*DECK DQK31 SUBROUTINE DQK31 (F, A, B, RESULT, ABSERR, RESABS, RESASC) C***BEGIN PROLOGUE DQK31 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 DOUBLE PRECISION (QK31-S, DQK31-D) C***KEYWORDS 31-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 Double precision version 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 calling program. C C A - Double precision C Lower limit of integration C C B - Double precision C Upper limit of integration C C ON RETURN C RESULT - Double precision C Approximation to the integral I C RESULT is computed by applying the 31-POINT C GAUSS-KRONROD RULE (RESK), obtained by optimal C addition of abscissae to the 15-POINT GAUSS C RULE (RESG). C C ABSERR - Double precision C Estimate of the modulus of the modulus, C which should not exceed ABS(I-RESULT) C C RESABS - Double precision C Approximation to the integral J C C RESASC - Double precision C Approximation to the integral of ABS(F-I/(B-A)) C over (A,B) C C***REFERENCES (NONE) C***ROUTINES CALLED D1MACH 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 DQK31 DOUBLE PRECISION A,ABSC,ABSERR,B,CENTR,DHLGTH, 1 D1MACH,EPMACH,F,FC,FSUM,FVAL1,FVAL2,FV1,FV2,HLGTH,RESABS,RESASC, 2 RESG,RESK,RESKH,RESULT,UFLOW,WG,WGK,XGK INTEGER J,JTW,JTWM1 EXTERNAL F C DIMENSION FV1(15),FV2(15),XGK(16),WGK(16),WG(8) 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 31-POINT KRONROD RULE C XGK(2), XGK(4), ... ABSCISSAE OF THE 15-POINT C GAUSS RULE C XGK(1), XGK(3), ... ABSCISSAE WHICH ARE OPTIMALLY C ADDED TO THE 15-POINT GAUSS RULE C C WGK - WEIGHTS OF THE 31-POINT KRONROD RULE C C WG - WEIGHTS OF THE 15-POINT GAUSS RULE C C C GAUSS QUADRATURE WEIGHTS AND KRONROD QUADRATURE ABSCISSAE AND WEIGHTS C AS EVALUATED WITH 80 DECIMAL DIGIT ARITHMETIC BY L. W. FULLERTON, C BELL LABS, NOV. 1981. C SAVE WG, XGK, WGK DATA WG ( 1) / 0.0307532419 9611726835 4628393577 204 D0 / DATA WG ( 2) / 0.0703660474 8810812470 9267416450 667 D0 / DATA WG ( 3) / 0.1071592204 6717193501 1869546685 869 D0 / DATA WG ( 4) / 0.1395706779 2615431444 7804794511 028 D0 / DATA WG ( 5) / 0.1662692058 1699393355 3200860481 209 D0 / DATA WG ( 6) / 0.1861610000 1556221102 6800561866 423 D0 / DATA WG ( 7) / 0.1984314853 2711157645 6118326443 839 D0 / DATA WG ( 8) / 0.2025782419 2556127288 0620199967 519 D0 / C DATA XGK ( 1) / 0.9980022986 9339706028 5172840152 271 D0 / DATA XGK ( 2) / 0.9879925180 2048542848 9565718586 613 D0 / DATA XGK ( 3) / 0.9677390756 7913913425 7347978784 337 D0 / DATA XGK ( 4) / 0.9372733924 0070590430 7758947710 209 D0 / DATA XGK ( 5) / 0.8972645323 4408190088 2509656454 496 D0 / DATA XGK ( 6) / 0.8482065834 1042721620 0648320774 217 D0 / DATA XGK ( 7) / 0.7904185014 4246593296 7649294817 947 D0 / DATA XGK ( 8) / 0.7244177313 6017004741 6186054613 938 D0 / DATA XGK ( 9) / 0.6509967412 9741697053 3735895313 275 D0 / DATA XGK ( 10) / 0.5709721726 0853884753 7226737253 911 D0 / DATA XGK ( 11) / 0.4850818636 4023968069 3655740232 351 D0 / DATA XGK ( 12) / 0.3941513470 7756336989 7207370981 045 D0 / DATA XGK ( 13) / 0.2991800071 5316881216 6780024266 389 D0 / DATA XGK ( 14) / 0.2011940939 9743452230 0628303394 596 D0 / DATA XGK ( 15) / 0.1011420669 1871749902 7074231447 392 D0 / DATA XGK ( 16) / 0.0000000000 0000000000 0000000000 000 D0 / C DATA WGK ( 1) / 0.0053774798 7292334898 7792051430 128 D0 / DATA WGK ( 2) / 0.0150079473 2931612253 8374763075 807 D0 / DATA WGK ( 3) / 0.0254608473 2671532018 6874001019 653 D0 / DATA WGK ( 4) / 0.0353463607 9137584622 2037948478 360 D0 / DATA WGK ( 5) / 0.0445897513 2476487660 8227299373 280 D0 / DATA WGK ( 6) / 0.0534815246 9092808726 5343147239 430 D0 / DATA WGK ( 7) / 0.0620095678 0067064028 5139230960 803 D0 / DATA WGK ( 8) / 0.0698541213 1872825870 9520077099 147 D0 / DATA WGK ( 9) / 0.0768496807 5772037889 4432777482 659 D0 / DATA WGK ( 10) / 0.0830805028 2313302103 8289247286 104 D0 / DATA WGK ( 11) / 0.0885644430 5621177064 7275443693 774 D0 / DATA WGK ( 12) / 0.0931265981 7082532122 5486872747 346 D0 / DATA WGK ( 13) / 0.0966427269 8362367850 5179907627 589 D0 / DATA WGK ( 14) / 0.0991735987 2179195933 2393173484 603 D0 / DATA WGK ( 15) / 0.1007698455 2387559504 4946662617 570 D0 / DATA WGK ( 16) / 0.1013300070 1479154901 7374792767 493 D0 / C C C LIST OF MAJOR VARIABLES 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 15-POINT GAUSS FORMULA C RESK - RESULT OF THE 31-POINT KRONROD FORMULA C RESKH - APPROXIMATION TO THE MEAN VALUE OF F OVER (A,B), C I.E. TO I/(B-A) C C MACHINE DEPENDENT CONSTANTS C --------------------------- C EPMACH IS THE LARGEST RELATIVE SPACING. C UFLOW IS THE SMALLEST POSITIVE MAGNITUDE. C***FIRST EXECUTABLE STATEMENT DQK31 EPMACH = D1MACH(4) UFLOW = D1MACH(1) C CENTR = 0.5D+00*(A+B) HLGTH = 0.5D+00*(B-A) DHLGTH = ABS(HLGTH) C C COMPUTE THE 31-POINT KRONROD APPROXIMATION TO C THE INTEGRAL, AND ESTIMATE THE ABSOLUTE ERROR. C FC = F(CENTR) RESG = WG(8)*FC RESK = WGK(16)*FC RESABS = ABS(RESK) DO 10 J=1,7 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 = 1,8 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*0.5D+00 RESASC = WGK(16)*ABS(FC-RESKH) DO 20 J=1,15 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.0.0D+00.AND.ABSERR.NE.0.0D+00) 1 ABSERR = RESASC*MIN(0.1D+01,(0.2D+03*ABSERR/RESASC)**1.5D+00) IF(RESABS.GT.UFLOW/(0.5D+02*EPMACH)) ABSERR = MAX 1 ((EPMACH*0.5D+02)*RESABS,ABSERR) RETURN END