NIntegrate Integration Rules
Introduction
An integration rule computes an estimate of an integral over a region using a weighted sum. In the context of
NIntegrate usage, an integration rule object provides both an integral estimate and an error estimate as a measure of the integral estimate's accuracy.
An integration rule samples the integrand with a set of points. These points are called sampling points. In the literature these are usually called abscissas. Corresponding to each sampling point
xi there is a weight number
wi. An integration rule estimates the integral
with the weighted sum
wif (xi). An integration rule is a functional, that is, it maps functions over the interval
[a, b] (or a more general region) into real numbers.
If a rule is applied over the region
V this will be denoted as
RV (f), where
f is the integrand.
The sampling points of the rules considered below are chosen to compute estimates for integrals either over the interval
[0, 1], or the unit cube
[0, 1]d, or the "centered" unit cube
, where
d is the dimension of the integral. So if
V is one of these regions,
R (f) will be used instead of
RV (f). When these rules are applied to other regions, their abscissas and estimates need to be scaled accordingly.
The integration rule
R is said to be exact for the function
f if
.
The application of an integration rule
R to a function
f will be referred as an integration of
f, for example, "when
f is integrated by
R, we get
R (f)."
A one-dimensional integration rule is said to be of degree
n if it integrates exactly all polynomials of degree
n or less, and will fail to do so for at least one polynomial of degree
n+1.
A multidimensional integration rule is said to be of degree
n if it integrates exactly all monomials of degree
n or less, and will fail to do so for at least one monomial of degree
n+1, that is, the rule is exact for all monomials of the form
, where
d is the dimension,
i≥ 0, and
i≤n.
A null rule of degree
m will integrate to zero all monomials of degree
≤ m and will fail to do so for at least one monomial of degree
m+1. Each null rule may be thought of as the difference between a basic integration rule and an appropriate integration rule of a lower degree.
If the set of sampling points of a rule
R1 of degree
n, contains the set of sampling points of a rule
R2 of a lower degree
m, that is,
n>m, then
R2 is said to be embedded in
R1. This will be denoted as
R2
R1.
An integration rule of degree
n that is a member of a family of rules with a common derivation and properties, but different degrees will be denoted as
R (f, n), where
R might be chosen to identify the family. (For example, trapezoidal rule of degree 4 might be referred to as
T (f, 4).)
If each rule in a family is embedded in another rule in the same family, then the rules of that family are called progressive. (For any given
m
there exists
n, n>m, for which
R (f, m)R (f, n)).
An integration rule is of open type if the integrand is not evaluated at the end points of the interval. It is of closed type if it uses integrand evaluations at the interval end points.
An
NIntegrate integration rule object has one integration rule for the integral estimate and one or several null rules for the error estimate. The sampling points of the integration rule and the null rules coincide. It should be clear from the context whether "integration rule" or "rule" would mean an
NIntegrate integration rule object, or an integration rule in the usual mathematical sense.
Integration Rule Specification
All integration rules described below, except
"MonteCarloRule", are to be used by the
adaptive strategies of
NIntegrate. In
NIntegrate, all Monte Carlo strategies,
crude and
adaptive, use
"MonteCarloRule". Changing the integration rule component of an integration strategy will make a different integration algorithm.
The way to specify what integration rule the adaptive strategies in
NIntegrate (see
"Global Adaptive Strategy" and
"Local Adaptive Strategy") should use is through a
Method suboption.
Here is an example of using an integration rule with a strategy ( "GlobalAdaptive").
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Here is an example of using the same integration rule as in the example above through a different strategy ( "LocalAdaptive").
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If
NIntegrate is given a method option that has only an integration rule specification other than
"MonteCarloRule", then that rule is used with the
"GlobalAdaptive" strategy. The two inputs below are equivalent.
For this integration only integration rule is specified.
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For this integration an integration strategy and an integration rule are specified.
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Similarly for
"MonteCarloRule", the adaptive Monte Carlo strategy is going to be used when the following two equivalent commands are executed.
For this Monte Carlo integration only the "MonteCarloRule" is specified.
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For this Monte Carlo integration a Monte Carlo integration strategy and "MonteCarloRule" are specified.
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"TrapezoidalRule"
The trapezoidal rule for integral estimation is one of the simplest and oldest rules (possibly used by the Babylonians and certainly by the ancient Greek mathematicians):
The compounded trapezoidal rule is a Riemann sum of the form
where
.
