maths_functions.f90 9.78 KB
 Luca committed May 21, 2019 1 2 3 4 5 `````` MODULE maths_functions use globals use utils implicit none `````` ulrich_y committed Jul 09, 2019 6 7 8 `````` interface polylog module procedure polylog1, polylog2 end interface polylog `````` Luca committed May 21, 2019 9 10 `````` CONTAINS `````` ulrich_y committed Jul 09, 2019 11 12 13 14 15 16 17 18 19 `````` FUNCTION zeta(n) real(kind=prec) :: values(9), zeta integer :: n values = (/1.6449340668482262, 1.2020569031595942, 1.0823232337111381, & 1.03692775514337, 1.0173430619844488, 1.008349277381923, & 1.0040773561979441, 1.0020083928260821, 1.000994575127818/) zeta = values(n-1) END FUNCTION zeta `````` Luca committed May 21, 2019 20 `````` `````` ulrich_y committed Jul 08, 2019 21 `````` FUNCTION naive_polylog(m,x) result(res) `````` Luca committed May 21, 2019 22 23 24 `````` ! Computes the classical polylogarithm Li_m(x) using series representation up to order n integer :: m complex(kind=prec) :: x, res `````` Luca Naterop committed Jul 10, 2019 25 `````` integer :: i `````` ulrich_y committed Jul 09, 2019 26 `````` res=0. `````` Luca Naterop committed Jul 10, 2019 27 `````` do i=1,PolylogInfinity `````` ulrich_y committed Jul 09, 2019 28 29 30 31 `````` if(i**m.lt.0) return ! roll over if(abs(x**i).lt.1.e-250) return res = res+x**i/i**m enddo `````` Luca committed May 21, 2019 32 `````` END FUNCTION naive_polylog `````` Luca committed May 21, 2019 33 `````` `````` Luca committed May 21, 2019 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 `````` FUNCTION Li2(x) !! Dilogarithm for arguments x < = 1.0 real (kind=prec):: X,Y,T,S,A,PI3,PI6,ZERO,ONE,HALF,MALF,MONE,MTWO real (kind=prec):: C(0:18),H,ALFA,B0,B1,B2,LI2_OLD real (kind=prec):: Li2 integer :: i DATA ZERO /0.0_prec/, ONE /1.0_prec/ DATA HALF /0.5_prec/, MALF /-0.5_prec/ DATA MONE /-1.0_prec/, MTWO /-2.0_prec/ DATA PI3 /3.289868133696453_prec/, PI6 /1.644934066848226_prec/ DATA C( 0) / 0.4299669356081370_prec/ DATA C( 1) / 0.4097598753307711_prec/ DATA C( 2) /-0.0185884366501460_prec/ DATA C( 3) / 0.0014575108406227_prec/ DATA C( 4) /-0.0001430418444234_prec/ DATA C( 5) / 0.0000158841554188_prec/ DATA C( 6) /-0.0000019078495939_prec/ DATA C( 7) / 0.0000002419518085_prec/ DATA C( 8) /-0.0000000319334127_prec/ DATA C( 9) / 0.0000000043454506_prec/ DATA C(10) /-0.0000000006057848_prec/ DATA C(11) / 0.0000000000861210_prec/ DATA C(12) /-0.0000000000124433_prec/ DATA C(13) / 0.0000000000018226_prec/ DATA C(14) /-0.0000000000002701_prec/ DATA C(15) / 0.0000000000000404_prec/ DATA C(16) /-0.0000000000000061_prec/ DATA C(17) / 0.0000000000000009_prec/ DATA C(18) /-0.0000000000000001_prec/ if(X > 1.00000000001_prec) then print*, 'crashes because Li called with bad arguments' elseif(X > 1.0_prec) then X = 1._prec endif IF(X > 0.999999_prec) THEN LI2_OLD=PI6 Li2 = Real(LI2_OLD,prec) RETURN `````` Luca Naterop committed Jul 08, 2019 78 `````` ELSE IF(abs(x-MONE) < zero) THEN `````` Luca committed May 21, 2019 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 `````` LI2_OLD=MALF*PI6 RETURN END IF T=-X IF(T .LE. MTWO) THEN Y=MONE/(ONE+T) S=ONE A=-PI3+HALF*(LOG(-T)**2-LOG(ONE+ONE/T)**2) ELSE IF(T .LT. MONE) THEN Y=MONE-T S=MONE A=LOG(-T) A=-PI6+A*(A+LOG(ONE+ONE/T)) ELSE IF(T .LE. MALF) THEN Y=(MONE-T)/T S=ONE A=LOG(-T) A=-PI6+A*(MALF*A+LOG(ONE+T)) ELSE IF(T .LT. ZERO) THEN Y=-T/(ONE+T) S=MONE A=HALF*LOG(ONE+T)**2 ELSE IF(T .LE. ONE) THEN Y=T S=ONE A=ZERO ELSE Y=ONE/T S=MONE A=PI6+HALF*LOG(T)**2 END IF H=Y+Y-ONE ALFA=H+H B1=ZERO B2=ZERO DO I = 18,0,-1 B0=C(I)+ALFA*B1-B2 B2=B1 B1=B0 ENDDO LI2_OLD=-(S*(B0-H*B2)+A) ! Artificial conversion Li2 = Real(LI2_OLD,prec) END FUNCTION Li2 `````` Luca committed May 21, 2019 124 `````` `````` Luca Naterop committed Jul 05, 2019 125 `````` RECURSIVE FUNCTION dilog(x) result(res) `````` ulrich_y committed Jul 10, 2019 126 `````` ! evaluates dilog for any argument |x|<1 `````` Luca committed May 21, 2019 127 128 `````` complex(kind=prec) :: res complex(kind=prec) :: x `````` Luca committed May 21, 2019 129 `````` `````` ulrich_y committed Jul 10, 2019 130 131 `````` if(abs(aimag(x)) < zero ) then res = Li2(real(x)) `````` Luca committed May 21, 2019 132 `````` else `````` ulrich_y committed Jul 10, 2019 133 134 `````` res = naive_polylog(2,x) endif `````` Luca committed May 21, 2019 135 136 `````` END FUNCTION dilog `````` Luca committed May 21, 2019 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 `````` FUNCTION Li3(x) ! Trilogarithm for arguments x < = 1.0 ! This was hacked from LI2 to also follow C332 ! In theory this could also produce Re[Li [x]] for x>1 real (kind=prec):: X,S,A real (kind=prec):: CA(0:18),HA,ALFAA,BA0,BA1,BA2, YA real (kind=prec):: CB(0:18),HB,ALFAB,BB0,BB1,BB2, YB DATA CA(0) / 0.4617293928601208/ DATA CA(1) / 0.4501739958855029/ DATA CA(2) / -0.010912841952292843/ DATA CA(3) / 0.0005932454712725702/ DATA CA(4) / -0.00004479593219266303/ DATA CA(5) / 4.051545785869334e-6/ DATA CA(6) / -4.1095398602619446e-7/ DATA CA(7) / 4.513178777974119e-8/ DATA CA(8) / -5.254661564861129e-9/ DATA CA(9) / 6.398255691618666e-10/ DATA CA(10) / -8.071938105510391e-11/ DATA CA(11) / 1.0480864927082917e-11/ DATA CA(12) / -1.3936328400075057e-12/ DATA CA(13) / 1.8919788723690422e-13/ DATA CA(14) / -2.6097139622039465e-14/ DATA CA(15) / 3.774985548158685e-15/ DATA CA(16) / -5.671361978114946e-16/ DATA CA(17) / 1.1023848202712794e-16/ DATA CA(18) / -5.0940525990875006e-17/ DATA CB(0) / -0.016016180449195803/ DATA CB(1) / -0.5036424400753012/ DATA CB(2) / -0.016150992430500253/ DATA CB(3) / -0.0012440242104245127/ DATA CB(4) / -0.00013757218124463538/ DATA CB(5) / -0.000018563818526041144/ DATA CB(6) / -2.841735345177361e-6/ DATA CB(7) / -4.7459967908588557e-7/ DATA CB(8) / -8.448038544563037e-8/ DATA CB(9) / -1.5787671270014e-8/ DATA CB(10) / -3.0657620579122164e-9/ DATA CB(11) / -6.140791949281482e-10/ DATA CB(12) / -1.2618831590198e-10/ DATA CB(13) / -2.64931268635803e-11/ DATA CB(14) / -5.664711482422879e-12/ DATA CB(15) / -1.2303909436235178e-12/ DATA CB(16) / -2.7089360852246495e-13/ DATA