3 3.1 n > 1 n = 1/2 x = 0 x = z 2 z 2n+1 F n (η) = 2 dz (5) e z2 η 3.2 G B G (G transform) B (B transform) (Gray and Atchison) [2] S(η) = S t (η
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- がんま かやぬま
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1 Cloutman [1] 11 FORTRAN C 2 3 n E 1/2 f(e ) n = 1 2π 2 = 1 2π 2 = 1 2π 2 x = E /k B T η = µ/k B T ( ) 3/2 2m h 2 E 1/2 f(e )de 0 ( ) 3/2 2m E 1/2 h 2 de (1) e (E µ)/kbt ( ) 3/2 2mkB T x 1/2 dx (2) 1 + ex η h 2 (1) E 1/2 E 3/2 F n (η) = 0 0 x n dx (3) 1 + ex η n(the Fermi-Dirac integral of order n) 1 η η Cloutman [1] 11 B G Aitken Cloutman η [3, 4] (3) η x 1 F n = F n(η) = nf n 1 (η) (4) Z 1 x n dx Γ(n + 1) ex η n 1
2 3 3.1 n > 1 n = 1/2 x = 0 x = z 2 z 2n+1 F n (η) = 2 dz (5) e z2 η 3.2 G B G (G transform) B (B transform) (Gray and Atchison) [2] S(η) = S t (η) = a t a f(η, x)dx (6) f(η, x)dx (7) S(η) = lim t S t (η) (8) G B G[S(η); t, k] = S t+k(η) R t (η, k)s t (η), R t 1 (9) 1 R t (η, k) f(η, t + k) R t (η, k) = k > 0 (10) f(η, t) B[S(η); t, k] = S tk(η) ρ t (η, k)s t (η), ρ t 1 (11) 1 ρ t (η, k) kf(η, kt) ρ t (η, k) = k > 1 (12) f(η, t) lim G[S(η); t, k] = lim B[S(η); t, k] = S(η) (13) t t Gray Atchison lim t R t (η, k) 0, 1G S t+k (η) B f = e x S t G[S(η); t, k] = Sf = x s B[S(η); t, k] = S 2
3 3.3 Aitken µ 2µ 4µ H H/2 H/4 I µ I 2µ I 4µ I I = I µ + ch p (14) I = I 2µ + c(h/2) p (15) I = I 4µ + c(h/4) p (16) I = I 4µ (I 4µ I 2µ ) 2 (17) I 4µ 2I 2µ + I µ [ ] 1 I p = log 10 (2) log Iµ 10 (18) I I 2µ c = I I µ H p (19) 3.4 η < 0 η 0 (Cox and Giuli, 1968) F n (η) = Γ(n + 1)e η r=0 n = 1/2 n = 3/2n = 5/2 F 1/2 (η) = π1/2 2 F 3/2 (η) = 3π1/2 4 F 5/2 (η) = 15π1/2 8 e rη ( 1) r, n > 1 (20) (r + 1) n+1 ( 1) j=1 ( 1) j=1 ( 1) j=1 j+1 ejη, (21) j3/2 j+1 ejη j 5/2 (22) j+1 ejη j 7/2 (23) 3.5 η > 25 I(η) = 0 φ (u)du e u η + 1 (24) I(η) φ(η) + 2 C 2j φ 2j (η) (25) j=1 3
4 C 2 = π2 12, C 4 = 7π4 720, C 6 = 31π , C 8 = 127π , C 10 = 511π (26) (3) [ F n (η) = ηn n + 1 r=1 n = 1/2 n = 3/2 n = 5/2 ( n+1 2C 2r k=n 2r+2 k ) η 2r ], n > 0, η 1 (27) F 1/2 (η) 2 3 η3/2 + π2 12 η 1/2 + 7π4 960 η 5/2 + 31π η 9/ π η 13/2 (28) F 3/2 (η) 2 5 η5/2 + π2 4 η1/2 7π4 960 η 3/2 31π η 7/2 381π η 11/2 (29) F 5/2 (η) 2 7 η7/2 + 5π2 12 η3/2 + 7π4 192 η 1/2 + 31π η 5/ π η 9/2 (30) [1] L. D. Cloutman, Numerical evaluation of the Fermi-Dirac integrals, Astrophys. J. Suppl. Ser. 71, (1989). [2] H. L. Gray and T. A. Atchison, Applications of the G and B transforms to the Laplace transform, Proceedings of the rd ACM national conference, pp.73-77, (1968). pdf [3] 5 (2016/6/30 ). [4] (2016/6/29 ). [5] (2016/7/4 ). [1] FORTRAN n = 1/2 n = 1/2, 1/2, 3/2, 5/2 5 η 25 n = 1/2 [5] 1. /* fermi_dirac_inetgral.c -o fermi_dirac_integral */ /* translated from FDTAB written in the FORTRAN language to C. */ /* by M. Suzuki */ /* This program is based on the programme by Lawrence D. Cloutman, */ /* described in his original paper, "Numerical Evaluation of the */ 4