If the
Method option is given the value
"TrapezoidalRule", the compounded trapezoidal rule is used to estimate each subinterval formed by the integration strategy.
A "TrapezoidalRule" integration:
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| "Points" | 5 | number of coarse trapezoidal points |
| "RombergQuadrature" | True | should Romberg quadrature be used or not |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic preprocessing |
"TrapezoidalRule" options.
The trapezoidal rule and its compounded (multipanel) extension are not very accurate. (The compounded trapezoidal rule is exact for linear functions and converges at least as fast as
n-2, if the integrand has continuous second derivative [
DavRab84].) The accuracy of the multipanel trapezoidal rule can be increased using the "
Romberg quadrature".
Since the abscissas of
T (f, n) are a subset of
T (f, 2n-1), the difference
|T (f, 2 n-1)-T (f, n)|, can be taken to be an error estimate of the integral estimate
T (f, 2n-1), and can be computed without extra integrand evaluations.
The option
"Points"->k can be used to specify how many coarse points are used. The total number of points used by
"TrapezoidalRule" is
2k-1.
This verifies that the sampling points are as in ( 1).
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"TrapezoidalRule" can be used for multidimensional integrals too.
Here is a multidimensional integration with "TrapezoidalRule". The exact result is  .
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Remark: NIntegrate has both a trapezoidal rule and a trapezoidal strategy; see
"Trapezoidal" Strategy in the tutorial
Integration Strategies. All internally implemented integration rules of
NIntegrate have the suffix -
Rule. So
"TrapezoidalRule" is used to specify the trapezoidal integration rule, and
"Trapezoidal" is used to specify the trapezoidal strategy.
Romberg Quadrature
The idea of the Romberg quadrature is to use a linear combination of
T (f, n) and
T (f, 2 n-1) that eliminates the same order terms of truncation approximation errors of
T (f, n) and
T (f, 2 n-1).
From the Euler-Maclaurin formula [
DavRab84] we have
The
h2 terms of the equations above can be eliminated if the first equation is subtracted from the second equation four times. The result is
This example shows that a trapezoidal rule using the Romberg quadrature gives better performance than the standard trapezoidal rule. Also, the result of the former is closer to the exact result,  .
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Here is an integration with a trapezoidal rule that does not use Romberg quadrature.
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"TrapezoidalRule" Sampling Points and Weights
The following calculates the trapezoidal sampling points, weights, and error weights for a given precision.
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Here is how the Romberg quadrature weights and error weights can be derived.
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"NewtonCotesRule"
Newton-Cotes integration formulas are formulas of interpolatory type with sampling points that are equally spaced.
The Newton-Cotes quadrature for NIntegrate can be specified with the Method option value "NewtonCotesRule".
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| "Points" | 3 | number of coarse Newton-Cotes points |
| "Type" | Closed | type of the Newton-Cotes rule |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic preprocessing |
"NewtonCotesRule" options.
Let the interval of integration,
[a, b], be divided into
n-1 subintervals of equal length by the points
Then the integration formula of interpolatory type is given by
When
n is large, the Newton-Cotes
n-point coefficients are large and are of mixed sign.
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Since this may lead to large losses of significance by cancellation, a high-order Newton-Cotes rule must be used with caution.
"NewtonCotesRule" Sampling Points and Weights
The following calculates the Newton-Cotes sampling points, weights, and error weights for a given precision.
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"GaussBerntsenEspelidRule"
Gaussian quadrature uses optimal sampling points (through polynomial interpolation) to form a weighted sum of the integrand values over these points. On a subset of these sampling points a lower order quadrature rule can be made. The difference between the two rules can be used to estimate the error. Berntsen and Espelid derived error estimation rules by removing the central point of Gaussian rules with odd number of sampling points.
The Gaussian quadrature for NIntegrate can be specified with the Method option value "GaussBerntsenEspelidRule".
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| "Points" | Automatic | number of Gauss points |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic preprocessing |
"GaussBerntsenEspelidRule" options.
A Gaussian rule
G (f, n) of
n points for integrand
f is exact for polynomials of degree
2n-1 (i.e.,
if
f (x) is a polynomial of degree
≤2n-1).
Gaussian rules are of open type since the integrand is not evaluated at the end points of the interval. (
Lobatto rules,
Clenshaw-Curtis rules, and the
trapezoidal rule are of closed type since they use integrand evaluations at the interval end points.)
