util.F 26.9 KB
 Markus Betzinger committed Apr 26, 2016 1 2 3 4 5 6 7 8 9 10 11 12 `````` MODULE m_util USE m_juDFT c error and warning codes for intgrf function INTEGER, PARAMETER :: NO_ERROR = 0 INTEGER, PARAMETER :: NEGATIVE_EXPONENT_WARNING = 1 INTEGER, PARAMETER :: NEGATIVE_EXPONENT_ERROR = 2 c return type of the pure intgrf function TYPE :: intgrf_out REAL :: value ! value of the integration INTEGER :: ierror ! error code END TYPE intgrf_out `````` Daniel Wortmann committed Jun 21, 2017 13 `````` INTERFACE derivative `````` Daniel Wortmann committed Jul 14, 2017 14 `````` MODULE PROCEDURE derivative_t,derivative_nt `````` Daniel Wortmann committed Jun 21, 2017 15 16 `````` END INTERFACE `````` Markus Betzinger committed Apr 26, 2016 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 `````` CONTAINS c Calculates Gaunt coefficients, i.e. the integrals of three spherical harmonics c integral ( conjg(Y(l1,m1)) * Y(l2,m2) * conjg(Y(l3,m3)) ) c They are also the coefficients C(l1,l2,l3,m1,m2,m3) in c conjg(Y(l1,m1)) * Y(l2,m2) = sum(l3,m3) C(l1,l2,l3,m1,m2,m3) Y(l3,m3) c fac contains factorial up to maxfac, i.e. fac(i)= i! C sfac contains square root of fac, i.e. sfac(i)= sqrt(i!) FUNCTION gaunt(l1,l2,l3,m1,m2,m3,maxfac,fac,sfac) USE m_constants ,ONLY: pimach IMPLICIT NONE REAL :: gaunt INTEGER, INTENT(IN) :: l1,l2,l3,m1,m2,m3,maxfac REAL , INTENT(IN) :: fac(0:maxfac) REAL , INTENT(IN) :: sfac(0:maxfac) gaunt = 0 IF(m3.ne.m2-m1) RETURN IF(abs(m1).gt.l1) RETURN IF(abs(m2).gt.l2) RETURN IF(abs(m3).gt.l3) RETURN IF(l3.lt.abs(l1-l2).or.l3.gt.l1+l2) RETURN gaunt = (-1)**(m1+m3) * & sqrt((2*l1+1)*(2*l2+1)*(2*l3+1)/pimach()/4)* & wigner3j(l1,l2,l3,-m1,m2,-m3,maxfac,fac,sfac)* & wigner3j(l1,l2,l3, 0, 0, 0,maxfac,fac,sfac) END FUNCTION gaunt c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c Calculates the Wigner 3j symbols using Racah's formula FUNCTION wigner3j(l1,l2,l3,m1,m2,m3,maxfac,fac,sfac) IMPLICIT NONE REAL :: wigner3j INTEGER, INTENT(IN) :: l1,l2,l3,m1,m2,m3,maxfac REAL , INTENT(IN) :: fac(0:maxfac) REAL , INTENT(IN) :: sfac(0:maxfac) c - local - INTEGER :: tmin,tmax,t,f1,f2,f3,f4,f5 wigner3j = 0 c The following IF clauses should be in the calling routine and commented here. c if(-m3.ne.m1+m2) return c if(abs(m1).gt.l1) return c if(abs(m2).gt.l2) return c if(abs(m3).gt.l3) return c if(l3.lt.abs(l1-l2).or.l3.gt.l1+l2) return f1 = l3-l2+m1 f2 = l3-l1-m2 f3 = l1+l2-l3 f4 = l1-m1 f5 = l2+m2 tmin = max(0,-f1,-f2) ! The arguments to fac (see below) tmax = min(f3,f4,f5) ! must not be negative. ! The following line is only for testing and should be removed at a later time. IF(tmax-tmin .ne. min(l1+m1,l1-m1,l2+m2,l2-m2,l3+m3,l3-m3, & l1+l2-l3,l1-l2+l3,-l1+l2+l3)) & STOP 'wigner3j: Number of terms incorrect.' IF(tmin.le.tmax) THEN DO t = tmin,tmax wigner3j = wigner3j + (-1)**t / & ( fac(t) * fac(f1+t) * fac(f2+t) & *fac(f3-t) * fac(f4-t) * fac(f5-t) ) END DO