naginterfaces.library.quad.dim1_​gauss_​vec

naginterfaces.library.quad.dim1_gauss_vec(key, a, b, n, f, data=None)

dim1_gauss_vec computes an estimate of the definite integral of a function of known analytical form, using a Gaussian quadrature formula with a specified number of abscissae. Formulae are provided for a finite interval (Gauss–Legendre), a semi-infinite interval (Gauss–Laguerre, rational Gauss), and an infinite interval (Gauss–Hermite).

For full information please refer to the NAG Library document for d01ua

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d01/d01uaf.html

Parameters

keyint

Indicates the quadrature formula.

$key=0$

Gauss–Legendre quadrature on a finite interval, using normal weights.

$key=3$

Gauss–Laguerre quadrature on a semi-infinite interval, using normal weights.

$key=-3$

Gauss–Laguerre quadrature on a semi-infinite interval, using adjusted weights.

$key=4$

Gauss–Hermite quadrature on an infinite interval, using normal weights.

$key=-4$

Gauss–Hermite quadrature on an infinite interval, using adjusted weights.

$key=-5$

Rational Gauss quadrature on a semi-infinite interval, using adjusted weights.

afloat

The quantities $a$ and $b$ as described in the appropriate subsection of Notes.

bfloat

The quantities $a$ and $b$ as described in the appropriate subsection of Notes.

nint

$n$, the number of abscissae to be used.

fcallable (fv, iflag) = f(x, iflag, data=None)

$f$ must return the value of the integrand $f$, or the normalized integrand $g$, at a set of points.

Parameters

xfloat, ndarray, shape $\left(\text{nx}\right)$

The abscissae, $x_i$, for $\text{i}=1,2,\dots ,n_x$ at which function values are required.

iflagint

$iflag=0$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fvfloat, array-like, shape $\left(\text{nx}\right)$

If adjusted weights are used, the values of the integrand $f$. $fv[\text{i}-1]=f\left(x_{\text{i}}\right)$, for $\text{i}=1,2,\dots ,n_x$.

Otherwise the values of the normalized integrand $g$. $fv[\text{i}-1]=g\left(x_{\text{i}}\right)$, for $\text{i}=1,2,\dots ,n_x$.

iflagint

Set $iflag<0$ if you wish to force an immediate exit from dim1_gauss_vec with $errno$ = -1.

dataarbitrary, optional

User-communication data for callback functions.

Returns

dinestfloat

The estimate of the definite integral.

Raises

NagValueError

(errno $11$)

On entry, $key=⟨value⟩$.

Constraint: $key=0$, $3$, $-3$, $4$, $-4$ or $-5$.

(errno $12$)

The value of $a$ and/or $b$ is invalid.

On entry, $key=⟨value⟩$.

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $\left|b\right|>0.0$.

(errno $12$)

The value of $a$ and/or $b$ is invalid.

On entry, $key=⟨value⟩$.

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $b>0.0$.

(errno $12$)

The value of $a$ and/or $b$ is invalid.

On entry, $key=⟨value⟩$.

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $|a+b|>0.0$.

(errno $14$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

Warns

NagAlgorithmicWarning

(errno $-1$)

Exit requested from $f$ with $iflag=⟨value⟩$.

(errno $1$)

The $n$-point rule is not among those stored.

On entry: $n=⟨value⟩$.

$n$-point rule used: $n=⟨value⟩$.

(errno $2$)

Underflow occurred in calculation of normal weights.

Reduce $n$ or use adjusted weights: $n=⟨value⟩$.

(errno $3$)

No nonzero weights were generated for the provided parameters.

Notes

General

dim1_gauss_vec evaluates an estimate of the definite integral of a function $f\left(x\right)$, over a finite or infinite range, by $n$-point Gaussian quadrature (see Davis and Rabinowitz (1975), Fröberg (1970), Ralston (1965) and Stroud and Secrest (1966)). The integral is approximated by a summation of the product of a set of weights and a set of function evaluations at a corresponding set of abscissae $x_i$. For adjusted weights, the function values correspond to the values of the integrand $f$, and hence the sum will be

$$\sum _{i=1}^n{w}_{i}f\left(x_i\right)$$

where the $w_i$ are called the weights, and the $x_i$ the abscissae. A selection of values of $n$ is available. (See Parameters.)

