*DECK QK15I SUBROUTINE QK15I (F, BOUN, INF, A, B, RESULT, ABSERR, RESABS, + RESASC) C***BEGIN PROLOGUE QK15I C***PURPOSE The original (infinite integration range is mapped C onto the interval (0,1) and (A,B) is a part of (0,1). C it is the purpose to compute C I = Integral of transformed integrand over (A,B), C J = Integral of ABS(Transformed Integrand) over (A,B). C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A3A2, H2A4A2 C***TYPE SINGLE PRECISION (QK15I-S, DQK15I-D) C***KEYWORDS 15-POINT GAUSS-KRONROD RULES, 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 Integration Rule C Standard Fortran subroutine C Real version C C PARAMETERS C ON ENTRY 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 calling program. C C BOUN - Real C Finite bound of original integration C Range (SET TO ZERO IF INF = +2) C C INF - Integer C If INF = -1, the original interval is C (-INFINITY,BOUND), C If INF = +1, the original interval is C (BOUND,+INFINITY), C If INF = +2, the original interval is C (-INFINITY,+INFINITY) AND C The integral is computed as the sum of two C integrals, one over (-INFINITY,0) and one over C (0,+INFINITY). C C A - Real C Lower limit for integration over subrange C of (0,1) C C B - Real C Upper limit for integration over subrange C of (0,1) C C ON RETURN C RESULT - Real C Approximation to the integral I C Result is computed by applying the 15-POINT C KRONROD RULE(RESK) obtained by optimal addition C of abscissae to the 7-POINT GAUSS RULE(RESG). C C ABSERR - Real C Estimate of the modulus of the absolute error, C WHICH SHOULD EQUAL or EXCEED ABS(I-RESULT) C C RESABS - Real C Approximation to the integral J C C RESASC - Real C Approximation to the integral of C ABS((TRANSFORMED INTEGRAND)-I/(B-A)) over (A,B) C C***REFERENCES (NONE) C***ROUTINES CALLED R1MACH C***REVISION HISTORY (YYMMDD) C 800101 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C***END PROLOGUE QK15I C REAL A,ABSC,ABSC1,ABSC2,ABSERR,B,BOUN,CENTR, 1 DINF,R1MACH,EPMACH,F,FC,FSUM,FVAL1,FVAL2,FV1, 2 FV2,HLGTH,RESABS,RESASC,RESG,RESK,RESKH,RESULT,TABSC1,TABSC2, 3 UFLOW,WG,WGK,XGK INTEGER INF,J EXTERNAL F C DIMENSION FV1(7),FV2(7),XGK(8),WGK(8),WG(8) C C THE ABSCISSAE AND WEIGHTS ARE SUPPLIED FOR THE INTERVAL C (-1,1). BECAUSE OF SYMMETRY ONLY THE POSITIVE ABSCISSAE AND C THEIR CORRESPONDING WEIGHTS ARE GIVEN. C C XGK - ABSCISSAE OF THE 15-POINT KRONROD RULE C XGK(2), XGK(4), ... ABSCISSAE OF THE 7-POINT C GAUSS RULE C XGK(1), XGK(3), ... ABSCISSAE WHICH ARE OPTIMALLY C ADDED TO THE 7-POINT GAUSS RULE C C WGK - WEIGHTS OF THE 15-POINT KRONROD RULE C C WG - WEIGHTS OF THE 7-POINT GAUSS RULE, CORRESPONDING C TO THE ABSCISSAE XGK(2), XGK(4), ... C WG(1), WG(3), ... ARE