*DECK QAG SUBROUTINE QAG (F, A, B, EPSABS, EPSREL, KEY, RESULT, ABSERR, + NEVAL, IER, LIMIT, LENW, LAST, IWORK, WORK) C***BEGIN PROLOGUE QAG 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 SINGLE PRECISION (QAG-S, DQAG-D) C***KEYWORDS AUTOMATIC INTEGRATOR, GAUSS-KRONROD RULES, C GENERAL-PURPOSE, GLOBALLY ADAPTIVE, INTEGRAND EXAMINATOR, 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 Real version C C F - Real 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 - Real C Lower limit of integration C C B - Real C Upper limit of integration C C EPSABS - Real C Absolute accuracy requested C EPSREL - Real 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 KEY - Integer C Key for choice of local integration rule C A GAUSS-KRONROD PAIR is used with C 7 - 15 POINTS If KEY.LT.2, C 10 - 21 POINTS If KEY = 2, C 15 - 31 POINTS If KEY = 3, C 20 - 41 POINTS If KEY = 4, C 25 - 51 POINTS If KEY = 5, C 30 - 61 POINTS If KEY.GT.5. C C ON RETURN C RESULT - Real C Approximation to the integral C C ABSERR - Real 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 RESULT 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 C subdivisions by increasing the value of C LIMIT (and taking the according dimension C adjustments into account). HOWEVER, If C this yield no improvement it is advised C to analyze the integrand in order to C determine the integration difficulties. C If the position of a local difficulty can C be determined (I.E. 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 C detected, which prevents the requested C tolerance from being achieved. C = 3 Extremely bad integrand behaviour occurs C at some points of the integration C interval. 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 C to zero. C EXCEPT when LENW is invalid, IWORK(1), C WORK(LIMIT*2+1) and WORK(LIMIT*3+1) are C set to zero, WORK(1) is set to A and C 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 with C IER = 6. C C LAST - Integer C On return, LAST equals the number of subintervals C produced in the subdivision process, which C determines the number of significant elements C actually in the WORK ARRAYS. C C WORK ARRAYS C IWORK - Integer C Vector of dimension at least limit, the first K C elements of which contain pointers to the error C estimates over the subintervals, such that C WORK(LIMIT*3+IWORK(1)),... , WORK(LIMIT*3+IWORK(K)) C form a decreasing sequence with K = LAST If C LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST otherwise C C WORK - Real C Vector of dimension at least LENW C on return C WORK(1), ..., WORK(LAST) contain the left end C points of the subintervals in the partition of C (A,B), C WORK(LIMIT+1), ..., WORK(LIMIT+LAST) contain the C 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) contain C the error estimates. C C***REFERENCES (NONE) C***ROUTINES CALLED QAGE, 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 QAG REAL A,ABSERR,B,EPSABS,EPSREL,F,RESULT,WORK INTEGER IER,IWORK,KEY,LENW,LIMIT,LVL,L1,L2,L3,NEVAL C DIMENSION IWORK(*),WORK(*) C EXTERNAL F C***FIRST EXECUTABLE STATEMENT QAG IER = 6 NEVAL = 0 LAST = 0 RESULT = 0.0E+00 ABSERR = 0.0E+00 IF (LIMIT.GE.1 .AND. LENW.GE.LIMIT*4) THEN C C PREPARE CALL FOR QAGE. C L1 = LIMIT+1 L2 = LIMIT+L1 L3 = LIMIT+L2 C CALL QAGE(F,A,B,EPSABS,EPSREL,KEY,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 ENDIF C IF (IER.EQ.6) LVL = 1 IF (IER .NE. 0) CALL XERMSG ('SLATEC', 'QAG', + 'ABNORMAL RETURN', IER, LVL) RETURN END