This defines the divided differences functional [
Ehrich2000]
For the Gaussian rule
G (f, 2n+1), with sampling points
x1, x2, ..., x2n+1, Berntsen and Espelid have derived the following error estimate functional (see [
Ehrich2000])
(The original formula in [
Ehrich2000] is for sampling points in
[-1, 1]. The formula above is for sampling points in
[0, 1].)
This example shows the number of sampling points used by NIntegrate with various values of "GaussBerntsenEspelidRule"'s option "Points".
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"GaussBerntsenEspelidRule" Sampling Points and Weights
The following calculates the Gaussian abscissas, weights, and Bernsen-Espelid error weights for a given number of coarse points and precision.
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The Berntsen-Espelid error weights are implemented below.
This implements the divided differences. |
This computes the abscissas and the weights of G (f, 2n+1). |
This computes the Berntsen-Espelid error weights.
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"GaussKronrodRule"
Gaussian quadrature uses optimal sampling points (through polynomial interpolation) to form a weighted sum of the integrand values over these points. The Kronrod extension of a Gaussian rule adds new sampling points in between the Gaussian points and forms a higher-order rule that reuses the Gaussian rule integrand evaluations.
The Gauss-Kronrod quadrature for NIntegrate can be specified with the Method option value "GaussKronrodRule".
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| "Points" | Automatic | number of Gauss points that will be extended with Kronrod points |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic processing |
"GaussKronrodRule" options.
A Gaussian rule
G (f, n) of
n points for integrand
f is exact for polynomials of degree
2n-1, that is,
if
f (x) is a polynomial of degree
≤2n-1.
Gauss-Kronrod rules are of open type since the integrand is not evaluated at the end points of the interval.
The Kronrod extension
GK (f, n) of a Gaussian rule with
n points
G (f, n) adds
n+1 points to
G (f, n) and the extended rule is exact for polynomials of degree
3n+1 if
n is even, or
3n+2 if
n is odd. The weights associated with a Gaussian rule change in its Kronrod extension.
Since the abscissas of
G (f, n) are a subset of
GK (f, n), the difference
GK (f, n)-G (f, n)
can be taken to be an error estimate of the integral estimate
GK (f, n), and can be computed without extra integrand evaluations.
This example shows the number of sampling points used by NIntegrate with various values of "GaussKronrodRule"'s option "Points".
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For an implementation description of Kronrod extensions of Gaussian rules, see [
PiesBrand74].
"GaussKronrodRule" Sampling Points and Weights
The following calculates the Gauss-Kronrod abscissas, weights, and error weights for a given number of coarse points and precision.
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The calculations below demonstrate the degree of the Gauss-Kronrod integration rule (see
above).
This computes the degree of the Gauss-Kronrod integration rule.
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The command below implements the integration rule weighted sums for the integral estimate,
wi f (xi), and the error estimate,
ei f (xi), where
are the abscissas,
are the weights, and
are the error weights.
These are the integral and error estimates for  computed with the rule.
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The integral estimate coincides with the exact result.
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The error estimate is not zero since the embedded Gauss rule is exact for polynomials of degree
≤2 n-1. If we integrate a polynomial of that degree, the error estimate becomes zero.
These are the integral and error estimates for  computed with the rule.
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"LobattoKronrodRule"
The Lobatto integration rule is a Gauss type rule with preassigned abscissas. It uses the end points of the integration interval and optimal sampling points inside the interval to form a weighted sum of the integrand values over these points. The Kronrod extension of a Lobatto rule adds new sampling points in between the Lobatto rule points and forms a higher-order rule that reuses the Lobatto rule integrand evaluations.
NIntegrate uses the Kronrod extension of the Lobatto rule if the Method option is given the value "LobattoKronrodRule".
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| "Points" | 5 | number of Gauss-Lobatto points that will be extended with Kronrod points |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic preprocessing |
"LobattoKronrodRule" options.
A Lobatto rule
L (f, n) of
n points for integrand
f is exact for polynomials of degree
2n-3, (i.e.,
if
f (x) is a polynomial of degree
≤2n-3).
The Kronrod extension
LK (f, n) of a Lobatto rule with
n points
L (f, n) adds
n-1 points to
L (f, n) and the extended rule is exact for polynomials of degree
3n-2 if
n is even, or
3n-3 if
n is odd. The weights associated with a Lobatto rule change in its Kronrod extension.