5 /* Fermi-Dirac Integrals",The Astrophysical Journal of Supplement */ /* Series vol.71, pp (1989). The paper is available at */ /* */ /* PROGRAM FDTAB */ /* COMPUTE ACCURATE TABLES OF FERMI-DIRAC INTEGRALS */ /* USING SIMPSON S RULE WITH EXTRAPOLATION TECHNIQUES */ #include <stdio.h> #include <stdlib.h> #include <math.h> #include <string.h> /* Evaluate the integrand at (x, eta) */ double fint(double x, double eta) double z; z=2*x*x/(exp(x*x-eta)+1); return z; int main(int argc, char *argv[]) FILE *fp; char filenameout[200], dummy[200]; int m2, m4, mpts, mptsav, n, ne, nfail, ngb, nm, itrap, N; int i, ii, j; double a, b, bad, bgt, binc, bk, bkt, bratio, bsave; double bxfrm, cextr, de, denom, digits, eta, eta0, extrap; double fmax, fsum, gk, gkt, gxfrm, h, pextr; double rgb, rob, rog, xmax; double relerr, seta, sgxfrm; double x[2000], f[2000], value[20]; /* INITIAL DATA */ /* (a, b) are the starting values fo the integration limits (0, t) */ a=0.0; b=1.0; /* binc is the increment in b used by the adaptive upper */ /* integration limit routine, bratio is the maximum value */ /* of f(b)/max(f),where the function f is the integrand. */ binc=0.5; bratio=1.0e-6; /* Incrementing factors for the B and G transforms */ gk=1.0; bk=1.1; /* mpts is the number of mesh points to which Simpson s rule */ /* is applied for the coarsest mesh, must be an odd integer */ mpts=201; /* eta0 is the first value of degeneracy parameter, de is the */ 5
6 /* increment in eta, and ne is the number of eta values */ eta0=-5.0; de=0.05; ne=601; /* nfail is a diagnostic that counts extrapolation failures */ nfail=0; /* itrap=1 for trapezoidal rule, */ /* otherwise Simpson s rule integration */ itrap=0; /* make sure mpts is even for Simpson s rule */ if((mpts % 2)!=1 && itrap==0) printf("%d should be even\n", mpts); exit(0); mptsav=mpts; bsave=b; /* Outer loop is over all values of eta */ for(n=0;n<ne;n++) eta=eta0+de*n; bad=bsave; printf("\neta=%lf\n", eta); /* This loop does the three integrations required for the */ /* B and G transforms, after which the transforms are computed. */ for(ngb=0;ngb<3;ngb++) /* Set appropriate upper integration limit */ b=bad; if(ngb==1) b=bad+gk; if(ngb==2) b=bad*bk; /* This loop increments the number of mesh points to do the */ /* three integrations required for each Aitken extrapolation. */ for(nm=0;nm<3;nm++) while(1) m2=(mpts-3)/2; m4=m2+1; h=(b-a)/(mpts-1); fmax=-1.0e100; for(j=0;j<mpts;j++) x[j]=a+h*j; f[j]=fint(x[j], eta); 6