wigner3j = wigner3j * (-1)**(l1-l2-m3) * sfac(l1+l2-l3) & * sfac(l1-l2+l3) * sfac(-l1+l2+l3) & / sfac(l1+l2+l3+1) * & sfac(l1+m1) * sfac(l1-m1) * & sfac(l2+m2) * sfac(l2-m2) * & sfac(l3+m3) * sfac(l3-m3) END IF END FUNCTION wigner3j c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c Integrates function f numerically (Lagrange and Simpson integration) c on grid(itype) and is much faster than intgr. c (Only normal outward integration.) c Before first use of this function it has to be initialized with c intgrf_init. FUNCTION intgrf(f,jri,jmtd,rmsh,dx,ntype,itype,gridf) IMPLICIT NONE REAL :: intgrf INTEGER, INTENT(IN) :: itype,ntype,jmtd INTEGER, INTENT(IN) :: jri(ntype) REAL , INTENT(IN) :: dx(ntype),rmsh(jmtd,ntype) REAL , INTENT(IN) :: gridf(jmtd,ntype) REAL , INTENT(IN) :: f(*) c - local - TYPE(intgrf_out) :: fct_res fct_res = pure_intgrf(f,jri,jmtd,rmsh,dx,ntype,itype,gridf) IF (fct_res%ierror == NEGATIVE_EXPONENT_WARNING) THEN write(6,*) 'intgrf: Warning!'// + 'Negative exponent x in extrapolation a+c*r**x' ELSEIF (fct_res%ierror == NEGATIVE_EXPONENT_ERROR) THEN write(6,*) + 'intgrf: Negative exponent x in extrapolation a+c*r**x' CALL juDFT_error( + 'intgrf: Negative exponent x in extrapolation a+c*r**x') END IF intgrf = fct_res%value END FUNCTION intgrf c pure wrapper for intgrf with same functionality c can be used within forall loops PURE FUNCTION pure_intgrf(f,jri,jmtd,rmsh,dx,ntype,itype,gridf) IMPLICIT NONE TYPE(intgrf_out) :: pure_intgrf INTEGER, INTENT(IN) :: itype,ntype,jmtd INTEGER, INTENT(IN) :: jri(ntype) REAL , INTENT(IN) :: dx(ntype),rmsh(jmtd,ntype) REAL , INTENT(IN) :: gridf(jmtd,ntype) REAL , INTENT(IN) :: f(*) c - local - INTEGER :: n REAL :: r1,h,a,x n = jri(itype) r1 = rmsh(1,itype) h = dx(itype) pure_intgrf%ierror = NO_ERROR ! integral from 0 to r1 approximated by leading term in power series expansion `````` Matthias Redies committed Jul 15, 2019 164 `````` IF (f(1)*f(2).gt.1e-10.and.h.gt.0) THEN `````` Markus Betzinger committed Apr 26, 2016 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 `````` IF(f(2).eq.f(1)) THEN pure_intgrf%value = r1*f(1) ELSE x = (f(3)-f(2)) / (f(2)-f(1)) a = (f(2)-x*f(1)) / (1-x) x = log(x)/h IF(x.lt.0) THEN IF(x.gt.-1) THEN pure_intgrf%ierror = NEGATIVE_EXPONENT_WARNING ELSE IF(x.le.-1) THEN pure_intgrf%ierror = NEGATIVE_EXPONENT_ERROR RETURN END IF END IF pure_intgrf%value = r1*(f(1)+x*a) / (x+1) ! x = f(2) / f(1) ! x = log(x)/h ! IF(x.lt.0) THEN ! IF(x.gt.-1) write(6,'(A,ES9.1)') 'intgrf: Warning!& ! & Negative exponent x in& ! & extrapolation c*r**x:',x ! IF(x.le.-1) write(6,'(A,ES9.1)') 'intgrf: Negative exponent& ! & x in extrapolation& ! & c*r**x:',x ! IF(x.le.