1 subroutine dqng(f,a,b,epsabs,epsrel,result,abserr,neval,ier) 2c***begin prologue dqng 3c***date written 800101 (yymmdd) 4c***revision date 810101 (yymmdd) 5c***category no. h2a1a1 6c***keywords automatic integrator, smooth integrand, 7c non-adaptive, gauss-kronrod(patterson) 8c***author piessens,robert,appl. math. & progr. div. - k.u.leuven 9c de doncker,elise,appl math & progr. div. - k.u.leuven 10c kahaner,david,nbs - modified (2/82) 11c***purpose the routine calculates an approximation result to a 12c given definite integral i = integral of f over (a,b), 13c hopefully satisfying following claim for accuracy 14c abs(i-result).le.max(epsabs,epsrel*abs(i)). 15c***description 16c 17c non-adaptive integration 18c standard fortran subroutine 19c double precision version 20c 21c f - double precision 22c function subprogram defining the integrand function 23c f(x). the actual name for f needs to be declared 24c e x t e r n a l in the driver program. 25c 26c a - double precision 27c lower limit of integration 28c 29c b - double precision 30c upper limit of integration 31c 32c epsabs - double precision 33c absolute accuracy requested 34c epsrel - double precision 35c relative accuracy requested 36c if epsabs.le.0 37c and epsrel.lt.max(50*rel.mach.acc.,0.5d-28), 38c the routine will end with ier = 6. 39c 40c on return 41c result - double precision 42c approximation to the integral i 43c result is obtained by applying the 21-point 44c gauss-kronrod rule (res21) obtained by optimal 45c addition of abscissae to the 10-point gauss rule 46c (res10), or by applying the 43-point rule (res43) 47c obtained by optimal addition of abscissae to the 48c 21-point gauss-kronrod rule, or by applying the 49c 87-point rule (res87) obtained by optimal addition 50c of abscissae to the 43-point rule. 51c 52c abserr - double precision 53c estimate of the modulus of the absolute error, 54c which should equal or exceed abs(i-result) 55c 56c neval - integer 57c number of integrand evaluations 58c 59c ier - ier = 0 normal and reliable termination of the 60c routine. it is assumed that the requested 61c accuracy has been achieved. 62c ier.gt.0 abnormal termination of the routine. it is 63c assumed that the requested accuracy has 64c not been achieved. 65c error messages 66c ier = 1 the maximum number of steps has been 67c executed. the integral is probably too 68c difficult to be calculated by dqng. 69c = 6 the input is invalid, because 70c epsabs.le.0 and 71c epsrel.lt.max(50*rel.mach.acc.,0.5d-28). 72c result, abserr and neval are set to zero. 73c 74c***references (none) 75c***routines called d1mach,xerror 76c***end prologue dqng 77c 78 double precision a,absc,abserr,b,centr,dabs,dhlgth,dmax1,dmin1, 79 * d1mach,epmach,epsabs,epsrel,f,fcentr,fval,fval1,fval2,fv1,fv2, 80 * fv3,fv4,hlgth,result,res10,res21,res43,res87,resabs,resasc, 81 * reskh,savfun,uflow,w10,w21a,w21b,w43a,w43b,w87a,w87b,x1,x2,x3,x4 82 integer ier,ipx,k,l,neval 83 external f 84c 85 dimension fv1(5),fv2(5),fv3(5),fv4(5),x1(5),x2(5),x3(11),x4(22), 86 * w10(5),w21a(5),w21b(6),w43a(10),w43b(12),w87a(21),w87b(23), 87 * savfun(21) 88c 89c the following data statements contain