CB(17) / -6.024075373994343e-14/ DATA CB(18) / -1.2894320641440237e-14/ real (kind=prec):: Li3 real (kind=prec), parameter :: zeta2 = 1.6449340668482264365 real (kind=prec), parameter :: zeta3 = 1.2020569031595942854 integer :: i if(x > 1.00000000001_prec) then print*, 'need to crash Li3, since not convergent' elseif(x > 1.0_prec) then x = 1._prec endif IF(X > 0.999999_prec) THEN LI3=zeta3 RETURN `````` Luca Naterop committed Jul 08, 2019 198 `````` ELSE IF( abs(x+1) < zero) THEN `````` Luca committed May 21, 2019 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 `````` LI3=-0.75_prec*zeta3 RETURN END IF IF(X .LE. -1._prec) THEN YA=1._prec/x ; YB=0._prec S=-1._prec A=-LOG(-X)*(zeta2+LOG(-x)**2/6._prec) ELSE IF(X .LE. 0._prec) THEN YA=x ; YB=0._prec S=-1._prec A=0._prec ELSE IF(X .LE. 0.5_prec) THEN YA=0._prec ; YB=x S=-1._prec A=0._prec ELSE IF(X .LE. 1._prec) THEN YA=(x-1._prec)/x ; YB=1._prec-x S=1._prec A=zeta3 + zeta2*Log(x) - (Log(1._prec - X)*Log(X)**2)/2._prec + Log(X)**3/6._prec ELSE IF(X .LE. 2._prec) THEN YA=1._prec - X ; YB=(X-1._prec)/X S=1._prec A=zeta3 + zeta2*Log(x) - (Log(X - 1._prec)*Log(X)**2)/2._prec + Log(X)**3/6._prec ELSE YA=0._prec ; YB=1._prec/X S=-1._prec A=2*zeta2*Log(x)-Log(x)**3/6._prec END IF HA=-2._prec*YA-1._prec ; HB= 2._prec*YB ALFAA=HA+HA ; ALFAB = HB+HB BA0 = 0. ; BA1=0. ; BA2=0. BB0 = 0. ; BB1=0. ; BB2=0. DO I = 18,0,-1 BA0=CA(I)+ALFAA*BA1-BA2 ; BA2=BA1 ; BA1=BA0 BB0=CB(I)+ALFAB*BB1-BB2 ; BB2=BB1 ; BB1=BB0 ENDDO Li3 = A + S * ( (BA0 - HA*BA2) + (BB0 - HB*BB2) ) END FUNCTION Li3 `````` Luca committed May 21, 2019 240 241 `````` FUNCTION trilog(x) result(res) `````` ulrich_y committed Jul 10, 2019 242 `````` ! evaluates trilog for any argument |x|<1 `````` Luca committed May 21, 2019 243 244 `````` complex(kind=prec) :: res complex(kind=prec) :: x `````` ulrich_y committed Jul 10, 2019 245 246 `````` if(abs(aimag(x)) < zero ) then res = Li3(real(x)) `````` Luca committed May 21, 2019 247 `````` else `````` ulrich_y committed Jul 10, 2019 248 249 `````` res = naive_polylog(3,x) endif `````` Luca committed May 21, 2019 250 251 `````` END FUNCTION trilog `````` ulrich_y committed Jul 09, 2019 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 `````` FUNCTION BERNOULLI_POLYNOMIAL(n, x) result(res) integer n complex(kind=prec) :: x, res select case(n) case(1) res = -1/2. + x case(2) res = 1/6. - x + x**2 case(3) res = x/2. - 3*x**2/2. + x**3 case(4) res = -1/30. + x**2 - 2*x**3 + x**4 case(5) res = -x/6. + 5*x**3/3 - 5*x**4/2 + x**5 case(6) res = 1/42. - x**2/2 + 5*x**4/2 - 3*x**5 + x**6 case(7) res = x/6. - 7*x**3/6 + 7*x**5/2 - 7*x**6/2 + x**7 case(8) res = -1/30. + 2*x**2/3 - 7*x**4/3 + 14*x**6/3 - 4*x**7 + x**8 case(9) res = -3*x/10 + 2*x**3 - 21*x**5/5 + 6*x**7 - 9*x**8/2 + x**9 case(10) res = 5/66. - 3*x**2/2 + 5*x**4 - 7*x**6 + 15*x**8/2 - 5*x**9 + x**10 case(11) res = 5*x/6 - 11*x**3/2 + 11*x**5 - 11*x**7 + 55*x**9/6 - 11*x**10/2 + x**11 case(12) res = -691/2730. + 5*x**2 - 33*x**4/2 + 22*x**6 - 33*x**8/2 + 11*x**10 - 6*x**11 + x**12 case(13) res = -691*x/210 + 65*x**3/3 - 429*x**5/10 + 286*x**7/7 - 143*x**9/6 + 13*x**11 - 13*x**12/2 + x**13 case(14) res = 7/6. - 691*x**2/30 + 455*x**4/6 - 1001*x**6/10 + 143*x**8/2 - 1001*x**10/30 + 91*x**12/6 - 7*x**13 + x**14 case(15) res = 35*x/2 - 691*x**3/6 + 455*x**5/2 - 429*x**7/2 + 715*x**9/6 - 91*x**11/2 + 35*x**13/2 - 15*x**14/2 + x**15 case default print*,"Bernoulli beyond 15 is not implemented" stop end select END FUNCTION `````` ulrich_y committed Jul 09, 2019 294 `````` RECURSIVE FUNCTION polylog1(m,x) result(res) `````` Luca Naterop committed Jul 07, 2019 295 296 `````` ! computes the polylog `````` Luca committed May 21, 2019 297 `````` integer :: m `````` ulrich_y committed Jul 09, 2019 298 `````` type(inum) :: x, inv `````` ulrich_y committed Jul 09, 2019 299 `````` complex(kind=prec) :: res `````` Luca Naterop committed Jul 05, 2019 300 `````` `````` ulrich_y committed Jul 09, 2019 301 302 `````` if(verb >= 70) print*, 'called polylog(',m,',',x%c,x%i0,')' if ((m.le.9).and.(abs(x%c-1.).lt.zero)) then `````` ulrich_y committed Jul 09, 2019 303 `````` res = zeta(m) `````` ulrich_y committed Jul 09, 2019 304 305 `````` return else if ((m.le.9).and.(abs(x%c+1.).lt.zero)) then `````` ulrich_y committed Jul 09, 2019 306 `````` res = -(1. - 2.**(1-m))*zeta(m) `````` ulrich_y committed Jul 09, 2019 307 308 `````` return else if (abs(x) .gt. 1) then `````` ulrich_y committed Jul 09, 2019 309 310 311 `````` inv = inum(1./x%c, x%i0) res = (-1)**(m-1)*polylog(m,inv) & - cmplx(0,2*pi)**m * bernoulli_polynomial(m, 0.5-cmplx(0.,1.)*conjg(log(-x%c))/2/pi) / factorial(m) `````` ulrich_y committed Jul 09, 2019 312 313 314 315 316 `````` return endif if(m == 2) then res = dilog(x%c) `````` Luca committed May 21, 2019 317 `````` else if(m == 3) then `````` ulrich_y committed Jul 09, 2019 318 `````` res = trilog(x%c) `````` Luca committed May 21, 2019 319 `````` else `````` ulrich_y committed Jul 09, 2019 320 `````` res = naive_polylog(m,x%c) `````` Luca committed May 21, 2019 321 `````` end if `````` ulrich_y committed Jul 09, 2019 322 323 324 325 326 327 328 329 330 `````` END FUNCTION polylog1 RECURSIVE FUNCTION polylog2(m,x,y) result(res) type(inum) :: x, y integer m complex(kind=prec) :: res `````` ulrich_y committed Jul 09, 2019 331 `````` !TODO!! `````` ulrich_y committed Jul 09, 2019 332 333 334 335 336 337 338 339 340 `````` res=polylog1(m,inum(x%c/y%c,di0)) END FUNCTION POLYLOG2 FUNCTION PLOG1(a,b) ! calculates log(1-a/b) implicit none type(inum) :: a,b complex(kind=prec) plog1 `````` ulrich_y committed Jul 09, 2019 341 `````` !TODO!! `````` ulrich_y committed Jul 09, 2019 342 343 `````` plog1 = log(1.-a%c/b%c) END FUNCTION `````` Luca committed May 21, 2019 344 `````` `````` Luca committed May 21, 2019 345 346 ``````END MODULE maths_functions `````` Luca committed May 21, 2019 347 348 349 350 351 352 353 ``````! PROGRAM test ! use maths_functions ! implicit none ! complex(kind=prec) :: res ! res = Li3(0.4d0) ! print*, res ! END PROGRAM test `````` Luca committed May 21, 2019 354