***BEGIN PROLOGUE DQAGS ***PURPOSE The routine calculates an approximation result to a given Definite integral I = Integral of F over (A,B), Hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). ***LIBRARY SLATEC (QUADPACK) ***CATEGORY H2A1A1 ***TYPE real(kind=wp) (QAGS-S, DQAGS-D) ***KEYWORDS AUTOMATIC INTEGRATOR, END POINT SINGULARITIES, EXTRAPOLATION, GENERAL-PURPOSE, GLOBALLY ADAPTIVE, QUADPACK, QUADRATURE ***AUTHOR Piessens, Robert Applied Mathematics and Programming Division K. U. Leuven de Doncker, Elise Applied Mathematics and Programming Division K. U. Leuven ***DESCRIPTION
Computation of a definite integral
Standard fortran subroutine
Double precision version
PARAMETERS
ON ENTRY
F - class(bound_user_function)
Function whose method 'eval' defines the integrand
Function F(X).
A - Double precision
Lower limit of integration
B - Double precision
Upper limit of integration
EPSABS - Double precision
Absolute accuracy requested
EPSREL - Double precision
Relative accuracy requested
If EPSABS.LE.0
And EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
The routine will end with IER = 6.
ON RETURN
RESULT - Double precision
Approximation to the integral
ABSERR - Double precision
Estimate of the modulus of the absolute error,
which should equal or exceed ABS(I-RESULT)
NEVAL - Integer
Number of integrand evaluations
IER - Integer
IER = 0 Normal and reliable termination of the
routine. It is assumed that the requested
accuracy has been achieved.
IER.GT.0 Abnormal termination of the routine
The estimates for integral and error are
less reliable. It is assumed that the
requested accuracy has not been achieved.
ERROR MESSAGES
IER = 1 Maximum number of subdivisions allowed
has been achieved. One can allow more sub-
divisions by increasing the value of LIMIT
(and taking the according dimension
adjustments into account. However, if
this yields no improvement it is advised
to analyze the integrand in order to
determine the integration difficulties. If
the position of a local difficulty can be
determined (E.G. SINGULARITY,
DISCONTINUITY WITHIN THE INTERVAL) one
will probably gain from splitting up the
interval at this point and calling the
integrator on the subranges. If possible,
an appropriate special-purpose integrator
should be used, which is designed for
handling the type of difficulty involved.
= 2 The occurrence of roundoff error is detec-
ted, which prevents the requested
tolerance from being achieved.
The error may be under-estimated.
= 3 Extremely bad integrand behaviour
occurs at some points of the integration
interval.
= 4 The algorithm does not converge.
Roundoff error is detected in the
Extrapolation table. It is presumed that
the requested tolerance cannot be
achieved, and that the returned result is
the best which can be obtained.
= 5 The integral is probably divergent, or
slowly convergent. It must be noted that
divergence can occur with any other value
of IER.
= 6 The input is invalid, because
(EPSABS.LE.0 AND
EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28)
OR LIMIT.LT.1 OR LENW.LT.LIMIT*4.
RESULT, ABSERR, NEVAL, LAST are set to
zero. Except when LIMIT or LENW is
invalid, IWORK(1), WORK(LIMIT*2+1) and
WORK(LIMIT*3+1) are set to zero, WORK(1)
is set to A and WORK(LIMIT+1) TO B.
DIMENSIONING PARAMETERS
LIMIT - Integer
DIMENSIONING PARAMETER FOR IWORK
LIMIT determines the maximum number of subintervals
in the partition of the given integration interval
(A,B), LIMIT.GE.1.
IF LIMIT.LT.1, the routine will end with IER = 6.
LENW - Integer
DIMENSIONING PARAMETER FOR WORK
LENW must be at least LIMIT*4.
If LENW.LT.LIMIT*4, the routine will end
with IER = 6.
LAST - Integer
On return, LAST equals the number of subintervals
produced in the subdivision process, determines the
number of significant elements actually in the WORK
Arrays.
WORK ARRAYS
IWORK - Integer
Vector of dimension at least LIMIT, the first K
elements of which contain pointers
to the error estimates over the subintervals
such that WORK(LIMIT*3+IWORK(1)),... ,
WORK(LIMIT*3+IWORK(K)) form a decreasing
sequence, with K = LAST IF LAST.LE.(LIMIT/2+2),
and K = LIMIT+1-LAST otherwise
WORK - Double precision
Vector of dimension at least LENW
on return
WORK(1), ..., WORK(LAST) contain the left
end-points of the subintervals in the
partition of (A,B),
WORK(LIMIT+1), ..., WORK(LIMIT+LAST) contain
the right end-points,
WORK(LIMIT*2+1), ..., WORK(LIMIT*2+LAST) contain
the integral approximations over the subintervals,
WORK(LIMIT*3+1), ..., WORK(LIMIT*3+LAST)
contain the error estimates.
***REFERENCES (NONE) ***ROUTINES CALLED DQAGSE, XERMSG ***REVISION HISTORY (YYMMDD) 800101 DATE WRITTEN 890831 Modified array declarations. (WRB) 890831 REVISION DATE from Version 3.2 891214 Prologue converted to Version 4.0 format. (BAB) 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) ***END PROLOGUE DQAGS