the 90c abscissae and weights of the integration rules used. 91c 92c x1 abscissae common to the 10-, 21-, 43- and 87- 93c point rule 94c x2 abscissae common to the 21-, 43- and 87-point rule 95c x3 abscissae common to the 43- and 87-point rule 96c x4 abscissae of the 87-point rule 97c w10 weights of the 10-point formula 98c w21a weights of the 21-point formula for abscissae x1 99c w21b weights of the 21-point formula for abscissae x2 100c w43a weights of the 43-point formula for abscissae x1, x3 101c w43b weights of the 43-point formula for abscissae x3 102c w87a weights of the 87-point formula for abscissae x1, 103c x2, x3 104c w87b weights of the 87-point formula for abscissae x4 105c 106c 107c gauss-kronrod-patterson quadrature coefficients for use in 108c quadpack routine qng. these coefficients were calculated with 109c 101 decimal digit arithmetic by l. w. fullerton, bell labs, nov 1981. 110c 111 data x1 ( 1) / 0.9739065285 1717172007 7964012084 452 d0 / 112 data x1 ( 2) / 0.8650633666 8898451073 2096688423 493 d0 / 113 data x1 ( 3) / 0.6794095682 9902440623 4327365114 874 d0 / 114 data x1 ( 4) / 0.4333953941 2924719079 9265943165 784 d0 / 115 data x1 ( 5) / 0.1488743389 8163121088 4826001129 720 d0 / 116 data w10 ( 1) / 0.0666713443 0868813759 3568809893 332 d0 / 117 data w10 ( 2) / 0.1494513491 5058059314 5776339657 697 d0 / 118 data w10 ( 3) / 0.2190863625 1598204399 5534934228 163 d0 / 119 data w10 ( 4) / 0.2692667193 0999635509 1226921569 469 d0 / 120 data w10 ( 5) / 0.2955242247 1475287017 3892994651 338 d0 / 121c 122 data x2 ( 1) / 0.9956571630 2580808073 5527280689 003 d0 / 123 data x2 ( 2) / 0.9301574913 5570822600 1207180059 508 d0 / 124 data x2 ( 3) / 0.7808177265 8641689706 3717578345 042 d0 / 125 data x2 ( 4) / 0.5627571346 6860468333 9000099272 694 d0 / 126 data x2 ( 5) / 0.2943928627 0146019813 1126603103 866 d0 / 127 data w21a ( 1) / 0.0325581623 0796472747 8818972459 390 d0 / 128 data w21a ( 2) / 0.0750396748 1091995276 7043140916 190 d0 / 129 data w21a ( 3) / 0.1093871588 0229764189 9210590325 805 d0 / 130 data w21a ( 4) / 0.1347092173 1147332592 8054001771 707 d0 / 131 data w21a ( 5) / 0.1477391049 0133849137 4841515972 068 d0 / 132 data w21b ( 1) / 0.0116946388 6737187427 8064396062 192 d0 / 133 data w21b ( 2) / 0.0547558965 7435199603 1381300244 580 d0 / 134 data w21b ( 3) / 0.0931254545 8369760553 5065465083 366 d0 / 135 data w21b ( 4) / 0.1234919762 6206585107 7958109831 074 d0 / 136 data w21b ( 5) / 0.1427759385 7706008079 7094273138 717 d0 / 137 data w21b ( 6) / 0.1494455540 0291690566 4936468389 821 d0 / 138c 139 data x3 ( 1) / 0.9993333609 0193208139 4099323919 911 d0 / 140 data x3 ( 2) / 0.9874334029 0808886979 5961478381 209 d0 / 141 data x3 ( 3) / 0.9548079348 1426629925 7919200290 473 d0 / 142 data x3 ( 4) / 0.9001486957 4832829362 5099494069 092 d0 / 143 data x3 ( 5) / 0.8251983149 8311415084 7066732588 520 d0 / 144 data x3 ( 6) / 0.7321483889 8930498261 2354848755 461 d0 / 145 data x3 ( 7) / 0.6228479705 3772523864 1159120344 323 d0 / 146 data x3 ( 8) / 0.4994795740 7105649995 2214885499 755 d0 / 147 data x3 ( 9) / 0.3649016613 4658076804 3989548502 644 d0 / 148 data