Questions on the numerical functions of Mathematica, implementing numerical methods and numerical computing with Mathematica.
31
votes
7answers
896 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 ...
26
votes
6answers
2k 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 ...
10
votes
1answer
279 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:
...
16
votes
5answers
456 views
Is this the most efficient way to round approximate integers to integers while leaving other Reals untouched?
This might seem like an overly simple question, but I need to specify custom plot tick marks as integers (no trailing decimal point) if they are approximately integers, but not if they are not. Using ...
8
votes
1answer
500 views
Combined numerical minimization and maximization
I want to numerically calculate the maximum of a function defined by the minimization of another function, like the following:
...
33
votes
1answer
884 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 ...
3
votes
1answer
333 views
Symbolic integration in the complex plane
Context
While answering this question, I defined (symbolic and numerical) path integrations as follows
...
19
votes
2answers
475 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 ...
14
votes
5answers
768 views
The difference between 0. and 0
I have a function for which 0 is a special case:
f[A___, 0, B___] := 0
But since I am doing numerics, sometimes in the course ...
10
votes
6answers
2k 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 ...
13
votes
3answers
299 views
Make mathematica treat $e_i^2$ as numeric
With NumericQ[symbol] = True, I can declare that a symbol is numeric. I want the expressions matching: $$e_{\text{i$\_$}?\text{IntegerQ}}^2$$ to be treated as ...
7
votes
2answers
249 views
Precision differences
I run this sum and get the symbolic answer below :
Sum[ (1/(k^2 - k) - 1/k^2), {k, 2, Infinity}]
$2 - \frac{\pi^2}{6}$
I look up the sequence on OEIS and ...
8
votes
1answer
885 views
Kramers-Kronig in Mathematica
I am trying to calculate the change of the refractive index from the change of the absorption coefficient using the Kramers-Kronig relations, in Mathematica.
...
7
votes
2answers
306 views
How to apply restrictions to the “integrated” variable, when using NDSolve?
I have to integrate an energy along a path. I know the energy at the "beginning" of the path (energy[0]), and I can determine the energy change (gain and loss) ...
13
votes
3answers
1k 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 ...
8
votes
1answer
380 views
How do I find all the solutions of three simultaneous equations within a given box?
Sometimes, one needs to find all the solutions of three simultaneous nonlinear equations in three unknowns
$$\begin{align*}f(x,y,z)&=0\\g(x,y,z)&=0\\h(x,y,z)&=0\end{align*}$$
within a ...
10
votes
2answers
350 views
Is it possible to use the LevenbergMarquardt algorithm for fitting a black-box residual function?
I have a black-box multiargument multiparametric function of the type SRD[dataPoint_List,params_List] which accepts experimental data along with the parameters of ...
12
votes
3answers
489 views
Strategies to avoid LessEqual::nord in NMinimize?
When using NMinimize on functions with complex intermediate expressions (but a real end result), quite often one gets the error ...
3
votes
1answer
611 views
How to discretize a nonlinear PDE fast?
I wish to numerically solve the following PDE. Although there are some complete discussions for solving PDEs in tutorial/NDSolvePDE, there is no hint for the nonlinear case by discretization. Thus, I ...
2
votes
3answers
596 views
How can I differentiate Numerically?
Mathematica has two ways to integrate: Integrate and NIntegrate.
But what about D? ...
9
votes
1answer
150 views
Converting to machine precision
There are multiple ways to convert an expression to machine precision, for example:
...
8
votes
1answer
234 views
Why is MainEvaluate being used when LinearSolve can be compiled?
According to this question LinearSolve can be Compiled. However, CompilePrint shows a MainEvaluate but no-warning is generated. It appears that LinearSolve is not ...
4
votes
1answer
155 views
Is there a way to globally set when to treat a very small number as zero?
I understand that I can use Chop to force a very small number to be treated as 0 and can use ...
10
votes
6answers
435 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
...
8
votes
1answer
234 views
ReplaceAll[] and Limit[] don't give correct results for this expression under extreme variables [duplicate]
Possible Duplicate:
Funny behaviour when plotting a polynomial of high degree and large coefficients
1/x^2 + (3 + x)/(6 (1 - Exp[x] + x))
——This is a ...
2
votes
2answers
139 views
Position function not always retuning an answer even with no apparent problems
I'm having some problems with Position.
