src/numerical/slatec/fortran/dqk21.f

   1 *DECK DQK21
   2       SUBROUTINE DQK21 (F, A, B, RESULT, ABSERR, RESABS, RESASC)
   3 C***BEGIN PROLOGUE  DQK21
   4 C***PURPOSE  To compute I = Integral of F over (A,B), with error
   5 C                           estimate
   6 C                       J = Integral of ABS(F) over (A,B)
   7 C***LIBRARY   SLATEC (QUADPACK)
   8 C***CATEGORY  H2A1A2
   9 C***TYPE      DOUBLE PRECISION (QK21-S, DQK21-D)
  10 C***KEYWORDS  21-POINT GAUSS-KRONROD RULES, QUADPACK, QUADRATURE
  11 C***AUTHOR  Piessens, Robert
  12 C             Applied Mathematics and Programming Division
  13 C             K. U. Leuven
  14 C           de Doncker, Elise
  15 C             Applied Mathematics and Programming Division
  16 C             K. U. Leuven
  17 C***DESCRIPTION
  18 C
  19 C           Integration rules
  20 C           Standard fortran subroutine
  21 C           Double precision version
  22 C
  23 C           PARAMETERS
  24 C            ON ENTRY
  25 C              F      - Double precision
  26 C                       Function subprogram defining the integrand
  27 C                       FUNCTION F(X). The actual name for F needs to be
  28 C                       Declared E X T E R N A L in the driver program.
  29 C
  30 C              A      - Double precision
  31 C                       Lower limit of integration
  32 C
  33 C              B      - Double precision
  34 C                       Upper limit of integration
  35 C
  36 C            ON RETURN
  37 C              RESULT - Double precision
  38 C                       Approximation to the integral I
  39 C                       RESULT is computed by applying the 21-POINT
  40 C                       KRONROD RULE (RESK) obtained by optimal addition
  41 C                       of abscissae to the 10-POINT GAUSS RULE (RESG).
  42 C
  43 C              ABSERR - Double precision
  44 C                       Estimate of the modulus of the absolute error,
  45 C                       which should not exceed ABS(I-RESULT)
  46 C
  47 C              RESABS - Double precision
  48 C                       Approximation to the integral J
  49 C
  50 C              RESASC - Double precision
  51 C                       Approximation to the integral of ABS(F-I/(B-A))
  52 C                       over (A,B)
  53 C
  54 C***REFERENCES  (NONE)
  55 C***ROUTINES CALLED  D1MACH
  56 C***REVISION HISTORY  (YYMMDD)
  57 C   800101  DATE WRITTEN
  58 C   890531  Changed all specific intrinsics to generic.  (WRB)
  59 C   890531  REVISION DATE from Version 3.2
  60 C   891214  Prologue converted to Version 4.0 format.  (BAB)
  61 C***END PROLOGUE  DQK21
  62 C
  63       DOUBLE PRECISION A,ABSC,ABSERR,B,CENTR,DHLGTH,
  64      1  D1MACH,EPMACH,F,FC,FSUM,FVAL1,FVAL2,FV1,FV2,HLGTH,RESABS,RESASC,
  65      2  RESG,RESK,RESKH,RESULT,UFLOW,WG,WGK,XGK
  66       INTEGER J,JTW,JTWM1
  67       EXTERNAL F
  68 C
  69       DIMENSION FV1(10),FV2(10),WG(5),WGK(11),XGK(11)
  70 C
  71 C           THE ABSCISSAE AND WEIGHTS ARE GIVEN FOR THE INTERVAL (-1,1).
  72 C           BECAUSE OF SYMMETRY ONLY THE POSITIVE ABSCISSAE AND THEIR
  73 C           CORRESPONDING WEIGHTS ARE GIVEN.
  74 C
  75 C           XGK    - ABSCISSAE OF THE 21-POINT KRONROD RULE
  76 C                    XGK(2), XGK(4), ...  ABSCISSAE OF THE 10-POINT
  77 C                    GAUSS RULE
  78 C                    XGK(1), XGK(3), ...  ABSCISSAE WHICH ARE OPTIMALLY
  79 C                    ADDED TO THE 10-POINT GAUSS RULE
  80 C
  81 C           WGK    - WEIGHTS OF THE 21-POINT KRONROD RULE
  82 C
  83 C           WG     - WEIGHTS OF THE 10-POINT GAUSS RULE
  84 C
  85 C
  86 C GAUSS QUADRATURE WEIGHTS AND KRONROD QUADRATURE ABSCISSAE AND WEIGHTS
  87 C AS EVALUATED WITH 80 DECIMAL DIGIT ARITHMETIC BY L. W. FULLERTON,
  88 C BELL LABS, NOV. 1981.
  89 C
  90       SAVE WG, XGK, WGK
  91       DATA WG  (  1) / 0.0666713443 0868813759 3568809893 332 D0 /
  92       DATA WG  (  2) / 0.1494513491 5058059314 5776339657 697 D0 /
  93       DATA WG  (  3) / 0.2190863625 1598204399 5534934228 163 D0 /
  94       DATA WG  (  4) / 0.2692667193 0999635509 1226921569 469 D0 /
  95       DATA WG  (  5) / 0.2955242247 1475287017 3892994651 338 D0 /