x3 ( 10) / 0.2222549197 7660129649 8260928066 212 d0 / 149 data x3 ( 11) / 0.0746506174 6138332204 3914435796 506 d0 / 150 data w43a ( 1) / 0.0162967342 8966656492 4281974617 663 d0 / 151 data w43a ( 2) / 0.0375228761 2086950146 1613795898 115 d0 / 152 data w43a ( 3) / 0.0546949020 5825544214 7212685465 005 d0 / 153 data w43a ( 4) / 0.0673554146 0947808607 5553166302 174 d0 / 154 data w43a ( 5) / 0.0738701996 3239395343 2140695251 367 d0 / 155 data w43a ( 6) / 0.0057685560 5976979618 4184327908 655 d0 / 156 data w43a ( 7) / 0.0273718905 9324884208 1276069289 151 d0 / 157 data w43a ( 8) / 0.0465608269 1042883074 3339154433 824 d0 / 158 data w43a ( 9) / 0.0617449952 0144256449 6240336030 883 d0 / 159 data w43a ( 10) / 0.0713872672 6869339776 8559114425 516 d0 / 160 data w43b ( 1) / 0.0018444776 4021241410 0389106552 965 d0 / 161 data w43b ( 2) / 0.0107986895 8589165174 0465406741 293 d0 / 162 data w43b ( 3) / 0.0218953638 6779542810 2523123075 149 d0 / 163 data w43b ( 4) / 0.0325974639 7534568944 3882222526 137 d0 / 164 data w43b ( 5) / 0.0421631379 3519181184 7627924327 955 d0 / 165 data w43b ( 6) / 0.0507419396 0018457778 0189020092 084 d0 / 166 data w43b ( 7) / 0.0583793955 4261924837 5475369330 206 d0 / 167 data w43b ( 8) / 0.0647464049 5144588554 4689259517 511 d0 / 168 data w43b ( 9) / 0.0695661979 1235648452 8633315038 405 d0 / 169 data w43b ( 10) / 0.0728244414 7183320815 0939535192 842 d0 / 170 data w43b ( 11) / 0.0745077510 1417511827 3571813842 889 d0 / 171 data w43b ( 12) / 0.0747221475 1740300559 4425168280 423 d0 / 172c 173 data x4 ( 1) / 0.9999029772 6272923449 0529830591 582 d0 / 174 data x4 ( 2) / 0.9979898959 8667874542 7496322365 960 d0 / 175 data x4 ( 3) / 0.9921754978 6068722280 8523352251 425 d0 / 176 data x4 ( 4) / 0.9813581635 7271277357 1916941623 894 d0 / 177 data x4 ( 5) / 0.9650576238 5838461912 8284110607 926 d0 / 178 data x4 ( 6) / 0.9431676131 3367059681 6416634507 426 d0 / 179 data x4 ( 7) / 0.9158064146 8550720959 1826430720 050 d0 / 180 data x4 ( 8) / 0.8832216577 7131650137 2117548744 163 d0 / 181 data x4 ( 9) / 0.8457107484 6241566660 5902011504 855 d0 / 182 data x4 ( 10) / 0.8035576580 3523098278 8739474980 964 d0 / 183 data x4 ( 11) / 0.7570057306 8549555832 8942793432 020 d0 / 184 data x4 ( 12) / 0.7062732097 8732181982 4094274740 840 d0 / 185 data x4 ( 13) / 0.6515894665 0117792253 4422205016 736 d0 / 186 data x4 ( 14) / 0.5932233740 5796108887 5273770349 144 d0 / 187 data x4 ( 15) / 0.5314936059 7083193228 5268948562 671 d0 / 188 data x4 ( 16) / 0.4667636230 4202284487 1966781659 270 d0 / 189 data x4 ( 17) / 0.3994248478 5921880473 2101665817 923 d0 / 190 data x4 ( 18) / 0.3298748771 0618828826 5053371824 597 d0 / 191 data x4 ( 19) / 0.2585035592 0216155180 2280975429 025 d0 / 192 data x4 ( 20) / 0.1856953965 6834665201 5917141167 606 d0 / 193 data x4 ( 21) / 0.1118422131 7990746817 2398359241 362 d0 / 194 data x4 ( 22) / 0.0373521233 9461987081 4998165437 704 d0 / 195 data w87a ( 1) / 0.0081483773 8414917290 0002878448 190 d0 / 196 data w87a ( 2) / 0.0187614382 0156282224 3935059003 794 d0 / 197 data w87a ( 3) / 0.0273474510 5005228616 1582829741 283 d0 / 198 data