

Fast Spherical Harmonics radiative transfer

This is a rather specific question and I apologize for spamming you with some lengthy code. But it could be interesting for some reader and maybe you can help out, so please bear with me. I am using ...
39
votes
1answer
2k views

How can I define a new symbolic constant like Pi?

There are a few builtin symbolic constants which behave like numbers, e.g. E, Pi, EulerGamma,...
39
votes
1answer
2k views

How to work with Experimental`NumericalFunction?

This question is intimately connected with previous one: "How to create internally optimized expression for computing with high WorkingPrecision?" Oleksandr R. correctly states in the comment: A ...
38
votes
2answers
3k views

Meaning of backtick in floating-point literal

If I compute, say, 1/3//N, Mathematica displays 0.333333 as the result. When I copy that output to use elsewhere, the paste ...
38
votes
2answers
1k views

Is it possible to make Mathematica reformulate an expression in a more numerically stable way?

I'm writing a numerical optimization, and I'm having a problem with an expression of the form $$ e^{-t} (1+\mathrm{erf}(t)) $$ The overall shape of the function looks correct, but when $t$ is small, $...
38
votes
1answer
1k views

Numerics with Mathematica

From time to time, I would like to use Mathematica purely numerically, e.g., plotting a function which is defined as an integral which cannot be solve analytically or a solution of a differential ...
37
votes
2answers
6k views

Efficient Langevin Equation Solver

This question is not about good algorithms for solving stochastic differential equations. It is about how to implement simple codes in Mathematica efficiently exploiting Mathematica's programming ...
36
votes
2answers
1k views

How to implement custom integration rules for use by NIntegrate?

How can NIntegrate be extended with custom implementation of integration rules? This answer of the question "Monte Carlo integration with random numbers generated ...
36
votes
1answer
4k views

AccuracyGoal, PrecisionGoal, WorkingPrecision and NDSolve

I'm trying to understand exactly what WorkingPrecision, AccuracyGoal and PrecisionGoal mean ...
35
votes
3answers
2k views

How to flush machine underflows to zero and prevent conversion to arbitrary precision?

I'm working on some pretty intense computation in Mathematica; when my code started running slowly, I tracked the source of the problem to Exp[]. I need to ...
34
votes
2answers
731 views

GroupBy twice gives different results

Bug introduced in 7.0 or earlier and persisting through 11.1 It took me quite a lot of time to finally trace down to this strange output. I really don't know why. First, I create a list ...
33
votes
3answers
7k views

Global precision setting

Coming from Maple I do not understand how the precision for numerical computations in Mathematica is specified. I understand that there are various options to commands such as ...
31
votes
4answers
2k views

How to use NDSolve to track equilibrium?

I am looking for an extension of NDSolve where integration runs until certain variables are settled at an equilibrium. Now I have a working solution in my sleeves ...
31
votes
3answers
2k views

Why is Plus so much slower than Total?

Why is Plus so much slower than Total for summing lists? ...
30
votes
8answers
1k views

Elegant high precision `log1p`?

Sometimes it is hard to understand how numerical expressions are evaluated. I remember reading claims by Wolfram on how smart the Kernel is to evaluate expressions trees numerically by recognizing ...
30
votes
2answers
6k views

How to fit 3 data sets to a model of 4 differential equations?

I'm a biologist and a newbie in Mathematica. I want to fit three data sets to a model consisting of four differential equations and 10 parameters. I want to find the parameters best fitting to my ...
30
votes
3answers
4k views

Numerical solution of coupled ODEs with boundary conditions

I have to solve the following set of ODEs and just can't get good results using Mathematica $$ r\frac{d}{dr}\left(\frac{1}{r}\frac{d}{dr}A(r)\right)-\xi^2F(r)^2\left(A(r)-1\right)=0 $$ $$ \frac{1}{r}\...
30
votes
0answers
577 views

Is MathieuC for moderately large imaginary arguments broken?