7 if(f[j] >= fmax) xmax=x[j]; fmax=f[j]; if((ngb!= 0) (nm!= 0) (f[mpts-1] < bratio*fmax) (b > 100.0)) break; b+=binc; bad=b; if(nm==0) printf("fmax %le at %lf, end point x=%lf f=%le\n", fmax,xmax,x[mpts-1],f[mpts-1]); if(itrap==1) /* Use trapezoidal rule integrations */ fsum=0.0; for(i=0;i<mpts;i++) fsum+=f[i]; value[nm]=h*(fsum-0.5*(f[0]+f[mpts-1])); else /* Integrate the f array with Simpson s rule */ fsum=0.0; for(ii=0;ii<m2;ii++) i=ii; // j descending order i=m2-ii+1; // j ascending order j=m4-i; fsum+=2*f[2*j+1]+f[2*j+2]; value[nm]=h*(2*fsum+f[0]+f[mpts-1]+4*f[mpts-2])/3; printf("integral=%20.14le for mpts=%d\n", value[nm], mpts); printf("%d\t%le\t%le\t%le\n", nm, value[0], value[1], value[2]); if(nm < 2) mpts=2*(mpts-1)+1; /* Aitken extrapolation */ denom=value[2]+value[0]-2*value[1]; if(denom!= 0.0) extrap=value[2]-pow((value[2]-value[1]),2)/denom; else 7
8 extrap=value[2]; nfail++; printf("cannot perform Aitken extrapolation\n"); if(extrap-value[1]!= 0.0) denom=(extrap-value[0])/(extrap-value[1]); else denom=0.0; if(denom > 0.0) pextr=log10(denom)/log10(2.0); h=(b-a)/(mptsav-1); cextr=(extrap-value[0])/pow(h, pextr); else nfail++; pextr= ; cextr=pextr; printf("cannot compute Aitken parameters p and c\n "); printf("extrap=%le, value[1,2]=%le, %le\n", extrap, value[0], value[1]); relerr=fabs((extrap-value[2])/extrap); if(relerr > 0.0) digits=-log10(relerr); else digits=-30.0; printf("extrapolated integral = %le, ", extrap); printf("p=%le, c=%le, h=%le, digits=%lf\n", pextr, cextr, h, digits); if(ngb==0) bgt=extrap; rgb=f[mpts-1]; if(ngb==1) gkt=extrap; rog=f[mpts-1]; if(ngb==2) bkt=extrap; rob=f[mpts-1]; mpts=mptsav; /* Calculate the B and G transforms */ rog=rog/rgb; rob=bk*rob/rgb; 8
9 printf("%le\t%le\t%20.12le\n", rog, rob, f[mpts-1]); /* Limited error checking */ if(rob <= 0.0 rog <=0.0) printf("eta=%le, rob=%le, rog=%le, Cannot do B and G transforms", eta, rob, rog); exit(0); gxfrm=(gkt-rog*bgt)/(1.0-rog); bxfrm=(bkt-rob*bgt)/(1.0-rob); printf("g transform=%le, B transform=%le\n", gxfrm, bxfrm); relerr=fabs((gxfrm-gkt)/gxfrm); if(relerr > 0.0) digits=-log10(relerr); else digits=-30; seta=eta; sgxfrm=gxfrm; printf("%lf\t%18.11le\n", seta, sgxfrm); printf("%lf\t%le, %lf digits G xfrm\n", seta, sgxfrm, digits); 2. /* FDSET.c -o FDSET */ /* translated from FDSET written in the FORTRAN language to C. */ /* by M. Suzuki */ /* This program is based on the programm written by Lawrence D. Cloutman, */ /* described in his original paper, "Numerical Evaluation of the */ /* Fermi-Dirac Integrals",The Astrophysical Journal of Supplement */ /* Series vol.71, pp (1989). The paper is available at */ /* */ /* SUBROUTINE FDSET */ /* THIS SUBROUTINE INITIALIZES ARRAYS NEEDED BY FUNCTION FD. */ /* FDSET IS CALLED ONCE (AND ONLY ONCE) BEFORE CALLING FD. */ #include <stdio.h> #include <stdlib.h> #include <math.h> #include <string.h> static double fdtab; static double F[601][4], ahi[5][3], alo[5][3]; /* SET UP F-D INTEGRAL ASSYMPTOTIC EXPANSION COEFFICIENTS */ int FDSET(void) FILE *fp; char filenamein[100]; int i; 9