As with
"GaussKronrodRule", the number of Gauss points is specified with the option
"GaussPoints". If
"LobattoKronrodRule" is invoked with
"Points"->n, the total number of rule points will be
2 n -1.
This example shows the number of sampling points used by NIntegrate with various values of "LobattoKronrodRule"'s option "Points".
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Since the Lobatto rule is a closed rule, the integrand needs to be evaluated at the end points of the interval. If there is a singularity at these end points,
NIntegrate will ignore it.
For an implementation description of Kronrod extensions of Lobatto rules, see [
PiesBrand74].
"LobattoKronrodRule" Sampling Points and Weights
The following calculates the Lobatto-Kronrod abscissas, weights, and error weights for a given number of coarse points and precision.
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The calculations below demonstrate the degree of the Lobatto-Kronrod integration rule (see
above).
This computes the degree of the Lobatto-Kronrod integration rule.
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The command below implements the integration rule weighted sums for the integral estimate,
wi f (xi), and the error estimate,
ei f (xi), where
are the abscissas,
are the weights, and
are the error weights.
These are the integral and error estimates for  computed with the rule.
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The preceding integral estimate coincides with the exact result.
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The preceding error estimate is not zero since the embedded Lobatto rule is exact for polynomials of degree
≤2 n-3. If we integrate a polynomial of that degree, the error estimate becomes zero.
These are the integral and error estimates for  computed with the rule.
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"ClenshawCurtisRule"
A Clenshaw-Curtis rule uses sampling points derived from the Chebyshev polynomial approximation of the integrand.
The Clenshaw-Curtis quadrature for NIntegrate can specified with the Method option value "ClenshawCurtisRule".
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| "Points" | 5 | number of coarse Clenshaw-Curtis points |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic processing |
"ClenshawCurtisRule" options.
Theoretically a Clenshaw-Curtis rule with
n sampling points is exact for polynomials of degree
n or less. In practice, though, Clenshaw-Curtis rules achieve the accuracy of the
Gaussian rules [
Evans93][
OHaraSmith68]. The error of the Clenshaw-Curtis formula is analyzed in [
OHaraSmith68].
The sampling points of the classical Clenshaw-Curtis rule are zeros of Chebyshev polynomials. The sampling points of a practical Clenshaw-Curtis rule are chosen to be Chebyshev polynomial extremum points. The classical Clenshaw-Curtis rules are not progressive, but the practical Clenshaw-Curtis rules are [
DavRab84][
KrUeb98].
Let
PCC (f, n) denote a practical Clenshaw-Curtis rule of
n sampling points for the function
f.
The progressive property means that the sampling points of
PCC (f, n) are a subset of the sampling points of
PCC (f, 2 n-1). Hence the difference
|PCC (f, 2 n-1)-PCC (f, n) can be taken to be an error estimate of the integral estimate
PCC (f, 2 n-1), and can be computed without extra integrand evaluations.
The NIntegrate option Method->{"ClenshawCurtisRule", "Points"->k} uses a practical Clenshaw-Curtis rule with 2n-1 points PCC (f, 2 n-1).
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This example shows the number of sampling points used by NIntegrate with various values of "ClenshawCurtisRule"'s option "Points".
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"ClenshawCurtisRule" Sampling Points and Weights
Here are the sampling points and the weights of the Clenshaw-Curtis rule for a given coarse number of points and precision.
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Here is another way to compute the sampling points of PCC (f, 2 n-1).
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These are the integral and error estimates for  computed with the rule.
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"MultiPanelRule"
"MultiPanelRule" combines into one rule the applications of a one-dimensional integration rule over two or more adjacent intervals. An application of the original rule to any of the adjacent intervals is called a panel.
Here is an example of an integration with "MultiPanelRule".
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| Method | "NewtonCotesRule" | integration rule specification that provides the abscissas, weights, and error weights for a single panel |
| "Panels" | 5 | number of panels |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic processing |
"MultiPanelRule" options.
