D01 Chapter Contents
D01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentD01RGF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

D01RGF is a general purpose integrator which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $\left[a,b\right]$:
 $I= ∫ab fx dx .$
The routine is suitable as a general purpose integrator, and can be used when the integrand has singularities and infinities. In particular, the routine can continue if the subroutine F explicitly returns a quiet or signalling NaN or a signed infinity.

## 2  Specification

 SUBROUTINE D01RGF ( A, B, F, EPSABS, EPSREL, DINEST, ERREST, NEVALS, IUSER, RUSER, IFAIL)
 INTEGER NEVALS, IUSER(*), IFAIL REAL (KIND=nag_wp) A, B, EPSABS, EPSREL, DINEST, ERREST, RUSER(*) EXTERNAL F

## 3  Description

D01RGF uses the algorithm described in Gonnet (2010). It is an adaptive algorithm, similar to the QUADPACK routine QAGS (see Piessens et al. (1983), see also D01ATF, D01AUF and D01RAF) but includes significant differences regarding how the integrand is represented, how the integration error is estimated and how singularities and divergent integrals are treated. The local error estimation is described in Gonnet (2010).
D01RGF requires a subroutine to evaluate the integrand at an array of different points and is therefore particularly efficient when the evaluation can be performed in vector mode on a vector-processing machine.

## 4  References

Gonnet P (2010) Increasing the reliability of adaptive quadrature using explicit interpolants ACM Trans. Math. software 37 26
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

## 5  Parameters

1:     A – REAL (KIND=nag_wp)Input
On entry: $a$, the lower limit of integration.
2:     B – REAL (KIND=nag_wp)Input
On entry: $b$, the upper limit of integration. It is not necessary that $a.
Note: if ${\mathbf{A}}={\mathbf{B}}$, the routine will immediately return ${\mathbf{DINEST}}=0.0$, ${\mathbf{ERREST}}=0.0$ and ${\mathbf{NEVALS}}=0$.
3:     F – SUBROUTINE, supplied by the user.External Procedure
F must return the value of the integrand $f$ at a set of points.
The specification of F is:
 SUBROUTINE F ( X, NX, FV, IFLAG, IUSER, RUSER)
 INTEGER NX, IFLAG, IUSER(*) REAL (KIND=nag_wp) X(NX), FV(NX), RUSER(*)
1:     X(NX) – REAL (KIND=nag_wp) arrayInput
On entry: the abscissae, ${x}_{i}$, for $\mathit{i}=1,2,\dots ,{\mathbf{NX}}$, at which function values are required.
2:     NX – INTEGERInput
On entry: the number of abscissae at which a function value is required.
3:     FV(NX) – REAL (KIND=nag_wp) arrayOutput
On exit: FV must contain the values of the integrand $f$. ${\mathbf{FV}}\left(i\right)=f\left({x}_{i}\right)$ for all $i=1,2,\dots ,{\mathbf{NX}}$.
4:     IFLAG – INTEGERInput/Output
On entry: ${\mathbf{IFLAG}}=0$.
On exit: set ${\mathbf{IFLAG}}<0$ to force an immediate exit with ${\mathbf{IFAIL}}=-{\mathbf{1}}$.
5:     IUSER($*$) – INTEGER arrayUser Workspace
6:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
F is called with the parameters IUSER and RUSER as supplied to D01RGF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.
F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01RGF is called. Parameters denoted as Input must not be changed by this procedure.
4:     EPSABS – REAL (KIND=nag_wp)Input
On entry: the absolute accuracy required.
If EPSABS is negative, $\left|{\mathbf{EPSABS}}\right|$ is used. See Section 7.
If ${\mathbf{EPSABS}}=0.0$, only the relative error will be used.
5:     EPSREL – REAL (KIND=nag_wp)Input
On entry: the relative accuracy required.
If EPSREL is negative, $\left|{\mathbf{EPSREL}}\right|$ is used. See Section 7.
If ${\mathbf{EPSREL}}=0.0$, only the absolute error will be used otherwise the actual value of EPSREL used by D01RGF is .
Constraint: at least one of EPSABS and EPSREL must be nonzero.
6:     DINEST – REAL (KIND=nag_wp)Output
On exit: the estimate of the definite integral F.
7:     ERREST – REAL (KIND=nag_wp)Output
On exit: the error estimate of the definite integral F.
8:     NEVALS – INTEGEROutput
On exit: the number of function evaluations.
9:     IUSER($*$) – INTEGER arrayUser Workspace
10:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
IUSER and RUSER are not used by D01RGF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.
11:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if ${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Note: D01RGF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
The requested accuracy was not achieved. Consider using larger values of EPSABS and EPSREL.
${\mathbf{IFAIL}}=2$
The integral is probably divergent or slowly convergent.
${\mathbf{IFAIL}}=14$
Both ${\mathbf{EPSABS}}=0.0$ and ${\mathbf{EPSREL}}=0.0$.
${\mathbf{IFAIL}}=-1$
Exit requested from F with ${\mathbf{IFLAG}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

## 7  Accuracy

D01RGF cannot guarantee, but in practice usually achieves, the following accuracy:
 $I-DINEST ≤ tol ,$
where
 $tol = max EPSABS , EPSREL × I ,$
and EPSABS and EPSREL are user-specified absolute and relative error tolerances. Moreover, it returns the quantity ERREST which, in normal circumstances, satisfies
 $I-DINEST ≤ ERREST ≤ tol .$

The time taken by D01RGF depends on the integrand and the accuracy required.
D01RGF is suitable for evaluating integrals that have singularities within the requested interval.
In particular, D01RGF accepts non-finite values on return from the user-supplied subroutine F, and will adapt the integration rule accordingly to eliminate such points. Non-finite values include NaNs and infinities.

## 9  Example

This example computes
 $∫ -1 1 sinx x ln101 - x .$

### 9.1  Program Text

Program Text (d01rgfe.f90)

None.

### 9.3  Program Results

Program Results (d01rgfe.r)