Sometimes it will give an empty list instead of the actual position of the element I am looking for when that element is ...
1
vote
2answers
284 views
Running a Numerical Simulation on a System of Differential Equations with unique initial conditions
I have a system of differential equations as follows:
...
21
votes
3answers
373 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) such as:
\begin{equation}
...
21
votes
4answers
548 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 ...
12
votes
2answers
428 views
Higher order periodic interpolation (curve fitting)
I have a list of points in 3D, and I want to get a smooth interpolation or curve fit (it is more for illustration) of these points such that the first and second derivatives at the start and end ...
20
votes
1answer
464 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 ...
17
votes
0answers
635 views
Identifying critical points 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 ...
8
votes
1answer
455 views
Poisson solver using Mathematica
I am looking for some help with a Poisson solver I am writing in Mathematica. The code is quite long with Arrays plugged in, so the full details can be found at http://pastebin.com/uSrSDcW6
I am ...
5
votes
3answers
770 views
NDSolve with Euler method
I want to solve this equation with NDSolve[] using the Euler method:
x'[t] == 0.5*x[t]-0.04*(x[t])^2
with initial condition ...
4
votes
3answers
222 views
How to cancel floating point factors?
I am currently having problems with some floating points.
I have a function, which gives as an intermediate result (for example)
...
10
votes
2answers
364 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 ...
10
votes
4answers
691 views
Numerical underflow for a scaled error function
I calculate scaled error function defined as
f[x_] := Erfc[x]*Exp[x^2]
but it can not calculate f[30000.]. ...
9
votes
2answers
197 views
Determining the default Method used in optimization and root-finding algorithms
Is it possible to extract the Method which is used in functions like NMinimize, FindRoot, ...
7
votes
1answer
238 views
Symbolic Integration along contour: branch cut problem?
Context
Following this question on path integrals in the complex plane, having defined again a numerical and symbolic integrator along a path as
...
7
votes
1answer
193 views
Converting other C++ classes to MTensor in LibraryLink
Hopefully this will be a quick question + a quick answer:
Say I have a C++ (or C) code using LibraryLink. I am using a library that defines a specific matrix class, as many numerical libraries ...
5
votes
3answers
482 views
Solution for equation system with piece-wise defined functions
As I could swear this worked just yesterday, I am probably just doing something stupid here and I am sorry to bother you :)
I am trying to find the point where a curve crosses a line. In this case, ...
4
votes
1answer
193 views
How can I get Mathematica to allow me to apply FindRoot to an expression that contains NIntegrate?
I am trying to run the following command in Mathematica:
FindRoot[NIntegrate[D[f[x], x] / Sqrt[1 - x^2], {x, 0, 1}] - d, {a, 245}]
As you might expect, a is ...
0
votes
1answer
171 views
How do I numerically solve a custom function?
Whenever I use functions like FindRoot or NDSolve, it sends x through the function and deals ...
9
votes
1answer
940 views
What method does NDSolve use for solving PDEs?
What is NDSolve's mode of operation?
I use it to solve partial differential equations and never gave it too much thought. Recently, I came across this question. ...
4
votes
2answers
251 views
Can the CholeskyDecomposition function in Mathematica be made to work on non-symmetric matrices?
The CholeskyDecomposition[m] function in Mathematica requires a symmetric and positive definite matrix m.
For instance, the ...
3
votes
1answer
382 views
MaxSteps and Computing time issue for Solving Differential equation in Mathematica
When we solve differential equation numerically using NDSolve then sometimes we get error like NDSolve::mxst: Maximum steps reached
According to Mathematica docs ...
2
votes
1answer
127 views
Numerical comparisons of matrices
I have a matrix which should be equal to a null matrix. However due to the numerical precision, a brutal equality test with a matrix initialized with zeros does not work.
How should I perform the ...
1
vote
1answer
222 views
1
vote
1answer
175 views
Why is NDSolve solving in term of two 1st order ODE slower than 2nd order?
As mentioned in the documentation for NDSolve it's often convenient to reduce a 2nd order ODE to a system of first order equations. When I do this however I seem to see a significant speed reduction ...
1
vote
1answer
287 views
NDSolve runs out of memory
I need to solve a second order ODE numerically. The ODE depends on two parameters (a,b). Things work fine when 'a' is small, but for large 'a' the solutions are oscillating rapidly and Mathematica ...
