Questions on the use of numerical functions NIntegrate and NDSolve.
Questions on the use of numerical functions NIntegrate and NDSolve.
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Questions tagged [numerical-integration]
3496 questions
46
votes
2 answers
Nested NIntegrate
Suppose that we have the given simple integral expression
$$
\int_{-5}^{5} x \int_{-\infty}^{x} e^{\int_{0}^{z} -y dy} dz dx
$$
Writing this out in Mathematica we obtain:
Integrate[x Integrate[Exp[Integrate[-y, {y, 0, z}]], {z, -∞, x}], {x, -5.,…
jmlopez
- 6,470
- 4
- 28
- 48
38
votes
2 answers
Determining which rule NIntegrate selects automatically
I need to numerically integrate a highly oscillatory function over the semi-infinite domain $(0,\infty)$:
$$\int_0^\infty \frac{\sin^2(x) \sin^2(1000 x)}{x^{5/2}}\mathrm dx$$
Since the Levin rule (which was recently added to Mathematica, starting in…
user7885
- 381
- 4
- 3
16
votes
1 answer
Obtaining an NIntegrate error estimate
Is there a way to extract the error that Mathematica estimates when calculating a numerical integral using NIntegrate?
Internally Mathematica must keep track of this error, because it is used to determine if the PrecisionGoal has been met.
The…
TimRias
- 3,160
- 13
- 17
16
votes
5 answers
This integral is divergent. How to use NIntegrate to see how it grows?
I am trying to get information on the following integral:
$$
\int_{\pi-0.3}^{\pi-\epsilon} \frac{1}{(3+\cos{x})\sqrt{(3+\cos{x})^2-4}}
$$
The lower limit is somewhat arbitrary; the point is that this integral is known to be divergent if the…
Gelasio Salazar
- 163
- 7
12
votes
1 answer
Boosting the performance of expensive NIntegrate by feeding in a cheap approximation of the integrand
I need to integrate an expensive likelihood L[x] over its n-dimensional domain.
I know that L[x] is decently approximated by a gaussian likelihood G[x], which is very cheap to evaluate.
In particular, L and G share the same maximum.
The…
Valerio
- 1,982
- 13
- 23
12
votes
2 answers
Efficient evaluation of functions defined by NIntegrate
I have a complicated function $f$ and I want to plot the function $F(x)$ defined by the definite integral of $f$ from $0$ to $x$:
$$
F(x) = \int_0^x f(y)\mathrm dy.
$$
Apparently $f$ cannot be integrated in closed-form, and I use NIntegrate[]…
Kakashi Hatake
- 123
- 1
- 5
10
votes
1 answer
NDSolve break condition
I'm solving a differential equation numerically by
NDSolve[{p'[r] == -function[r,p[r]], p[0] == pcenter}, p,{r, 0, rmax}]
with function>0. At some r, p[r] becomes negative. I want NDSolve to stop as soon as this happens and save the value of r. Any…
dan-ros
- 415
- 3
- 8
10
votes
4 answers
NIntegrate doesn't evaluate
The integral
Integrate[(t^4 x^3)/Sqrt[-t + x], {x, 0, 1}, {t, 0, x}]
(* 512/5355 *)
can be solved analytically.
Trying to apply NIntegrate
NIntegrate[(t^4 x^3)/Sqrt[-t + x], {x, 0, 1}, {t, 0, x}]
fails (although with Method ->…
Ulrich Neumann
- 53,729
- 2
- 23
- 55
10
votes
2 answers
NIntegrate appears to give incorrect results
I am trying to specify a bivariate probability density function in Mathematica. As a check, I would like to confirm that it integrates to one. Here is the function:
f[x1_, x2_, u1_, u2_, v11_, v22_, v12_] := Det[2*Pi*{{v11, v12}, {v12,…
Miguel
- 201
- 1
- 5
9
votes
2 answers
Unify the sampling of NIntegrate[ {f, g, h} w ]
I'm trying to numerically integrate a function which has a vector-valued slow part and a much faster component which is shared by all the components, i.e. an integral of the form
$$
\int_a^b\begin{pmatrix}f(x)\\ g(x) \\ h(x)\end{pmatrix}w(x)\,\text…
Emilio Pisanty
- 10,255
- 1
- 36
- 69
9
votes
3 answers
NIntegrate does not converge for this integrand but Simpson's Rule does
I am trying to numerically integrate a function that flips sign at a certain location. In the vicinity of this location the function is approximately asymmetric so there are large negative and positive contributions to the integral which presumably…
kotozna
- 319
- 1
- 6
8
votes
3 answers
highly oscillatory integral
Hi am trying to compute the following integral in Mathematica:
NIntegrate[(Sign[Cos[q]] Sqrt[Abs[Cos[q]]])/(q+100),{q,0,Infinity}]
But when I run that command I get the following error:
NIntegrate::ncvb: NIntegrate failed to converge to…
Andrea
- 521
- 2
- 10
8
votes
3 answers
Certain integral over torus
Let $F(t_1,s_1,t_2,s_2)=$
$$\big((2+\cos t_1)\cos s_1 - (2+\cos t_2)\cos s_2\big)^2 + \big((2+\cos t_1)\sin s_1-(2+\cos t_2)\sin s_2\big)^2 + (\sin t_1 -\sin t_2 )^2.$$
I am interested in computing the following…
BigM
- 257
- 3
- 10
7
votes
1 answer
Strange Behaviour of NIntegrate
I found some of the values remained unevaluated using the following code
Table[NIntegrate[Sin[i x]/((2^x + 1) (Sin[x])), {x, -Pi/2, Pi/2}], {i,70, 90}]
Pick them out, the unevaluated ones remain unevaluated. For example,
NIntegrate[Sin[81 x]/((2^x…
vapor
- 7,911
- 2
- 22
- 55
7
votes
4 answers
Conditional numerical integration boundaries
I have a multidimensional integration of the form:
somefunc[t] = NIntegrate[ otherfunc[x, y, z, t ],
{z, z1[t], z2[t]}, {y, y1[t, z], y2[t, z]}, {x, x1[t, z, y], x2[t, z, y]}];
This needs to be evaluated only when $z2 > z1$, $y2 > y1$ and $x2 >…
Hsn
- 413
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