-1) STOP 'intgrf: Negative exponent& ! & x in extrapolation & ! & c*r**x' ! END IF ! intgrf = (r1*f(1))/(x+1) END IF ELSE pure_intgrf%value = 0 END IF ! integrate from r(1) to r(n) by multiplying with gridf pure_intgrf%value = pure_intgrf%value + + dot_product(gridf(:n,itype),f(:n)) END FUNCTION pure_intgrf c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c Initializes fast numerical integration intgrf (see below). SUBROUTINE intgrf_init(ntype,jmtd,jri,dx,rmsh,gridf) IMPLICIT NONE INTEGER, INTENT(IN) :: ntype,jmtd INTEGER, INTENT(IN) :: jri(ntype) REAL, INTENT(IN) :: dx(ntype),rmsh(jmtd,ntype) `````` Daniel Wortmann committed Jul 11, 2017 220 `````` REAL, INTENT(OUT), ALLOCATABLE :: gridf(:,:) `````` Markus Betzinger committed Apr 26, 2016 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 `````` c - local - INTEGER :: i,j,itype INTEGER :: n,nstep,n0 = 6 INTEGER, PARAMETER :: simpson(7) = (/41,216,27,272,27,216,41/) REAL :: r1,h,dr REAL :: r(7) REAL, PARAMETER :: lagrange(7,6)= reshape( & (/19087.,65112.,-46461., 37504.,-20211., 6312., -863., & -863.,25128., 46989.,-16256., 7299.,-2088., 271., & 271.,-2760., 30819., 37504., -6771., 1608., -191., & -191., 1608., -6771., 37504., 30819.,-2760., 271., & 271.,-2088., 7299.,-16256., 46989.,25128., -863., & -863., 6312.,-20211., 37504.,-46461.,65112.,19087./), ! The last row is actually never used. & (/7,6/) ) n = jmtd ALLOCATE ( gridf(n,ntype) ) gridf = 0 DO itype = 1,ntype n = jri(itype) r1 = rmsh(1,itype) h = dx(itype) nstep = (n-1)/6 n0 = n-6*nstep dr = exp(h) ! Calculate Lagrange-integration coefficients from r(1) to r(n0) r(1)=r1 IF(n0.gt.1) THEN DO i=2,7 r(i) = r(i-1)*dr END DO DO i=1,7 gridf(i,itype) = h/60480 * r(i) * sum(lagrange(i,1:n0-1)) END DO r(1) = r(n0) END IF ! Calculate Simpson-integration coefficients from r(n0) to r(n) DO i = 1,nstep DO j = 2,7 r(j) = r(j-1)*dr END DO DO j = n0,n0+6 gridf(j,itype) = gridf(j,itype) + h/140 * r(j-n0+1) * & simpson(j-n0+1) END DO n0 = n0 + 6 r(1) = r(7) END DO END DO END SUBROUTINE intgrf_init c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c Calculates the primitive of f, on grid(itypein): c c r c primf(r) = integral f(r') dr' ( r = grid point ) c 0 c c If itypein is negative, the primitive c c R c primf(r) = integral f(r') dr' ( R = MT sphere radius ) c r c c is calculated instead. c c ----------------------------- c Fast calculation of primitive. c Only Lagrange integration is used SUBROUTINE primitivef(primf,fin,rmsh,dx,jri,jmtd,itypein,ntype) IMPLICIT NONE c - scalars - INTEGER, INTENT(IN) :: itypein,jmtd,ntype c - arrays - INTEGER, INTENT(IN) :: jri(ntype) REAL, INTENT(OUT) :: primf( jri( abs(itypein) ) ) REAL, INTENT(IN) :: fin( jri( abs(itypein) ) ) REAL, INTENT(IN) :: rmsh(jmtd,ntype),dx(ntype) c - local scalars - INTEGER :: itype,n,i,n0 REAL :: h,x,h1 REAL :: intgr,r1,a,dr c - local arrays - REAL :: fr(7) REAL :: f( jri( abs(itypein) )) REAL :: r( jri( abs(itypein) ) ) REAL, PARAMETER :: lagrange(7,6)= reshape( & (/19087.,65112.,-46461., 37504.,-20211., 6312., -863., & -863.,25128., 46989.,-16256., 7299.,-2088., 271., & 271.,-2760., 30819., 37504., -6771., 