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# Integral Error Mathematica

## Contents

Hence an adaptive strategy has these components [MalcSimp75]: (i) an integration rule to compute the integral and error estimates over a region; (ii) a method for deciding which elements of a In[14]:= Out[14]= Here is the Cauchy principal value of . Mathematica Stack Exchange works best with JavaScript enabled current community blog chat Mathematica Mathematica Meta your communities Sign up or log in to customize your list. Ignoring the singularity may not work with oscillating integrands. http://offsethq.com/error-function/integral-of-the-error.html

at this answer How to numerically integrate this integral where you can find an automatic error estimate and two recipes how to deal with this problematic issue. –Artes Feb 19 '15 M.♦ Oct 12 '12 at 8:32 1 @fpghost You can include the information you gave in the comment into your question. I'm sorry for being unclear. The double exponential quadrature was proposed by Mori and Takahasi in 1974 and it was inspired by the so-called IMT rule and TANH rule.

## Complementary Error Function

Browse other questions tagged calculus-and-analysis or ask your own question. If is the dimension of integration and , then Duffy's coordinates is a suitable technique for singularities of the following type (see again [Duffy82]): 1. , , ; 2. , , The strategy "GlobalAdaptive" keeps track of the number of times the total error estimate has not decreased after the bisection of the region with the largest error estimate. Here is an example of a switch from "DoubleExponential" to IMT singularity handling.

In[1]:= Here are the points at which the integrand has been evaluated. there are no numbers of the specified working precision between region's boundaries; 2. Adaptive strategies differ by the way they partition the integration region into disjoint subregions. Error Function Matlab Contact the MathWorld Team © 1999-2016 Wolfram Research, Inc. | Terms of Use THINGS TO TRY: erf error function (1) error function (0) WolframAlpha.com WolframCloud.com All Sites & Public Resources...

For theoretical justification and practical recommendations of ignoring the singularity see [DavRab65IS] and [DavRab84]. Error Function Calculator NIntegrate is very robust (see my remark). The patterns of the sampling points over show the change from Gaussian quadrature () to double exponential quadrature (), which later is replaced by Gaussian quadrature using the IMT variable transformation The function NIntegrateProfile defined earlier is used in the following examples.

In[228]:= Out[228]= Convergence Rate This section demonstrates that the asymptotic error of the double exponential quadrature in terms of the number of evaluation points used is where is a positive constant. Error Function Excel It is known that for an integral like (14) of an analytic integrand the trapezoidal rule is an optimal rule [Mori74]. How do you prove that mirrors aren't parallel universes? In[124]:= The result can be checked using NIntegrate with singular range specification with the zeros of BesselJ[2,x] (see BesselJZero).

1. How do I catch a Ditto?
2. The trapezoidal rule is applied to (15): The terms in (16) decay double exponentially for large enough .
3. For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990).
4. There's "DoubleExponential" or "ClenshawCurtisOscillatoryRule" which you could have used...
5. Trends Internet of Things High-Performance Computing Hackathons All Solutions » Support & Learning Learning Wolfram Language Documentation Fast Introduction for Programmers Training Videos & Screencasts Wolfram Language Introductory Book Virtual
6. In these oscillating kernel forms, , , and are real constants, and is a positive integer.

## Error Function Calculator

The double exponential algorithm for NIntegrate is specified with the Method option value "DoubleExponential". The result would be 0 if everything were done exactly. Complementary Error Function Increasing the precision goal does not change the number of sampling points used by "DoubleExponential". Inverse Error Function Browse other questions tagged numerical-integration or ask your own question.

In[166]:= Out[168]= Double Exponential Strategy The double exponential quadrature consists of applying the trapezoidal rule after a variable transformation. http://offsethq.com/error-function/integral-over-error-function.html In[1]:= Out[2]= These are the sampling points used by "LocalAdaptive" integration, which partitions the regions with large error into three subregions. In[133]:= Out[134]= Timings for the integral with singularity handlers Automatic, "DuffyCoordinates", and IMT, and with no singularity handling. This loads a package that defines the profiling function NIntegrateProfile that gives the number of sampling points and the time needed by a numerical integration command. Error Function Table

Another factor is the ability of "LocalAdaptive" to reuse the integral values of all points already sampled. "GlobalAdaptive" has the ability to reuse very few integral values (at most 3 per Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the Then the integration over that subregion is done with the singularity handler specified by "SingularityHandler". http://offsethq.com/error-function/integral-de-error.html The patterns formed clearly show the three next recursion level subregions of each region of the first and second recursion levels.

The following example generates the NIntegrate::maxp message (obviously this very integral has an analytical solution): NIntegrate[Sin[x]/Sqrt[x], {x, 0, 100}, Method -> "MonteCarlo",PrecisionGoal -> 6] NIntegrate::maxp: The integral failed to converge after Error Function Python Double exponential quadrature and Gaussian quadrature for . The error estimate of a region, regionError, is considered insignificant if The stopping criteria (3) will compute the integral to the working precision.

## Now I want to estimate the error bound of the integration.

Products & Services Mathematica Mathematica Online Development Platform Programming Lab Data Science Platform Finance Platform SystemModeler Enterprise Private Cloud Enterprise Mathematica Wolfram|Alpha Appliance Enterprise Solutions Corporate Consulting Technical Services Wolfram|Alpha Business Wolfram Engine Software engine implementing the Wolfram Language. In[20]:= Out[20]= The number of integrand evaluations and timings for an integrand that has a singularity at a corner of the integration region. Erf(1) In[21]:= Out[21]= Local Adaptive Strategy In order to reach the required precision and accuracy goals of the integral estimate, a local adaptive strategy recursively partitions the subregion into smaller disjoint subregions

As explained earlier, this region will not be partitioned further as long as the "DoubleExponential" singularity handler works over it. and Oldham, K.B. "The Error Function and Its Complement ." Ch.40 in An Atlas of Functions. The strategy also stops when the number of recursive bisections of a region exceeds a certain number (see "MinRecursion and MaxRecursion"), or when the global integration error oscillates too much (see http://offsethq.com/error-function/integral-of-error.html For example, consider the sampling points and weights of the Gauss-Kronrod rule.

By guessing an error bound equal to the precision goal we my be massively overestimating the actual error. (Which is not good if the result is to be used in a The following example uses all the different specialized oscillatory strategies on different subregions. For multidimensional integrals "GlobalAdaptive" is much faster because "LocalAdaptive" does partitioning along each axis, so the number of regions can explode combinatorically. How long does it take for a fact to become a legend? (In a medieval setting) Help me I'm lost in the ocean!