w87a ( 4) / 0.0336777073 1163793004 6581056957 588 d0 / 199 data w87a ( 5) / 0.0369350998 2042790761 4589586742 499 d0 / 200 data w87a ( 6) / 0.0028848724 3021153050 1334156248 695 d0 / 201 data w87a ( 7) / 0.0136859460 2271270188 8950035273 128 d0 / 202 data w87a ( 8) / 0.0232804135 0288831112 3409291030 404 d0 / 203 data w87a ( 9) / 0.0308724976 1171335867 5466394126 442 d0 / 204 data w87a ( 10) / 0.0356936336 3941877071 9351355457 044 d0 / 205 data w87a ( 11) / 0.0009152833 4520224136 0843392549 948 d0 / 206 data w87a ( 12) / 0.0053992802 1930047136 7738743391 053 d0 / 207 data w87a ( 13) / 0.0109476796 0111893113 4327826856 808 d0 / 208 data w87a ( 14) / 0.0162987316 9678733526 2665703223 280 d0 / 209 data w87a ( 15) / 0.0210815688 8920383511 2433060188 190 d0 / 210 data w87a ( 16) / 0.0253709697 6925382724 3467999831 710 d0 / 211 data w87a ( 17) / 0.0291896977 5647575250 1446154084 920 d0 / 212 data w87a ( 18) / 0.0323732024 6720278968 5788194889 595 d0 / 213 data w87a ( 19) / 0.0347830989 5036514275 0781997949 596 d0 / 214 data w87a ( 20) / 0.0364122207 3135178756 2801163687 577 d0 / 215 data w87a ( 21) / 0.0372538755 0304770853 9592001191 226 d0 / 216 data w87b ( 1) / 0.0002741455 6376207235 0016527092 881 d0 / 217 data w87b ( 2) / 0.0018071241 5505794294 8341311753 254 d0 / 218 data w87b ( 3) / 0.0040968692 8275916486 4458070683 480 d0 / 219 data w87b ( 4) / 0.0067582900 5184737869 9816577897 424 d0 / 220 data w87b ( 5) / 0.0095499576 7220164653 6053581325 377 d0 / 221 data w87b ( 6) / 0.0123294476 5224485369 4626639963 780 d0 / 222 data w87b ( 7) / 0.0150104473 4638895237 6697286041 943 d0 / 223 data w87b ( 8) / 0.0175489679 8624319109 9665352925 900 d0 / 224 data w87b ( 9) / 0.0199380377 8644088820 2278192730 714 d0 / 225 data w87b ( 10) / 0.0221949359 6101228679 6332102959 499 d0 / 226 data w87b ( 11) / 0.0243391471 2600080547 0360647041 454 d0 / 227 data w87b ( 12) / 0.0263745054 1483920724 1503786552 615 d0 / 228 data w87b ( 13) / 0.0282869107 8877120065 9968002987 960 d0 / 229 data w87b ( 14) / 0.0300525811 2809269532 2521110347 341 d0 / 230 data w87b ( 15) / 0.0316467513 7143992940 4586051078 883 d0 / 231 data w87b ( 16) / 0.0330504134 1997850329 0785944862 689 d0 / 232 data w87b ( 17) / 0.0342550997 0422606178 7082821046 821 d0 / 233 data w87b ( 18) / 0.0352624126 6015668103 3782717998 428 d0 / 234 data w87b ( 19) / 0.0360769896 2288870118 5500318003 895 d0 / 235 data w87b ( 20) / 0.0366986044 9845609449 8018047441 094 d0 / 236 data w87b ( 21) / 0.0371205492 6983257611 4119958413 599 d0 / 237 data w87b ( 22) / 0.0373342287 5193504032 1235449094 698 d0 / 238 data w87b ( 23) / 0.0373610737 6267902341 0321241766 599 d0 / 239c 240c list of major variables 241c ----------------------- 242c 243c centr - mid point of the integration interval 244c hlgth - half-length of the integration interval 245c fcentr - function value at mid point 246c absc - abscissa 247c fval - function value 248c savfun - array of function values which have already been 249c computed 250c res10 - 10-point gauss result 251c res21 - 21-point kronrod result 252c res43 - 43-point result 253c res87 - 87-point result 254c resabs - approximation to the integral of abs(f) 255c resasc - approximation to the integral of abs(f-i/(b-a)) 256c 257c machine dependent constants 258c --------------------------- 259c 260c epmach is the largest relative spacing. 