1608., -191., & -191., 1608., -6771., 37504., 30819.,-2760., 271., & 271.,-2088., 7299.,-16256., 46989.,25128., -863., & -863., 6312.,-20211., 37504.,-46461.,65112.,19087./), & (/7,6/) ) itype = abs(itypein) primf = 0 n = jri(itype) h = dx(itype) IF(itypein.gt.0) THEN r1 = rmsh(1,itype) ! perform outward integration f = fin ! (from 0 to r) ELSE r1 = rmsh(jri(itype),itype) ! perform inward integration h = -h ! (from MT sphere radius to r) f = fin(n:1:-1) ! END IF ! integral from 0 to r1 approximated by leading term in power series expansion (only if h>0) `````` Matthias Redies committed Jul 15, 2019 352 `````` IF(h.gt.0.and.f(1)*f(2).gt.1e-10) THEN `````` Markus Betzinger committed Apr 26, 2016 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 `````` IF(f(2).eq.f(1)) THEN intgr = r1*f(1) ELSE x = (f(3)-f(2))/(f(2)-f(1)) a = (f(2)-x*f(1)) / (1-x) x = log(x)/h IF(x.lt.0) THEN IF(x>-1) WRITE(6,'(A,ES9.1)') + '+intgr: Warning! Negative &exponent x in'// + 'extrapolation a+c*r**x:',x IF(x<=-1) WRITE(6,'(A,ES9.1)')'intgr: Negative exponent,'// + 'x in extrapolation a+c*r**x:',x IF(x<=-1) STOP 'intgr:Negative exponent x in extrapolation' END IF intgr = r1*(f(1)+x*a) / (x+1) END IF ELSE intgr = 0 END IF primf(1) = intgr dr = exp(h) r(1) = r1 n0 = 1 h1 = h/60480 ! Lagrange integration from r(n0) to r(n0+5) 1 DO i=2,7 r(i) = r(i-1)*dr END DO fr = f(n0:n0+6) * r(:7) DO i=1,6 intgr = intgr + h1 * dot_product(lagrange(:,i),fr) IF(primf(n0+i).eq.0) primf(n0+i) = intgr ! avoid double-definition END DO IF(n0+12.le.n) THEN r(1) = r(7) n0 = n0 + 6 GOTO 1 ELSE IF(n0+6.lt.n) THEN r(1) = r(n-5-n0) n0 = n-6 intgr = primf(n-6) GOTO 1 END IF IF(itypein.lt.0) THEN ! primf = -primf(n:1:-1) ! Inward integration END IF ! END SUBROUTINE primitivef c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - !function modulo1 maps kpoint into first BZ FUNCTION modulo1(kpoint,nkpt3) IMPLICIT NONE INTEGER,INTENT(IN) :: nkpt3(3) REAL ,INTENT(IN) :: kpoint(3) REAL :: modulo1(3) INTEGER :: help(3) modulo1 = kpoint*nkpt3 help = nint(modulo1) `````` Matthias Redies committed Jul 15, 2019 419 `````` IF(any(abs(help-modulo1).gt.1e-10)) THEN `````` Markus Betzinger committed Apr 26, 2016 420 421 `````` WRITE(6,'(A,F5.3,2('','',F5.3),A)')'modulo1: argument (',kpoint, + ') is not an element of the k-point set.' `````` Gregor Michalicek committed Sep 21, 2017 422 423 424 `````` CALL juDFT_error( + 'modulo1: argument not an element of k-point set.', + calledby = 'util:modulo1') `````` Markus Betzinger committed Apr 26, 2016 425 `````` END IF `````` Matthias Redies committed Jul 15, 2019 426 `````` modulo1 = modulo(help,nkpt3)*1.0/nkpt3 `````` Markus Betzinger committed Apr 26, 2016 427 428 429 430 431 432 433 434 435 `````` END FUNCTION modulo1 c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c Returns derivative of f in df. `````` Daniel Wortmann committed Jun 21, 2017 436 437 438 439 440 441 442 443 444 