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Local ex1 = (a1 + ... + a10)^20; ***Print; .end kap_5_apb3.frm000600 000765 000024 00000000153 14555773464 013265 0ustar00msstaff000000 000000 * * VL: Rechnernutzung in der Physik * WS 23/24 * * apb3.frm Symbol a,b; Local ex1 = (a+b)^3; Print; .end kap_5_fibonacci.frm000600 000765 000024 00000000666 14555773767 014374 0ustar00msstaff000000 000000 * * Rechnernutzung in der Physik * WS 23/24 * * Fibonacci Zahlen * #define NN "27" s n; cfunction f; l fib = f(`NN'); repeat; id f(1)=1; id f(2)=1; id f(n?) = f(n-1) + f(n-2); endrepeat; .sort print; .store s letzte,zweitletzte,dummy; cfunction f; l fib = f(1,1) * dummy^(`NN'-2); repeat; id f(letzte?,zweitletzte?)*dummy = f(letzte+zweitletzte,letzte); endrepeat; id f(letzte?,zweitletzte?) = letzte; .sort print; .end kap_5_fibonacci_2.frm000600 000765 000024 00000000336 14555774053 014574 0ustar00msstaff000000 000000 * * Rechnernutzung in der Physik * WS 23/24 * * Fibonacci Zahlen * Symbol n,x1,x2; CFunction f,fib; Local F = f(10,1,1); repeat; id f(0,?a) = fib(?a); id f(n?,?a,x1?,x2?) = f(n-1,?a,x1,x2,x1+x2); endrepeat; Print; .end kap_5_gamma.frm000600 000765 000024 00000000562 14555775331 013521 0ustar00msstaff000000 000000 * * Rechnernutzung in der Physik * WS 23/24 * * Gamma-Matrizen * index mu,nu,rho,si; index mu1,...,mu20; vector p,pp; vector p1,p2,p3,p4; symbol m,beta,E,[sin(the/2)]; local t1 = g_(1,mu)*g_(1,nu); local t1a = g_(1,mu,nu); local t2 = g_(1,mu,nu,rho,si); ***local t3 = g_(1,mu1,...,mu20); local t4 = g_(1,5_, p1,p2,p3,p4); print; .sort trace4,1; *print +s; .end kap_5_gram.frm000600 000765 000024 00000000471 14555203714 013353 0ustar00msstaff000000 000000 * * VL: Rechnernutzung in der Physik * WS 23/24 * * gram.frm * * Gram-Determinante einer Matrix * * ------------------------------------------------------------ Vector v1,...,v10; Local G2 = e_(v1,v2)^2; Local G3 = e_(v1,v2,v3)^2; Local G10 = e_(v1,...,v10)^2; contract; AntiBracket v1,...,v10; Print[]; .end kap_5_id.frm000600 000765 000024 00000000305 14555203714 013015 0ustar00msstaff000000 000000 * * VL: Rechnernutzung in der Physik * WS 23/24 * * id.frm * * Ein erstes FORM-Beispiel * s a,b,x; l expr = a*x+x^2; id x = a + b; .sort if(count(b,1)==1); multiply 4*a/b; endif; print; .end kap_5_kommutator.frm000600 000765 000024 00000000675 14555775011 014641 0ustar00msstaff000000 000000 * * Rechnernutzung in der Physik * WS 23/24 * * Berechne Kommutator in QM * * Deklariere Symbole: symbol hbar,m; * Deklariere nicht-kommutierende Variablen: functions x,p,H; * Deklariere Ausdruck, der berechnet werden soll: local [H,x] = H*x - x*H; * Spezifiziere Hamilton-Operator: id H = p^2/(2*m); *id H = p^4/(m^3); print; .sort * Verwende (so oft wie moeglich) [x,p] = i*hbar repeat; id x*p = p*x + hbar*i_; endrepeat; print; .end kap_5_kommutator_2.frm000600 000765 000024 00000001372 14555203714 015051 0ustar00msstaff000000 000000 * * Rechnernutzung in der Physik * WS 23/24 * * Berechne Kommutator in QM * * Deklariere Symbole: symbol hbar,m; * Deklariere nicht-kommutierende Variablen: functions x,y,z,px,py,pz,H,Lx,Ly,Lz; * Deklariere Ausdruck, der berechnet werden soll: local [H,Lx] = H*Lx - Lx*H; local [H,Ly] = H*Ly - Ly*H; local [H,Lz] = H*Lz - Lz*H; * Spezifiziere Hamilton- und Drehimpls-Operator: id H = (px^2+py^2+pz^2)/(2*m); id Lx = y*pz - pz*y; id Ly = z*px - px*z; id Lz = x*py - py*x; print; .sort * Verwende (so oft wie moeglich) [x,p] = i*hbar repeat; id x*px = px*x + hbar*i_; id y*py = py*y + hbar*i_; id z*pz = pz*z + hbar*i_; id x*py = py*x; id x*pz = pz*x; id y*px = px*y; id y*pz = pz*y; id z*px = px*z; id z*py = py*z; endrepeat; print; .end kap_5_levi_civita.frm000600 000765 000024 00000001447 14555203714 