*DECK QK41 SUBROUTINE QK41 (F, A, B, RESULT, ABSERR, RESABS, RESASC) C***BEGIN PROLOGUE QK41 C***PURPOSE To compute I = Integral of F over (A,B), with error C estimate C J = Integral of ABS(F) over (A,B) C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A1A2 C***TYPE SINGLE PRECISION (QK41-S, DQK41-D) C***KEYWORDS 41-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 rules 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 A - Real C Lower limit of integration C C B - Real C Upper limit of integration C C ON RETURN C RESULT - Real C Approximation to the integral I C RESULT is computed by applying the 41-POINT C GAUSS-KRONROD RULE (RESK) obtained by optimal C addition of abscissae to the 20-POINT GAUSS C RULE (RESG). C C ABSERR - Real C Estimate of the modulus of the absolute error, C which should not exceed ABS(I-RESULT) C C RESABS - Real C Approximation to the integral J C C RESASC - Real C Approximation to the integral of ABS(F-I/(B-A)) C 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 QK41 C REAL A,ABSC,ABSERR,B,CENTR,DHLGTH,EPMACH,F,FC,FSUM,FVAL1,FVAL2, 1 FV1,FV2,HLGTH,RESABS, 2 RESASC,RESG,RESK,RESKH,RESULT,R1MACH,UFLOW, 3 WG,WGK,XGK INTEGER J,JTW,JTWM1 EXTERNAL F C DIMENSION FV1(20),FV2(20),XGK(21),WGK(21),WG(10) C C THE ABSCISSAE AND WEIGHTS ARE GIVEN FOR THE INTERVAL (-1,1). C BECAUSE OF SYMMETRY ONLY THE POSITIVE ABSCISSAE AND THEIR C CORRESPONDING WEIGHTS ARE GIVEN. C C XGK - ABSCISSAE OF THE 41-POINT GAUSS-KRONROD RULE C XGK(2), XGK(4), ... ABSCISSAE OF THE 20-POINT C GAUSS RULE C XGK(1), XGK(3), ... ABSCISSAE WHICH ARE OPTIMALLY C ADDED TO THE 20-POINT GAUSS RULE C C WGK - WEIGHTS OF THE 41-POINT GAUSS-KRONROD RULE C C WG - WEIGHTS OF THE 20-POINT GAUSS RULE C SAVE XGK, WGK, WG DATA XGK(1),XGK(2),XGK(3),XGK(4),XGK(5),XGK(6),XGK(7),XGK(8), 1 XGK(9),XGK(10),XGK(11),XGK(12),XGK(13),XGK(14),XGK(15), 2 XGK(16),XGK(17),XGK(18),XGK(19),XGK(20),XGK(21)/ 3 0.9988590315882777E+00, 0.9931285991850949E+00, 4 0.9815078774502503E+00, 0.9639719272779138E+00, 5 0.9408226338317548E+00, 0.9122344282513259E+00, 6 0.8782768112522820E+00, 0.8391169718222188E+00, 7 0.7950414288375512E+00, 0.7463319064601508E+00, 8 0.6932376563347514E+00, 0.6360536807265150E+00, 9 0.5751404468197103E+00, 0.5108670019508271E+00, 1 0.4435931752387251E+00, 0.3737060887154196E+00, 2 0.3016278681149130E+00, 0.2277858511416451E+00, 3 0.1526054652409227E+00, 0.7652652113349733E-01, 4 0.0E+00 / DATA WGK(1),WGK(2),WGK(3),WGK(4),WGK(5),WGK(6),WGK(7),WGK(8), 1 WGK(9),WGK(10),WGK(11),WGK(12),WGK(13),WGK(14),WGK(15),WGK(16), 2 WGK(17),WGK(18),WGK(19),WGK(20),WGK(21)/ 3 0.3073583718520532E-02, 0.8600269855642942E-02, 4 0.1462616925697125E-01, 0.2038837346126652E-01, 5 0.2588213360495116E-01, 0.3128730677703280E-01, 6 0.3660016975820080E-01, 0.4166887332797369E-01, 7 0.4643482186749767E-01, 0.5094457392372869E-01, 8 0.5519510534828599E-01, 0.5911140088063957E-01, 9 0.6265323755478117E-01, 0.6583459713361842E-01, 1 0.6864867292852162E-01, 0.7105442355344407E-01, 2 0.7303069033278667E-01, 0.7458287540049919E-01, 3 0.7570449768455667E-01, 0.7637786767208074E-01, 4 0.7660071191799966E-01/ DATA WG(1),WG(2),WG(3),WG(4),WG(5),WG(6),WG(7),WG(8),WG(9),WG(10)/ 1 0.1761400713915212E-01, 0.4060142980038694E-01, 2 0.6267204833410906E-01, 0.8327674157670475E-01, 3 0.1019301198172404E+00, 0.1181945319615184E+00, 4 0.1316886384491766E+00, 0.1420961093183821E+00, 5 0.1491729864726037E+00, 0.1527533871307259E+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 FVAL* - FUNCTION VALUE C RESG - RESULT OF THE 20-POINT GAUSS FORMULA C RESK - RESULT OF THE 41-POINT KRONROD FORMULA C RESKH - APPROXIMATION TO MEAN VALUE OF F OVER (A,B), I.E. C 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 QK41 EPMACH = R1MACH(4) UFLOW = R1MACH(1) C CENTR = 0.5E+00*(A+B) HLGTH = 0.5E+00*(B-A) DHLGTH = ABS(HLGTH) C C COMPUTE THE 41-POINT GAUSS-KRONROD APPROXIMATION TO C THE INTEGRAL, AND ESTIMATE THE ABSOLUTE ERROR. C RESG = 0.0E+00 FC = F(CENTR) RESK = WGK(21)*FC RESABS = ABS(RESK) DO 10 J=1,10 JTW = J*2 ABSC = HLGTH*XGK(JTW) FVAL1 = F(CENTR-ABSC) FVAL2 = F(CENTR+ABSC) FV1(JTW) = FVAL1 FV2(JTW) = FVAL2 FSUM = FVAL1+FVAL2 RESG = RESG+WG(J)*FSUM RESK = RESK+WGK(JTW)*FSUM RESABS = RESABS+WGK(JTW)*(ABS(FVAL1)+ABS(FVAL2)) 10 CONTINUE DO 15 J = 1,10 JTWM1 = J*2-1 ABSC = HLGTH*XGK(JTWM1) FVAL1 = F(CENTR-ABSC) FVAL2 = F(CENTR+ABSC) FV1(JTWM1) = FVAL1 FV2(JTWM1) = FVAL2 FSUM = FVAL1+FVAL2 RESK = RESK+WGK(JTWM1)*FSUM RESABS = RESABS+WGK(JTWM1)*(ABS(FVAL1)+ABS(FVAL2)) 15 CONTINUE RESKH = RESK*0.5E+00 RESASC = WGK(21)*ABS(FC-RESKH) DO 20 J=1,20 RESASC = RESASC+WGK(J)*(ABS(FV1(J)-RESKH)+ABS(FV2(J)-RESKH)) 20 CONTINUE RESULT = RESK*HLGTH RESABS = RESABS*DHLGTH RESASC = RESASC*DHLGTH ABSERR = ABS((RESK-RESG)*HLGTH) IF(RESASC.NE.0.0E+00.AND.ABSERR.NE.0.E+00) 1 ABSERR = RESASC*MIN(0.1E+01, 2 (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