445 446 447 448 `````` SUBROUTINE derivative_t(df,f,atoms,itype) USE m_types IMPLICIT NONE REAL, INTENT(IN) :: f(:) REAL, INTENT(OUT) :: df(:) TYPE(t_atoms),INTENT(IN)::atoms INTEGER,INTENT(IN) :: itype call derivative_nt(df,f,atoms%jmtd,atoms%jri,atoms%dx,atoms%rmsh, + atoms%ntype,itype) END SUBROUTINE SUBROUTINE derivative_nt(df,f,jmtd,jri,dx,rmsh,ntype,itype) `````` Markus Betzinger committed Apr 26, 2016 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 `````` IMPLICIT NONE INTEGER,INTENT(IN) :: ntype,itype,jmtd INTEGER,INTENT(IN) :: jri(ntype) REAL, INTENT(IN) :: dx(ntype),rmsh(jmtd,ntype) REAL, INTENT(IN) :: f(jri(itype)) REAL, INTENT(OUT) :: df(jri(itype)) REAL :: x,h,r,d21,d32,d43,d31,d42,d41,df1,df2 INTEGER :: i,n n = jri(itype) h = dx(itype) r = rmsh(1,itype) ! use power series expansion a+c**x for first point IF(f(2).eq.f(1)) THEN `````` Matthias Redies committed Jul 15, 2019 465 `````` df(1) = 0.0 `````` Markus Betzinger committed Apr 26, 2016 466 467 468 469 470 471 472 473 474 `````` ELSE x = (f(3)-f(2))/(f(2)-f(1)) df(1) = (f(2)-f(1)) / (x-1) * log(x)/h / r END IF ! use Lagrange interpolation of 3rd order for all other points (and averaging) d21 = r * (exp(h)-1) ; d32 = d21 * exp(h) ; d43 = d32 * exp(h) d31 = d21 + d32 ; d42 = d32 + d43 d41 = d31 + d43 df(2) = - d32*d42 / (d21*d31*d41) * f(1) `````` Matthias Redies committed Jul 15, 2019 475 `````` & + ( 1.0/d21 - 1.0/d32 - 1.0/d42) * f(2) `````` Markus Betzinger committed Apr 26, 2016 476 477 478 479 `````` & + d21*d42 / (d31*d32*d43) * f(3) & - d21*d32 / (d41*d42*d43) * f(4) df1 = d32*d43 / (d21*d31*d41) * f(1) & - d31*d43 / (d21*d32*d42) * f(2) `````` Matthias Redies committed Jul 15, 2019 480 `````` & + ( 1.0/d31 + 1.0/d32 - 1.0/d43 ) * f(3) `````` Markus Betzinger committed Apr 26, 2016 481 482 483 484 485 486 `````` & + d31*d32 / (d41*d42*d43) * f(4) DO i = 3,n-2 d21 = d32 ; d32 = d43 ; d43 = d43 * exp(h) d31 = d42 ; d42 = d42 * exp(h) d41 = d41 * exp(h) df2 = - d32*d42 / (d21*d31*d41) * f(i-1) `````` Matthias Redies committed Jul 15, 2019 487 `````` & + ( 1.0/d21 - 1.0/d32 - 1.0/d42) * f(i) `````` Markus Betzinger committed Apr 26, 2016 488 489 490 491 492 `````` & + d21*d42 / (d31*d32*d43) * f(i+1) & - d21*d32 / (d41*d42*d43) * f(i+2) df(i) = ( df1 + df2 ) / 2 df1 = d32*d43 / (d21*d31*d41) * f(i-1) & - d31*d43 / (d21*d32*d42) * f(i) `````` Matthias Redies committed Jul 15, 2019 493 `````` & + ( 1.0/d31 + 1.0/d32 - 1.0/d43 ) * f(i+1) `````` Markus Betzinger committed Apr 26, 2016 494 495 496 497 498 499 `````` & + d31*d32 / (d41*d42*d43) * f(i+2) END DO df(n-1) = df1 df(n) = - d42*d43 / (d21*d31*d41) * f(n-3) & + d41*d43 / (d21*d32*d42) * f(n-2) & - d41*d42 / (d31*d32*d43) * f(n-1) `````` Matthias Redies committed Jul 15, 2019 500 `````` & + ( 1.0/d41 + 1.0/d42 + 1.0/d43 ) * f(n) `````` Markus Betzinger committed Apr 26, 2016 501 `````` `````` Daniel Wortmann committed Jun 21, 2017 