*DECK DQAGS SUBROUTINE DQAGS (F, A, B, EPSABS, EPSREL, RESULT, ABSERR, NEVAL, + IER, LIMIT, LENW, LAST, IWORK, WORK) C***BEGIN PROLOGUE DQAGS C***PURPOSE The routine calculates an approximation result to a given C Definite integral I = Integral of F over (A,B), C Hopefully satisfying following claim for accuracy C ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A1A1 C***TYPE DOUBLE PRECISION (QAGS-S, DQAGS-D) C***KEYWORDS AUTOMATIC INTEGRATOR, END POINT SINGULARITIES, C EXTRAPOLATION, GENERAL-PURPOSE, GLOBALLY ADAPTIVE, C QUADPACK, QUADRATURE C***AUTHOR Piessens, Robert C Applied Mathematics and Programming Division C K. U. Leuven C de Doncker, Elise C Applied Mathematics and Programming Division C K. U. Leuven C***DESCRIPTION C C Computation of a definite integral C Standard fortran subroutine C Double precision version C C C PARAMETERS C ON ENTRY C F - Double precision C Function subprogram defining the integrand C Function F(X). The actual name for F needs to be C Declared E X T E R N A L in the driver program. C C A - Double precision C Lower limit of integration C C B - Double precision C Upper limit of integration C C EPSABS - Double precision C Absolute accuracy requested C EPSREL - Double precision C Relative accuracy requested C If EPSABS.LE.0 C And EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28), C The routine will end with IER = 6. C C ON RETURN C RESULT - Double precision C Approximation to the integral C C ABSERR - Double precision C Estimate of the modulus of the absolute error, C which should equal or exceed ABS(I-RESULT) C C NEVAL - Integer C Number of integrand evaluations C C IER - Integer C IER = 0 Normal and reliable termination of the C routine. It is assumed that the requested C accuracy has been achieved. C IER.GT.0 Abnormal termination of the routine C The estimates for integral and error are C less reliable. It is assumed that the C requested accuracy has not been achieved. C ERROR MESSAGES C IER = 1 Maximum number of subdivisions allowed C has been achieved. One can allow more sub- C divisions by increasing the value of LIMIT C (and taking the according dimension C adjustments into account. However, if C this yields no improvement it is advised C to analyze the integrand in order to C determine the integration difficulties. If C the position of a local difficulty can be C determined (E.G. SINGULARITY, C DISCONTINUITY WITHIN THE INTERVAL) one C will probably gain from splitting up the C interval at this point and calling the C integrator on the subranges. If possible, C an appropriate special-purpose integrator C should be used, which is designed for C handling the type of difficulty involved. C = 2 The occurrence of roundoff error is detec- C ted, which prevents the requested C tolerance from being achieved. C The error may be under-estimated. C = 3 Extremely bad integrand behaviour C occurs at some points of the integration C interval. C = 4 The algorithm does not converge. C Roundoff error is detected in the C Extrapolation table. It is presumed that C the requested tolerance cannot be C achieved, and that the returned result is C the best which can be obtained. C = 5 The integral is probably divergent, or C slowly convergent. It must be noted that C divergence can occur with any other value C of IER. C = 6 The input is invalid, because C (EPSABS.LE.0 AND C EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28) C OR LIMIT.LT.1 OR LENW.LT.LIMIT*4. C RESULT, ABSERR, NEVAL, LAST are set to C zero. Except when LIMIT or LENW is C invalid, IWORK(1), WORK(LIMIT*2+1) and C WORK(LIMIT*3+1) are set to zero, WORK(1) C is set to A and WORK(LIMIT+1) TO B. C C DIMENSIONING PARAMETERS C LIMIT - Integer C DIMENSIONING PARAMETER FOR IWORK C LIMIT determines the maximum number of subintervals C in the partition of the given integration interval C (A,B), LIMIT.GE.1. C IF LIMIT.LT.1, the routine will end with IER = 6. C C LENW - Integer C DIMENSIONING PARAMETER FOR WORK C LENW must be at least LIMIT*4. C If LENW.LT.LIMIT*4, the routine will end C with IER = 6. C C LAST - Integer C On return, LAST equals the number of subintervals C produced in the subdivision process, determines the C number of significant elements actually in the WORK C Arrays. C C WORK ARRAYS C IWORK - Integer C Vector of dimension at least LIMIT, the first K C elements of which contain pointers C to the error estimates over the subintervals C such that WORK(LIMIT*3+IWORK(1)),... , C WORK(LIMIT*3+IWORK(K)) form a decreasing C sequence, with K = LAST IF LAST.LE.(LIMIT/2+2), C and K = LIMIT+1-LAST otherwise C C WORK - Double precision C Vector of dimension at least LENW C on return C WORK(1), ..., WORK(LAST) contain the left C end-points of the subintervals in the C partition of (A,B), C WORK(LIMIT+1), ..., WORK(LIMIT+LAST) contain C the right end-points, C WORK(LIMIT*2+1), ..., WORK(LIMIT*2+LAST) contain C the integral approximations over the subintervals, C WORK(LIMIT*3+1), ..., WORK(LIMIT*3+LAST) C contain the error estimates. C C***REFERENCES (NONE) C***ROUTINES CALLED DQAGSE, XERMSG C***REVISION HISTORY (YYMMDD) C 800101 DATE WRITTEN C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C***END PROLOGUE DQAGS C C DOUBLE PRECISION A,ABSERR,B,EPSABS,EPSREL,F,RESULT,WORK INTEGER IER,IWORK,LAST,LENW,LIMIT,LVL,L1,L2,L3,NEVAL C DIMENSION IWORK(*),WORK(*) C EXTERNAL F C C CHECK VALIDITY OF LIMIT AND LENW. C C***FIRST EXECUTABLE STATEMENT DQAGS IER = 6 NEVAL = 0 LAST = 0 RESULT = 0.0D+00 ABSERR = 0.0D+00 IF(LIMIT.LT.1.OR.LENW.LT.LIMIT*4) GO TO 10 C C PREPARE CALL FOR DQAGSE. C L1 = LIMIT+1 L2 = LIMIT+L1 L3 = LIMIT+L2 C CALL DQAGSE(F,A,B,EPSABS,EPSREL,LIMIT,RESULT,ABSERR,NEVAL, 1 IER,WORK(1),WORK(L1),WORK(L2),WORK(L3),IWORK,LAST) C C CALL ERROR HANDLER IF NECESSARY. C LVL = 0 10 IF(IER.EQ.6) LVL = 1 IF (IER .NE. 0) CALL XERMSG ('SLATEC', 'DQAGS', + 'ABNORMAL RETURN', IER, LVL) RETURN END