SET TO ZERO. C SAVE XGK, WGK, WG DATA XGK(1),XGK(2),XGK(3),XGK(4),XGK(5),XGK(6),XGK(7), 1 XGK(8)/ 2 0.9914553711208126E+00, 0.9491079123427585E+00, 3 0.8648644233597691E+00, 0.7415311855993944E+00, 4 0.5860872354676911E+00, 0.4058451513773972E+00, 5 0.2077849550078985E+00, 0.0000000000000000E+00/ C DATA WGK(1),WGK(2),WGK(3),WGK(4),WGK(5),WGK(6),WGK(7), 1 WGK(8)/ 2 0.2293532201052922E-01, 0.6309209262997855E-01, 3 0.1047900103222502E+00, 0.1406532597155259E+00, 4 0.1690047266392679E+00, 0.1903505780647854E+00, 5 0.2044329400752989E+00, 0.2094821410847278E+00/ C DATA WG(1),WG(2),WG(3),WG(4),WG(5),WG(6),WG(7),WG(8)/ 1 0.0000000000000000E+00, 0.1294849661688697E+00, 2 0.0000000000000000E+00, 0.2797053914892767E+00, 3 0.0000000000000000E+00, 0.3818300505051189E+00, 4 0.0000000000000000E+00, 0.4179591836734694E+00/ C C C LIST OF MAJOR VARIABLES C ----------------------- C C CENTR - MID POINT OF THE INTERVAL C HLGTH - HALF-LENGTH OF THE INTERVAL C ABSC* - ABSCISSA C TABSC* - TRANSFORMED ABSCISSA C FVAL* - FUNCTION VALUE C RESG - RESULT OF THE 7-POINT GAUSS FORMULA C RESK - RESULT OF THE 15-POINT KRONROD FORMULA C RESKH - APPROXIMATION TO THE MEAN VALUE OF THE TRANSFORMED C INTEGRAND OVER (A,B), I.E. TO I/(B-A) C C MACHINE DEPENDENT CONSTANTS C --------------------------- C C EPMACH IS THE LARGEST RELATIVE SPACING. C UFLOW IS THE SMALLEST POSITIVE MAGNITUDE. C C***FIRST EXECUTABLE STATEMENT QK15I EPMACH = R1MACH(4) UFLOW = R1MACH(1) DINF = MIN(1,INF) C CENTR = 0.5E+00*(A+B) HLGTH = 0.5E+00*(B-A) TABSC1 = BOUN+DINF*(0.1E+01-CENTR)/CENTR FVAL1 = F(TABSC1) IF(INF.EQ.2) FVAL1 = FVAL1+F(-TABSC1) FC = (FVAL1/CENTR)/CENTR C C COMPUTE THE 15-POINT KRONROD APPROXIMATION TO C THE INTEGRAL, AND ESTIMATE THE ERROR. C RESG = WG(8)*FC RESK = WGK(8)*FC RESABS = ABS(RESK) DO 10 J=1,7 ABSC = HLGTH*XGK(J) ABSC1 = CENTR-ABSC ABSC2 = CENTR+ABSC TABSC1 = BOUN+DINF*(0.1E+01-ABSC1)/ABSC1 TABSC2 = BOUN+DINF*(0.1E+01-ABSC2)/ABSC2 FVAL1 = F(TABSC1) FVAL2 = F(TABSC2) IF(INF.EQ.2) FVAL1 = FVAL1+F(-TABSC1) IF(INF.EQ.2) FVAL2 = FVAL2+F(-TABSC2) FVAL1 = (FVAL1/ABSC1)/ABSC1 FVAL2 = (FVAL2/ABSC2)/ABSC2 FV1(J) = FVAL1 FV2(J) = FVAL2 FSUM = FVAL1+FVAL2 RESG = RESG+WG(J)*FSUM RESK = RESK+WGK(J)*FSUM RESABS = RESABS+WGK(J)*(ABS(FVAL1)+ABS(FVAL2)) 10 CONTINUE RESKH = RESK*0.5E+00 RESASC = WGK(8)*ABS(FC-RESKH) DO 20 J=1,7 RESASC = RESASC+WGK(J)*(ABS(FV1(J)-RESKH)+ABS(FV2(J)-RESKH)) 20 CONTINUE RESULT = RESK*HLGTH RESASC = RESASC*HLGTH RESABS = RESABS*HLGTH ABSERR = ABS((RESK-RESG)*HLGTH) IF(RESASC.NE.0.0E+00.AND.ABSERR.NE.0.E0) ABSERR = RESASC* 1 MIN(0.1E+01,(0.2E+03*ABSERR/RESASC)**1.5E+00) IF(RESABS.GT.UFLOW/(0.5E+02*EPMACH)) ABSERR = MAX 1 ((EPMACH*0.5E+02)*RESABS,ABSERR) RETURN END