502 `````` END SUBROUTINE derivative_nt `````` Markus Betzinger committed Apr 26, 2016 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 `````` c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c Orders iarr(1:n) according to size and returns a correspondingly defined pointer in pnt SUBROUTINE iorderp(pnt,iarr,n) IMPLICIT NONE INTEGER, INTENT(IN) :: n INTEGER, INTENT(OUT) :: pnt(1:n) INTEGER, INTENT(IN) :: iarr(1:n) INTEGER :: i,j,k DO i=1,n pnt(i) = i DO j=1,i-1 IF(iarr(pnt(j)).gt.iarr(i)) THEN DO k=i,j+1,-1 pnt(k) = pnt(k-1) END DO pnt(j) = i EXIT END IF END DO END DO END SUBROUTINE iorderp c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c Orders rarr(1:n) according to size and returns a correspondingly defined pointer in pnt SUBROUTINE rorderp(pnt,rarr,n) IMPLICIT NONE INTEGER, INTENT(IN) :: n INTEGER, INTENT(OUT) :: pnt(1:n) REAL, INTENT(IN) :: rarr(1:n) INTEGER :: i,j,k DO i=1,n pnt(i) = i DO j=1,i-1 IF(rarr(pnt(j)).gt.rarr(i)) THEN DO k=i,j+1,-1 pnt(k) = pnt(k-1) END DO pnt(j) = i EXIT END IF END DO END DO END SUBROUTINE rorderp c Same as rorderp but divides the problem in halves np times (leading to 2**np intervals) and is c much faster than rorderp (devide and conquer algorithm). c There is an optimal np, while for larger np the overhead (also memory-wise) outweights the speed-up. `````` Matthias Redies committed Jul 15, 2019 563 ``````c np = max(0,int(log(n*0.001)/log(2.0))) should be a safe choice. `````` Markus Betzinger committed Apr 26, 2016 564 565 566 567 568 569 `````` RECURSIVE SUBROUTINE rorderpf(pnt,rarr,n,np) IMPLICIT NONE INTEGER, INTENT(IN) :: n,np INTEGER, INTENT(OUT) :: pnt(n) `````` Daniel Wortmann committed Feb 09, 2017 570 571 572 `````` REAL , INTENT(IN) :: rarr(n) REAL :: rarr1(n) REAL :: ravg `````` Markus Betzinger committed Apr 26, 2016 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 `````` INTEGER :: pnt1(n),pnt2(n) INTEGER :: n1,n2,i IF(np.eq.0) THEN CALL rorderp(pnt,rarr,n) RETURN ELSE IF(np.lt.0) THEN STOP 'rorderpf: fourth argument must be non-negative (bug?).' END IF ravg = sum(rarr)/n ! first half n1 = 0 DO i = 1,n IF(rarr(i).le.ravg) THEN n1 = n1 + 1 rarr1(n1) = rarr(i) pnt1(n1) = i END IF END DO CALL rorderpf(pnt2,rarr1,n1,np-1) pnt(:n1) = pnt1(pnt2(:n1)) ! second half n2 = 0 DO i = 1,n IF(rarr(i).gt.ravg) THEN n2 = n2 + 1 rarr1(n2) = rarr(i) pnt1(n2) = i END IF END DO CALL rorderpf(pnt2,rarr1,n2,np-1) pnt(n1+1:) = pnt1(pnt2(:n2)) END SUBROUTINE rorderpf c Calculates the spherical Bessel functions of orders 0 to l at x c by backward recurrence using j_l(x) = (2l+3)/x j_l+1(x) - j_l+2(x) . c (Starting points are calculated according to Zhang, Min, c "Computation of Special Functions".) pure SUBROUTINE sphbessel(sphbes,x,l) IMPLICIT NONE INTEGER, INTENT(IN) :: l REAL , INTENT(IN) :: x REAL , INTENT(INOUT) :: sphbes(0:l) REAL :: s0,s1,f,f0,f1,cs INTEGER :: ll,lsta,lmax,msta2 ! IF( x.lt.0 ) THEN ! STOP 