Where applicable, normal weights may instead be used, in which case the corresponding weight function $\omega$ is factored out of the integrand as $f\left(x\right)=\omega \left(x\right)g\left(x\right)$ and hence the sum will be

$$\sum _{-1}^n{\overline{w}}_{i}g\left(x\right)\text{,}$$

where the normal weights ${\overline{w}}_i=w_i\omega \left(x_i\right)$ are computed internally.

dim1_gauss_vec uses a vectorized $f$ to evaluate the integrand or normalized integrand at a set of abscissae, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n_x$. If adjusted weights are used, the integrand $f\left(x_i\right)$ must be evaluated otherwise the normalized integrand $g\left(x_i\right)$ must be evaluated.

Both Limits Finite

$${\int }_a^{b}f\left(x\right)dx\text{.}$$

The Gauss–Legendre weights and abscissae are used, and the formula is exact for any function of the form:

$$f\left(x\right)=\sum _{i=0}^{2n-1}{c}_i{x}^i\text{.}$$

The formula is appropriate for functions which can be well approximated by such a polynomial over $[a,b]$. It is inappropriate for functions with algebraic singularities at one or both ends of the interval, such as ${(1+x)}^{-1/2}$ on $[-1,1]$.

One Limit Infinite

$${\int }_a^{\infty }f\left(x\right)dx\text{or}{\int }_{-\infty }^{a}f\left(x\right)dx\text{.}$$

Two quadrature formulae are available for these integrals.

1. The Gauss–Laguerre formula is exact for any function of the form:

   $$f\left(x\right)=e^{-bx}\sum _{i=0}^{2n-1}{c}_i{x}^i\text{.}$$

   This formula is appropriate for functions decaying exponentially at infinity; the argument $b$ should be chosen if possible to match the decay rate of the function.

   If the adjusted weights are selected, the complete integrand $f\left(x\right)$ should be provided through $f$.

   If the normal form is selected, the contribution of $e^{-bx}$ is accounted for internally, and $f$ should only return $g\left(x\right)$, where $f\left(x\right)=e^{-bx}g\left(x\right)$.

   If $b<0$ is supplied, the interval of integration will be $[a,\infty )$. Otherwise if $b>0$ is supplied, the interval of integration will be $(-\infty ,a]$.

2. The rational Gauss formula is exact for any function of the form:

   $$f\left(x\right)=\sum _{i=2}^{2n+1}\frac{c_i}{{(x+b)}^i}=\frac{{\sum }_{i=0}^{2n-1}{c}_{2n+1-i}{(x+b)}^i}{{(x+b)}^{2n+1}}\text{.}$$

   This formula is likely to be more accurate for functions having only an inverse power rate of decay for large $x$. Here the choice of a suitable value of $b$ may be more difficult; unfortunately a poor choice of $b$ can make a large difference to the accuracy of the computed integral.

   Only the adjusted form of the rational Gauss formula is available, and as such, the complete integrand $f\left(x\right)$ must be supplied in $f$.

   If $a+b<0$, the interval of integration will be $[a,\infty )$. Otherwise if $a+b>0$, the interval of integration will be $(-\infty ,a]$.

Both Limits Infinite

$${\int }_{-\infty }^{+\infty }f\left(x\right)dx\text{.}$$

The Gauss–Hermite weights and abscissae are used, and the formula is exact for any function of the form:

$$f\left(x\right)=e^{-b{(x-a)}^2}\sum _{i=0}^{2n-1}{c}_i{x}^i\text{,}$$

where $b>0$. Again, for general functions not of this exact form, the argument $b$ should be chosen to match if possible the decay rate at $\pm \infty$.

If the adjusted weights are selected, the complete integrand $f\left(x\right)$ should be provided through $f$.

If the normal form is selected, the contribution of $e^{-b{(x-a)}^2}$ is accounted for internally, and $f$ should only return $g\left(x\right)$, where $f\left(x\right)=e^{-b{(x-a)}^2}g\left(x\right)$.

References

Davis, P J and Rabinowitz, P, 1975, Methods of Numerical Integration, Academic Press

Fröberg, C E, 1970, Introduction to Numerical Analysis, Addison–Wesley

Ralston, A, 1965, A First Course in Numerical Analysis, pp. 87–90, McGraw–Hill

Stroud, A H and Secrest, D, 1966, Gaussian Quadrature Formulas, Prentice–Hall