Bug introduced in 3.0 and persisting through 12.0 (reported as CASE:3208982) I'm trying to plot MathieuC[-3,0.3,I x] for $x\in[0,10]$, and here's what I get even ...
29
votes
2answers
916 views

Symbolic derivatives are being calculated numerically

Update: (1) By V11, not sure of the exact version, the derivative IntegerPart' has been given a symbolic definition. (2) The numeric derivative computed has changed ...
26
votes
5answers
10k views

How can I differentiate numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
26
votes
3answers
2k views

Computing polynomial eigenvalues in Mathematica

MATLAB offers a function polyeig for computing polynomial eigenvalues, which appear, for instance, in quadratic eigenvalue problems (see here for some applications) ...
26
votes
1answer
390 views

FindMinimum doesn't increase step size when necessary

(Cross-posted on Wolfram Community.) I've spent much time finding a minimal example demonstrating this problem with FindMinimum. Normally one faces this problem ...
25
votes
3answers
1k views

How to improve performance of BesselJ to the level of GSL?

Consider the following code: zs = N /@ Range[0, 12, 10^-5]; AbsoluteTiming[bessels = BesselJ[1, #] & /@ zs;] Length @ zs I've tried to measure only ...
25
votes
3answers
3k views

Can Mathematica Handle Open Intervals? Interval complements?

Open Intervals Following up on this question, I was wondering whether Mma can handle open intervals. For example, the union of the intervals, $$1<x<5$$ and $$5<x<8$$ should not ...
25
votes
3answers
954 views

Exponential fitting - isn't actually a BUG there?

There are many questions on this site about wrong exponential fitting in Mathematica but no one considers this well-known problem as a potential bug. Usually people suggest well-known workarounds ...
25
votes
1answer
2k views

How to determine BLAS/LAPACK implementation used internally for numerical matrix operations?

Is there a command which reveals which implementation of BLAS and LAPACK are used in Mathematica's matrix operations such as Eigensystem? I asked a related question ...
24
votes
6answers
4k views

Annoying display truncation of numerical results

I have a lot of data to inspect. An example of a number in my program is 123.189094 This gets displayed as 123.189 ...
24
votes
1answer
17k views

What do the X and Y axis stand for in the Fourier transform domain?

In Wolfram Mathematica the function Fourier has the following declaration Fourier[list] And after a list is given to the ...
24
votes
2answers
1k views

Obtain approximate Hessian using FindMinimum

According to the documentation, when FindMinimum is told to use the method "QuasiNewton" on a unconstrained problem, it uses the ...
24
votes
2answers
1k views

Different floating-point numbers equal?

Let's define two different numbers. x = 1. y = 1. + 2^-52 (* equivalently, 1 + $MachineEpsilon *) Let's make sure they're different with ...
24
votes
4answers
692 views

How does Internal`CompareNumeric work?

In this answer, Oleksandr R. mentioned an undocumented function Internal`CompareNumeric and briefly explained its usage as follows: ...
24
votes
1answer
726 views

Is there an NDSolve`ProcessEquations analog for NIntegrate?

NDSolve has an interface for repeatedly solving an equation with different initial conditions without having to analyze the equation and set up the solving ...
23
votes
3answers
6k views

Solving a Volterra integral equation numerically

I would like to solve for $P(t)$, in Mathematica, a Volterra integral equation of the 2nd kind. It is: $$P(t) = R_0(t) + \int_0^t P(t') R_0(t-t')dt'$$ I know the function $R_0$ and would ...
23
votes
2answers
3k views

Optimizing a Numerical Laplace Equation Solver

Laplace's Equation is an equation on a scalar in which, given the value of the scalar on the boundaries (the boundary conditions), one can determine the value of the scalar at any point in the region ...
23
votes
3answers
1k views

Real Numbers in the Wolfram Language

Epilog: Much of the discussion in the answers below revolves around the distinction between Real as a data type and real numbers as a domain or class of numbers (irrespective of the form in which ...