maths_functions.f90 9.51 KB
 Luca committed May 21, 2019 1 2 3 4 5 6 7 `````` MODULE maths_functions use globals use utils implicit none CONTAINS `````` ulrich_y committed Jul 09, 2019 8 9 10 11 12 13 14 15 16 `````` 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 17 `````` `````` ulrich_y committed Jul 08, 2019 18 `````` FUNCTION naive_polylog(m,x) result(res) `````` Luca committed May 21, 2019 19 20 21 22 `````` ! Computes the classical polylogarithm Li_m(x) using series representation up to order n integer :: m complex(kind=prec) :: x, res integer :: i,n `````` Luca Naterop committed Jul 05, 2019 23 `````` n = 1000 `````` ulrich_y committed Jul 09, 2019 24 25 26 27 28 29 `````` res=0. do i=1,n 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 30 `````` END FUNCTION naive_polylog `````` Luca committed May 21, 2019 31 `````` `````` Luca committed May 21, 2019 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 `````` 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 76 `````` ELSE IF(abs(x-MONE) < zero) THEN `````` Luca committed May 21, 2019 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 `````` 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 122 `````` `````` Luca Naterop committed Jul 05, 2019 123 `````` RECURSIVE FUNCTION dilog(x) result(res) `````` Luca committed May 21, 2019 124 125 126 `````` ! evaluates dilog for any argument complex(kind=prec) :: res complex(kind=prec) :: x `````` Luca committed May 21, 2019 127 `````` `````` Luca committed May 21, 2019 128 `````` if(abs(x) <= 1.0) then `````` Luca Naterop committed Jul 05, 2019 129 130 131 132 133 `````` if(abs(aimag(x)) < zero ) then res = Li2(real(x)) else res = naive_polylog(2,x) endif `````` Luca committed May 21, 2019 134 `````` else `````` Luca Naterop committed Jul 08, 2019 135 `````` res = -dilog(1/x) - (pi**2) /6 - log(add_ieps(-x))**2 / 2 `````` Luca committed May 21, 2019 136 137 138 `````` end if END FUNCTION dilog `````` Luca committed May 21, 2019 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 `````` 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 200 `````` ELSE IF( abs(x+1) < zero) THEN `````` Luca committed May 21, 2019 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 `````` 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 242 243 244 245 246 247 `````` FUNCTION trilog(x) result(res) ! evaluates trilog for any argument complex(kind=prec) :: res complex(kind=prec) :: x if(abs(x) <= 1.0) then `````` Luca Naterop committed Jul 05, 2019 248 249 250 251 252 `````` if(abs(aimag(x)) < zero ) then res = Li3(real(x)) else res = naive_polylog(3,x) endif `````` Luca committed May 21, 2019 253 254 255 256 257 `````` else res = naive_polylog(3,sub_ieps(x)**(-1)) - (log(-sub_ieps(x)))**3/6 - pi**2/6 * log(-sub_ieps(x)) end if END FUNCTION trilog `````` ulrich_y committed Jul 09, 2019 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 `````` 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 `````` Luca committed May 21, 2019 300 `````` FUNCTION polylog(m,x) result(res) `````` Luca Naterop committed Jul 07, 2019 301 302 `````` ! computes the polylog `````` Luca committed May 21, 2019 303 304 `````` integer :: m complex(kind=prec) :: x,res `````` Luca Naterop committed Jul 05, 2019 305 `````` `````` Luca committed May 22, 2019 306 `````` if(verb >= 70) print*, 'called polylog(',m,',',x,')' `````` ulrich_y committed Jul 09, 2019 307 308 309 310 311 `````` if ((m.le.9).and.(abs(x-1.).lt.zero)) then res = zeta(m) else if ((m.le.9).and.(abs(x+1.).lt.zero)) then res = -(1. - 2.**(1-m))*zeta(m) else if(m == 2) then `````` Luca committed May 21, 2019 312 `````` res = dilog(x) `````` Luca committed May 21, 2019 313 314 315 `````` else if(m == 3) then res = trilog(x) else `````` ulrich_y committed Jul 09, 2019 316 317 318 319 320 321 `````` if (abs(x).gt.1) then res = (-1)**(m-1)*naive_polylog(m,1./x) & - cmplx(0,2*pi)**m * bernoulli_polynomial(m, 0.5-cmplx(0.,1.)*clog(-x)/2/pi) / factorial(m) else res = naive_polylog(m,x) endif `````` Luca committed May 21, 2019 322 323 `````` end if END FUNCTION polylog `````` Luca committed May 21, 2019 324 `````` `````` Luca committed May 21, 2019 325 326 ``````END MODULE maths_functions `````` Luca committed May 21, 2019 327 328 329 330 331 332 333 ``````! 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 334