'sphbes: negative argument (bug?).' ! ELSE IF( x.eq.0 ) THEN `````` Matthias Redies committed Jul 15, 2019 627 `````` sphbes(0) = 1.0 `````` Markus Betzinger committed Apr 26, 2016 628 `````` DO ll = 1,l `````` Matthias Redies committed Jul 15, 2019 629 `````` sphbes(ll) = 0.0 `````` Markus Betzinger committed Apr 26, 2016 630 631 632 633 634 635 636 637 638 639 640 641 642 `````` END DO RETURN ENDIF sphbes(0) = sin(x) / x IF( l .eq. 0 ) RETURN sphbes(1) = ( sphbes(0) - cos(x) ) / x ! IF(l.le.1) RETURN s0 = sphbes(0) s1 = sphbes(1) lsta = lsta1(x,200) ! lmax = l ! IF(lsta.lt.l) THEN ! lmax = lsta ! determine starting point lsta `````` Matthias Redies committed Jul 15, 2019 643 `````` sphbes(lmax+1:) = 0.0 ! for backward recurrence `````` Markus Betzinger committed Apr 26, 2016 644 645 646 `````` ELSE ! lsta = lsta2(x,l,15) ! END IF ! `````` Matthias Redies committed Jul 15, 2019 647 `````` f0 = 0.0 ! `````` Matthias Redies committed Jul 15, 2019 648 `````` f1 = 1e-100 ! `````` Markus Betzinger committed Apr 26, 2016 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 `````` DO ll = lsta,0,-1 ! backward recurrence f = f1 / x * (2*ll+3) - f0 ; IF(ll.le.lmax) sphbes(ll) = f ! with arbitrary start values f0 = f1 ! f1 = f ! END DO ! IF(abs(s0).gt.abs(s1)) THEN ; cs = s0 / f ! ELSE ; cs = s1 / f0 ! scale to correct values END IF ! sphbes = cs * sphbes ! CONTAINS c Test starting point PURE FUNCTION lsta0(x,mp) IMPLICIT NONE INTEGER :: lsta0 INTEGER, INTENT(IN) :: mp REAL , INTENT(IN) :: x REAL :: f,lgx lgx = log10(x) lsta0 = 0 f = lgx DO WHILE(f.gt.-mp) lsta0 = lsta0 + 1 `````` Matthias Redies committed Jul 15, 2019 676 `````` f = f + lgx - log10(2.0*lsta0+1) `````` Markus Betzinger committed Apr 26, 2016 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 `````` END DO END FUNCTION lsta0 c Returns starting point lsta1 for backward recurrence such that sphbes(lsta1) approx. 10^(-mp). PURE FUNCTION lsta1(x,mp) IMPLICIT NONE INTEGER :: lsta1 INTEGER, INTENT(IN) :: mp REAL , INTENT(IN) :: x REAL :: f0,f1,f INTEGER :: n0,n1,nn,it n0 = int(1.1*x) + 1 f0 = envj(n0,x) - mp n1 = n0 + 5 f1 = envj(n1,x) - mp DO it = 1,20 `````` Matthias Redies committed Jul 15, 2019 696 `````` nn = n1 - (n1-n0) / (1.0-f0/f1) `````` Markus Betzinger committed Apr 26, 2016 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 `````` f = envj(nn,x) - mp IF(abs(nn-n1).lt.1) EXIT n0 = n1 f0 = f1 n1 = nn f1 = f END DO lsta1 = nn END FUNCTION lsta1 c Returns the starting point lsta2 for backward recurrence such that all sphbes(l) have mp significant digits. PURE FUNCTION lsta2(x,l,mp) IMPLICIT NONE INTEGER :: lsta2 INTEGER, INTENT(IN) :: l,mp REAL , INTENT(IN) :: x REAL :: f0,f1,f,hmp,ejn,obj INTEGER :: n0,n1,nn,it `````` Matthias Redies committed Jul 15, 2019 718 `````` hmp = 0.5 * mp `````` Markus Betzinger committed Apr 26, 2016 719 720 721 722 723 724 725 726 727 728 729 730 `````` ejn = envj(l,x) IF( ejn.le.hmp ) THEN obj = mp n0 = int(1.1*x) + 1 ELSE obj = hmp + ejn n0 = l END IF f0 = envj(n0,x) - obj n1 = n0 + 5 f1 = envj(n1,x) - obj DO it = 1,20 `````` Matthias Redies committed Jul 15, 2019 731 `````` nn = n1 - (n1-n0) / (10-f0/f1) `````` Markus Betzinger committed Apr 26, 2016 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 `````` f = envj(nn,x) - obj IF(abs(nn-n1).lt.1) EXIT n0 = n1 f0 = f1 n1 = nn f1 = f END DO lsta2 = nn + 10 END FUNCTION lsta2 PURE FUNCTION envj(l,x) IMPLICIT NONE REAL :: envj REAL , INTENT(IN) :: x INTEGER, INTENT(IN) :: l `````` Matthias Redies committed Jul 15, 2019 751 `````` envj = 0.5 * log10(6.28*l) - l*log10(1.36*x/l) `````` Markus Betzinger committed Apr 26, 2016 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 `````` END FUNCTION envj END SUBROUTINE sphbessel c Returns the spherical harmonics Y_lm(^rvec) c for l = 0,...,ll in Y(1,...,(ll+1)**2). PURE SUBROUTINE harmonicsr(Y,rvec,ll) IMPLICIT NONE integer , intent(in) :: ll real , intent(in) :: rvec(3) complex , intent(out) :: Y((ll+1)**2) complex :: c real :: stheta,ctheta,sphi,cphi,r,rvec1(3) integer :: l,m,lm `````` Matthias Redies committed Jul 15, 2019 772 `````` complex , parameter :: img = (0.0,1.0) `````` Markus Betzinger committed Apr 26, 2016 773 `````` `````` Matthias Redies committed Jul 15, 2019 774 `````` Y(1) = 0.282094791773878 `````` Markus Betzinger committed Apr 26, 2016 775 776 777 778 779 780 781 `````` IF(ll.eq.0) RETURN stheta = 0 ctheta = 0 sphi = 0 cphi = 0 r = sqrt(sum(rvec**2)) `````` Matthias Redies committed Jul 15, 2019 782 `````` IF(r.gt.1e-16) THEN `````` Markus Betzinger committed Apr 26, 2016 783 784 785 `````` rvec1 = rvec / r ctheta = rvec1(3) stheta = sqrt(rvec1(1)**2+rvec1(2)**2) `````` Matthias Redies committed Jul 15, 2019 786 `````` IF(stheta.gt.1e-16) THEN `````` Markus Betzinger committed Apr 26, 2016 787 788 789 790 `````` cphi = rvec1(1) / stheta sphi = rvec1(2) / stheta END IF ELSE `````` Matthias Redies committed Jul 15, 2019 791 `````` Y(2:) = 0.0 `````` Markus Betzinger committed Apr 26, 2016 792 793 794 795 796 797 798 `````` RETURN END IF c define Y,l,-l and Y,l,l r = Y(1) c = 1 DO l=1,ll `````` Matthias Redies committed Jul 15, 2019 799 `````` r = r*stheta*sqrt(1.0+1.0/(2*l)) `````` Markus Betzinger committed Apr 26, 2016 800 801 802 803 804 805 `````` c = c * (cphi + img*sphi) Y(l**2+1) = r*conjg(c) ! l,-l Y((l+1)**2) = r*c*(-1)**l ! l,l END DO c define Y,l,-l+1 and Y,l,l-1 `````` Matthias Redies committed Jul 15, 2019 806 `````` Y(3) = 0.48860251190292*ctheta `````` Markus Betzinger committed Apr 26, 2016 807 `````` DO l=2,ll `````` Matthias Redies committed Jul 15, 2019 808 `````` r = sqrt(2.0*l+1) * ctheta `````` Markus Betzinger committed Apr 26, 2016 809 810 811 812 813 814 815 816 817 `````` Y(l**2+2) = r*Y((l-1)**2+1) ! l,-l+1 Y(l*(l+2)) = r*Y(l**2) ! l,l-1 END DO c define Y,l,m, |m|