261c uflow is the smallest positive magnitude. 262c 263c***first executable statement dqng 264 epmach = d1mach(4) 265 uflow = d1mach(1) 266c 267c test on validity of parameters 268c ------------------------------ 269c 270 result = 0.0d+00 271 abserr = 0.0d+00 272 neval = 0 273 ier = 6 274 if(epsabs.le.0.0d+00.and.epsrel.lt.dmax1(0.5d+02*epmach,0.5d-28)) 275 * go to 80 276 hlgth = 0.5d+00*(b-a) 277 dhlgth = dabs(hlgth) 278 centr = 0.5d+00*(b+a) 279 fcentr = f(centr) 280 neval = 21 281 ier = 1 282c 283c compute the integral using the 10- and 21-point formula. 284c 285 do 70 l = 1,3 286 go to (5,25,45),l 287 5 res10 = 0.0d+00 288 res21 = w21b(6)*fcentr 289 resabs = w21b(6)*dabs(fcentr) 290 do 10 k=1,5 291 absc = hlgth*x1(k) 292 fval1 = f(centr+absc) 293 fval2 = f(centr-absc) 294 fval = fval1+fval2 295 res10 = res10+w10(k)*fval 296 res21 = res21+w21a(k)*fval 297 resabs = resabs+w21a(k)*(dabs(fval1)+dabs(fval2)) 298 savfun(k) = fval 299 fv1(k) = fval1 300 fv2(k) = fval2 301 10 continue 302 ipx = 5 303 do 15 k=1,5 304 ipx = ipx+1 305 absc = hlgth*x2(k) 306 fval1 = f(centr+absc) 307 fval2 = f(centr-absc) 308 fval = fval1+fval2 309 res21 = res21+w21b(k)*fval 310 resabs = resabs+w21b(k)*(dabs(fval1)+dabs(fval2)) 311 savfun(ipx) = fval 312 fv3(k) = fval1 313 fv4(k) = fval2 314 15 continue 315c 316c test for convergence. 317c 318 result = res21*hlgth 319 resabs = resabs*dhlgth 320 reskh = 0.5d+00*res21 321 resasc = w21b(6)*dabs(fcentr-reskh) 322 do 20 k = 1,5 323 resasc = resasc+w21a(k)*(dabs(fv1(k)-reskh)+dabs(fv2(k)-reskh)) 324 * +w21b(k)*(dabs(fv3(k)-reskh)+dabs(fv4(k)-reskh)) 325 20 continue 326 abserr = dabs((res21-res10)*hlgth) 327 resasc = resasc*dhlgth 328 go to 65 329c 330c compute the integral using the 43-point formula. 331c 332 25 res43 = w43b(12)*fcentr 333 neval = 43 334 do 30 k=1,10 335 res43 = res43+savfun(k)*w43a(k) 336 30 continue 337 do 40 k=1,11 338 ipx = ipx+1 339 absc = hlgth*x3(k) 340 fval = f(absc+centr)+f(centr-absc) 341 res43 = res43+fval*w43b(k) 342 savfun(ipx) = fval 343 40 continue 344c 345c test for convergence. 346c 347 result = res43*hlgth 348 abserr = dabs((res43-res21)*hlgth) 349 go to 65 350c 351c compute the integral using the 87-point formula. 352c 353 45 res87 = w87b(23)*fcentr 354 neval = 87 355 do 50 k=1,21 356 res87 = res87+savfun(k)*w87a(k) 357 50 continue 358 do 60 k=1,22 359 absc = hlgth*x4(k) 360 res87 = res87+w87b(k)*(f(absc+centr)+f(centr-absc)) 361 60 continue 362 result = res87*hlgth 363 abserr = dabs((res87-res43)*hlgth) 364 65 if(resasc.ne.0.0d+00.and.abserr.ne.0.0d+00) 365 * abserr = resasc*dmin1(0.1d+01,(0.2d+03*abserr/resasc)**1.5d+00) 366 if (resabs.gt.uflow/(0.5d+02*epmach)) abserr = dmax1 367 * ((epmach*0.5d+02)*resabs,abserr) 368 if (abserr.le.dmax1(epsabs,epsrel*dabs(result))) ier = 0 369c ***jump out of do-loop 370 if (ier.eq.0) go to 999 371 70 continue 372 80 call xerror('abnormal return from dqng ',26,ier,0) 373 999 return 374 end 375