maths_functions.f90 7.38 KB
 Luca committed May 21, 2019 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 `````` MODULE maths_functions use globals use utils implicit none CONTAINS FUNCTION naive_polylog(m,x,n_passed) result(res) ! Computes the classical polylogarithm Li_m(x) using series representation up to order n integer :: m integer, optional :: n_passed complex(kind=prec) :: x, res integer :: i,n integer, allocatable :: j(:) `````` Luca committed Jun 04, 2019 16 `````` n = 300 `````` Luca committed May 21, 2019 17 18 19 `````` j = (/(i, i=1,n,1)/) res = sum(x**j / j**m) END FUNCTION naive_polylog `````` Luca committed May 21, 2019 20 `````` `````` Luca committed May 21, 2019 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 `````` 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 ELSE IF(X .EQ. MONE) THEN 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 111 `````` `````` Luca committed May 21, 2019 112 113 114 115 116 `````` FUNCTION dilog_in_unit_circle(x) result(res) ! evaluates for any argument x in unit circle complex(kind=prec) :: x, res res = naive_polylog(2,x) END FUNCTION dilog_in_unit_circle `````` Luca committed May 21, 2019 117 `````` `````` Luca committed May 21, 2019 118 119 120 121 `````` FUNCTION dilog(x) result(res) ! evaluates dilog for any argument complex(kind=prec) :: res complex(kind=prec) :: x `````` Luca committed May 21, 2019 122 `````` `````` Luca committed May 21, 2019 123 124 125 `````` if(abs(x) <= 1.0) then res = naive_polylog(2,x) else `````` Luca committed May 21, 2019 126 `````` res = -dilog_in_unit_circle(1/x) - pi**2/6 - log(add_ieps(-x))**2 / 2 `````` Luca committed May 21, 2019 127 128 129 `````` end if END FUNCTION dilog `````` Luca committed May 21, 2019 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 `````` 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 ELSE IF(X .EQ. -1._prec) THEN 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 233 234 235 236 237 238 239 240 241 242 243 244 245 `````` FUNCTION trilog(x) result(res) ! evaluates trilog for any argument complex(kind=prec) :: res complex(kind=prec) :: x if(abs(x) <= 1.0) then res = naive_polylog(3,x) 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 `````` Luca committed May 21, 2019 246 247 248 249 `````` FUNCTION polylog(m,x) result(res) ! computes the polylog for now naively (except for dilog half-naively) integer :: m complex(kind=prec) :: x,res `````` Luca committed May 22, 2019 250 `````` if(verb >= 70) print*, 'called polylog(',m,',',x,')' `````` Luca committed May 21, 2019 251 252 `````` if(m == 2) then res = dilog(x) `````` Luca committed May 21, 2019 253 254 255 `````` else if(m == 3) then res = trilog(x) else `````` Luca committed May 21, 2019 256 257 258 `````` res = naive_polylog(m,x) end if END FUNCTION polylog `````` Luca committed May 21, 2019 259 `````` `````` Luca committed May 21, 2019 260 261 ``````END MODULE maths_functions `````` Luca committed May 21, 2019 262 263 264 265 266 267 268 ``````! 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 269