  96 C
  97       DATA XGK (  1) / 0.9956571630 2580808073 5527280689 003 D0 /
  98       DATA XGK (  2) / 0.9739065285 1717172007 7964012084 452 D0 /
  99       DATA XGK (  3) / 0.9301574913 5570822600 1207180059 508 D0 /
 100       DATA XGK (  4) / 0.8650633666 8898451073 2096688423 493 D0 /
 101       DATA XGK (  5) / 0.7808177265 8641689706 3717578345 042 D0 /
 102       DATA XGK (  6) / 0.6794095682 9902440623 4327365114 874 D0 /
 103       DATA XGK (  7) / 0.5627571346 6860468333 9000099272 694 D0 /
 104       DATA XGK (  8) / 0.4333953941 2924719079 9265943165 784 D0 /
 105       DATA XGK (  9) / 0.2943928627 0146019813 1126603103 866 D0 /
 106       DATA XGK ( 10) / 0.1488743389 8163121088 4826001129 720 D0 /
 107       DATA XGK ( 11) / 0.0000000000 0000000000 0000000000 000 D0 /
 108 C
 109       DATA WGK (  1) / 0.0116946388 6737187427 8064396062 192 D0 /
 110       DATA WGK (  2) / 0.0325581623 0796472747 8818972459 390 D0 /
 111       DATA WGK (  3) / 0.0547558965 7435199603 1381300244 580 D0 /
 112       DATA WGK (  4) / 0.0750396748 1091995276 7043140916 190 D0 /
 113       DATA WGK (  5) / 0.0931254545 8369760553 5065465083 366 D0 /
 114       DATA WGK (  6) / 0.1093871588 0229764189 9210590325 805 D0 /
 115       DATA WGK (  7) / 0.1234919762 6206585107 7958109831 074 D0 /
 116       DATA WGK (  8) / 0.1347092173 1147332592 8054001771 707 D0 /
 117       DATA WGK (  9) / 0.1427759385 7706008079 7094273138 717 D0 /
 118       DATA WGK ( 10) / 0.1477391049 0133849137 4841515972 068 D0 /
 119       DATA WGK ( 11) / 0.1494455540 0291690566 4936468389 821 D0 /
 120 C
 121 C
 122 C           LIST OF MAJOR VARIABLES
 123 C           -----------------------
 124 C
 125 C           CENTR  - MID POINT OF THE INTERVAL
 126 C           HLGTH  - HALF-LENGTH OF THE INTERVAL
 127 C           ABSC   - ABSCISSA
 128 C           FVAL*  - FUNCTION VALUE
 129 C           RESG   - RESULT OF THE 10-POINT GAUSS FORMULA
 130 C           RESK   - RESULT OF THE 21-POINT KRONROD FORMULA
 131 C           RESKH  - APPROXIMATION TO THE MEAN VALUE OF F OVER (A,B),
 132 C                    I.E. TO I/(B-A)
 133 C
 134 C
 135 C           MACHINE DEPENDENT CONSTANTS
 136 C           ---------------------------
 137 C
 138 C           EPMACH IS THE LARGEST RELATIVE SPACING.
 139 C           UFLOW IS THE SMALLEST POSITIVE MAGNITUDE.
 140 C
 141 C***FIRST EXECUTABLE STATEMENT  DQK21
 142       EPMACH = D1MACH(4)
 143       UFLOW = D1MACH(1)
 144 C
 145       CENTR = 0.5D+00*(A+B)
 146       HLGTH = 0.5D+00*(B-A)
 147       DHLGTH = ABS(HLGTH)
 148 C
 149 C           COMPUTE THE 21-POINT KRONROD APPROXIMATION TO
 150 C           THE INTEGRAL, AND ESTIMATE THE ABSOLUTE ERROR.
 151 C
 152       RESG = 0.0D+00
 153       FC = F(CENTR)
 154       RESK = WGK(11)*FC
 155       RESABS = ABS(RESK)
 156       DO 10 J=1,5
 157         JTW = 2*J
 158         ABSC = HLGTH*XGK(JTW)
 159         FVAL1 = F(CENTR-ABSC)
 160         FVAL2 = F(CENTR+ABSC)
 161         FV1(JTW) = FVAL1
 162         FV2(JTW) = FVAL2
 163         FSUM = FVAL1+FVAL2
 164         RESG = RESG+WG(J)*FSUM
 165         RESK = RESK+WGK(JTW)*FSUM
 166         RESABS = RESABS+WGK(JTW)*(ABS(FVAL1)+ABS(FVAL2))
 167    10 CONTINUE
 168       DO 15 J = 1,5
 169         JTWM1 = 2*J-1
 170         ABSC = HLGTH*XGK(JTWM1)
 171         FVAL1 = F(CENTR-ABSC)
 172         FVAL2 = F(CENTR+ABSC)
 173         FV1(JTWM1) = FVAL1
 174         FV2(JTWM1) = FVAL2
 175         FSUM = FVAL1+FVAL2
 176         RESK = RESK+WGK(JTWM1)*FSUM
 177         RESABS = RESABS+WGK(JTWM1)*(ABS(FVAL1)+ABS(FVAL2))
 178    15 CONTINUE
 179       RESKH = RESK*0.5D+00
 180       RESASC = WGK(11)*ABS(FC-RESKH)
 181       DO 20 J=1,10
 182         RESASC = RESASC+WGK(J)*(ABS(FV1(J)-RESKH)+ABS(FV2(J)-RESKH))
 183    20 CONTINUE
 184       RESULT = RESK*HLGTH
 185       RESABS = RESABS*DHLGTH
 186       RESASC = RESASC*DHLGTH
 187       ABSERR = ABS((RESK-RESG)*HLGTH)
 188       IF(RESASC.NE.0.0D+00.AND.ABSERR.NE.0.0D+00)
 189      1  ABSERR = RESASC*MIN(0.1D+01,(0.2D+03*ABSERR/RESASC)**1.5D+00)
 190       IF(RESABS.GT.UFLOW/(0.5D+02*EPMACH)) ABSERR = MAX
 191      1  ((EPMACH*0.5D+02)*RESABS,ABSERR)
 192       RETURN
 193       END