10 double a, pi, e0, f0, f1, f2, f3; pi=m_pi; strcpy(filenamein, "FD_table.txt"); a=sqrt(pi)/2; alo[0][0]=a; alo[1][0]=-a/pow(sqrt(2),3); alo[2][0]=a/pow(sqrt(3),3); alo[3][0]=-a/8; alo[4][0]=a/pow(sqrt(5),3); a=3*sqrt(pi)/4; alo[0][1]=a; alo[1][1]=-a/pow(sqrt(2),5); alo[2][1]=a/pow(sqrt(3),5); alo[3][1]=-a/32; alo[4][1]=a/pow(sqrt(5),5); a=15*sqrt(pi)/8; alo[0][2]=a; alo[1][2]=-a/pow(sqrt(2),7); alo[2][2]=a/pow(sqrt(3),7); alo[3][2]=-a/128; alo[4][2]=a/pow(sqrt(5),7); ahi[0][0]=2/3; ahi[1][0]=pow(pi, 2)/12; ahi[2][0]=pow(pi, 4)*7/960; ahi[3][0]=pow(pi, 6)*31/4608; ahi[4][0]=pow(pi, 8)*1397/81920; ahi[0][1]=0.4; ahi[1][1]=pow(pi, 2)/4; ahi[2][1]=-ahi[2][0]; ahi[3][1]=-pow(pi, 6)*31/10752; ahi[4][1]=-pow(pi, 8)*381/81920; ahi[0][2]=2/7; ahi[1][2]=pow(pi, 2)*5/12; ahi[2][2]=pow(pi, 4)*7/192; ahi[3][2]=pow(pi, 6)*31/10752; ahi[4][2]=pow(pi, 8)*127/49152; /* READ IN FERMI-DIRAC INTEGRAL TABLES GIVEN IN TABLE 5 */ if((fp=fopen(filenamein, "r"))==0) exit(0); i=0; while(i<601) fscanf(fp, "%lf\t%le\t%le\t%le\t%le\n", &e0,&f0,&f1,&f2,&f3); F[i][0]=f0; F[i][1]=f1; F[i][2]=f2; F[i][3]=f3; i++; fclose(fp); return 0; 10
11 /* COMPUTES THE FERMI-DIRAC INTEGRAL FD FOR DEGENERACY */ /* PARAMETER ETA. n=1 FOR ORDER 1/2; n=2 FOR ORDER 3/2; n=3 FOR ORDER 3/2. */ double FD(double eta, int n) int j, k, kk, l; double u, z; double x0, x1, x2, y0, y1, y2, s0, s1, s2; double HERMITE5(); if(eta >= -5 && eta <= 25) /* FIFTH ORDER HERMITE INTERPOLATION FOR INTERMEDIATE VALUES OF ETA. */ j=(eta+5)*20; if(j < 1) j=1; if(j > 599) j=599; x0= *(j-1); x1=x0+0.05; x2=x0+0.1; y0=f[j-1][n]; y1=f[j][n]; y2=f[j+1][n]; s0=f[j-1][n-1]*(n-0.5); s1=f[j][n-1]*(n-0.5); s2=f[j+1][n-1]*(n-0.5); z=hermite5(eta, x0, y0, s0, x1, y1, s1, x2, y2, s2); return z; if(eta <-5) z=0; for(k=0;k<5;k++) kk=k+1; z+=alo[k][n]*exp(k*eta); return z; if(eta >25) u=sqrt(eta); l=1+2*n; z=0; for(k=0;k<5;k++) z+=ahi[k][n]*pow(u,l-4*k); return z; 11
12 /* FIFTH ORDER HERMITE INTERPOLATION */ /* x = value of the independent variable where the function value */ /* Hermite5 is desired. */ /* x0, x1, x2 = values of the independent variable at the three */ /* interpolation nodes. */ /* p0, p1, p2 = function values at the interpolation nodes. */ /* dp0, dp1, dp2 = first derivatives of the function */ /* at the interpolation nodes. */ double HERMITE5(double x, double x0, double p0, double dp0, double x1, double p1, double dp1, double x2, double p2, double dp2) double xp, h; double a0, a1, a2, a3, a4, a5; h=x1-x0; xp=x-x1; a0=p1; a1=dp1; a4=(h*(dp2-dp0)-2*(p0+p2-p1-p1)); a2=((p0+p2-p1-p1)*-a4)/(4*h*h); a4*=0.25/pow(h, 4); a5=0.25*(h*(dp2+4*dp1+dp0)-3*(p2-p0)); a3=(0.5*(p2-p0)-h*dp1-a5)/pow(h, 3); a5/=pow(h, 5); return ((((a5*xp+a4)*xp+a3)*xp+a2)*xp+a1)*xp+a0; int main(int argc, char *argv[]) FILE *fp; int i, n; double eta, z; if(argc<3) printf("usage; a.out eta n\n"); exit(0); i=fdset(); eta=atof(argv[1]); n=atoi(argv[2]); z=fd(eta, n); printf("%8.5lf\t%2d\t%19.11le\n", eta, n, z); 12
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