Let the unit interval
[0, 1] be partitioned into
k sub-intervals with the points
0=y0<y1<...<yk=1.
it can be transformed into a rule for the interval
[yj-1, yj],
Let
xi j=xi (yj-yj-1)+yj-1, and
yj-yj-1=1/k,
j =1, ..., k. Then the
k-panel integration rule based on
R (f) can be written explicitly as
If
R (f) is closed, that is,
R (f) has
0 and
1 as sampling points, then
xn j-1=x1 j, and the number of sampling points of
k×R (f) can be reduced to
k (n-1)+1. (This is done in the implementation of
"MultiPanelRule".)
More about the theory of multi-panel rules, also referred to as compounded or composite rules, can be found in [
KrUeb98] and [
DavRab84].
"MultiPanelRule" Sampling Points and Weights
The sampling points and the weights of the "MultiPanelRule" can be obtained with this command.
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Here are the abscissas and weights of a Gauss-Kronrod rule.
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The multi-panel rule abscissas can be obtained using Rescale.
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This shows how to derive the multi-panel rule weights from the original weights.
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"CartesianRule"
A
d-dimensional Cartesian rule has sampling points that are a Cartesian product of the sampling points of
d one-dimensional rules. The weight associated with a Cartesian rule sampling point is the product of the one-dimensional rule weights that correspond to its coordinates.
The Cartesian product integration for NIntegrate can be specified with the Method option value "CartesianRule".
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| Method | "GaussKronrodRule" | a rule or a list of rules with which the Cartesian product rule will be formed |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic preprocessing |
"CartesianRule" options.
For example, suppose we have the formulas:
that are exact for polynomials of degree
d1,
d2, and
d3, respectively. Then it is not difficult to see that the formula with
n1×n2×n3 points,
is exact for polynomials in
x1, x2, x3 of degree
min (d1, d2, d3). Note that the weight associated with the abscissa
is
.
The general Cartesian product formula for
D one-dimensional rules the
i of which has
ni sampling points
and weights
is
Clearly (
2) can be written as
where
n=
nk , and for each integer
k[1, n],
and
.
Here is a visualization of a Cartesian product rule integration. Along the x axis "TrapezoidalRule" is used; along the y axis "GaussKronrodRule" is used.
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Cartesian rules are applicable for relatively low dimensions (
≤ 4), since for higher dimensions they are subject to "combinatorial explosion." For example, a
5-dimensional Cartesian product of
5 identical one-dimensional rules each having
10 sampling points would have
10^5 sampling points.
NIntegrate uses Cartesian product rule if the integral is multidimensional and the
Method option is given a one-dimensional rule or a list of one-dimensional rules.
Here is an example specifying Cartesian product rule integration with GaussKronrodRule.
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Here is an example specifying Cartesian product rule integration with a list of one-dimensional integration rules.
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Another example specifying Cartesian product rule integration with a list of one-dimensional integration rules.
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More about Cartesian rules can be found in [
Stroud71].
"CartesianRule" Sampling Points and Weights
The sampling points and the weights of the "CartesianRule" rule can be obtained with the command NIntegrate`CartesianRuleData.
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NIntegrate`CartesianRuleData keeps the abscissas and the weights of each rule separated. Otherwise, as it can be seen from (
3) the result might be too big for higher dimensions.
The results of NIntegrate`CartesianRuleData can be put into the form of ( 4) with this function.
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"MultiDimensionalRule"
A fully symmetric integration rule for the cube
,
d, d>1 consists of sets of points with the properties: (i) all points in a set can be generated by permutations and/or sign changes of the coordinates of any fixed point from that set, (ii) all points in a set have the same weight associated with them.
The fully symmetric multidimensional integration (fully symmetric cubature) for NIntegrate can be specified with the Method option value "MultiDimensionalRule".
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A set of points of a fully symmetric integration rule that satisfies the preceding properties is called an orbit. A point of an orbit,
{x1, x2, ..., xd}, for the coordinates of which the inequality
x1≥x2≥...≥ xd holds, is called a generator. (See [
KrUeb98][
GenzMalik83].)
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| "Generators" | 5 | number of generators of the fully symmetric rule |
| "SymbolicProcessing" | Automatic | number of seconds to do symbolic preprocessing |
"MultiDimensionalRule" options.
If an integration rule has
K orbits denoted
1, 2, ..., K, and the
ith of them,
i, has a weight
wi associated with it, then the integral estimate is calculated with the formula
A null rule of degree
m will integrate to zero all monomials of degree
≤ m and will fail to do so for at least one monomial of degree
m+1. Each null rule may be thought of as the difference between a basic integration rule and an appropriate integration of lower degree.