014727 0ustar00msstaff000000 000000 * * VL: Rechnernutzung in der Physik * WS 23/24 * * levi_civita.frm * * 3- und 4-dimensionaler \epsilon-Tensor * * ------------------------------------------------------------ * * \epsilon_ijk * Dimension 3; *** Default: Dimension 4; Indices i,j,k,p,q,r; Local ex0 = e_(i,j,k) * e_(p,q,r); Local ex1 = e_(i,j,k) * e_(p,q,k); Local ex2 = e_(i,j,k) * e_(p,j,k); Local ex3 = e_(i,j,k) * e_(i,j,k); contract; Print +s; * "+s" => ein Ausdruck pro Zeile .sort * compiliere Ausdruck bis zu diesem Punkt .clear * entspricht "reset" * ------------------------------------------------------------ * * \epsilon_ijkl * Indices i,j,k,l,p,q,r,s; Local ex0 = e_(i,j,k,l) * e_(p,q,r,s); Local ex2 = e_(i,j,k,l) * e_(p,q,k,l); Local ex4 = e_(i,j,k,l) * e_(i,j,k,l); contract; Print +s; .end kap_5_pattern.frm000600 000765 000024 00000000302 14555203714 014073 0ustar00msstaff000000 000000 * * VL: Rechnernutzung in der Physik * WS 23/24 * * pattern.frm * * Beispiel zu "pattern matching" * Symbol x,y,z,a,b; CFunction f; Local F = f(x)+f(y)+f(4)+f(a+b); id f(z?) = z^2; Print; .end kap_5_pauli.frm000600 000765 000024 00000000575 14555203714 013544 0ustar00msstaff000000 000000 * * Rechnernutzung in der Physik * WS 23/24 * * Pauli-Matrizen * functions s; index i,j,k; local f = ( s(1)*s(2) + s(1) + s(2) + s(3) )^4; print; .sort repeat; id s(2)*s(1) = - s(1)*s(2); id s(3)*s(1) = - s(1)*s(3); id s(3)*s(2) = - s(2)*s(3); id s(i?)^2 = 1; id s(1)*s(2) = i_*s(3); id s(2)*s(3) = i_*s(1); id s(1)*s(3) = -i_*s(2); endrepeat; .sort print; .end kap_5_gram.m000600 000765 000024 00000000453 14555774321 013032 0ustar00msstaff000000 000000 (* VL: Rechnernutzung in der Physik, WS23/24 gram.m Berechne Gram-Determinante einer Matrix *) (* *** < #include using namespace std; using namespace GiNaC; int main() { symbol x("x"), y("y"); ex poly; for (int i=0; i<3; ++i) poly += factorial(i+16)*pow(x,i)*pow(y,2-i); cout << poly << endl; return 0; } kap_5_ginac_2.cc000644 000765 000024 00000001165 14555203714 013541 0ustar00msstaff000000 000000 /* Rechnernutzung in der Physik WS 23/24 Beispiel zu ginac c++ kap_5_ginac_2.cc -lcln -lginac (use bash) g++ kap_5_ginac_2.cc -lginac -I/export/pc/usr/include/ -L/export/pc/usr/lib64/ */ #include #include using namespace std; using namespace GiNaC; ex HermitePoly(const symbol & x, int n) { ex HKer=exp(-pow(x, 2)); // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2) return normal(pow(-1, n) * diff(HKer, x, n) / HKer); } int main() { symbol z("z"); for (int i=0; i<6; ++i) cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl; return 0; } kap_5_ginsh.txt000644 000765 000024 00000000434 14555203714 013577 0ustar00msstaff000000 000000 x=5^101; y=5^100; x/y; a=Pi^2; evalf(a); Digits = 100; evalf(a); lsolve(a+x*y*z==1,z); a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4; b = x^2 + 4*x*y - y^2; a*b; expand(a*b); collect(a+b,y); diff(log(x),x); series(sin(x)/x^2,x==0,4); Digits=50; fsolve(sin(x)==1-x^2,x,0,2); kap_5_rpol.ferl000600 000765 000024 00000001123 14555203714 013540 0ustar00msstaff000000 000000 # VL: Rechnernutzung in der Physik, WS23/24 # Fermat: ferl # - Befehl beginnt mit "&" # - Fermat arbeitet auf Ringen # Default: rationale Zahlen # ------------------------------------------------------------ # "d" ist eine Polynom-Variable # Ring ueber Polynome in Variable d wird betrachtet. &(J=d) # "x" ist eine Polynom-Variable # Ring ueber Polynome in Variable x wird betrachtet. # Koeffizienten sind Polynome in "d". &(J=x) # "num" ist eine Variable num := d*((x + d + 2)^250 - 23*d + x^2) # "den" ist eine Variable den := (d - 1)*((x + d + 2)^250 - 23*d + x^2) num/den &q kap_5_rpol.m000600 000765 000024 00000000475 14555203714 013055 0ustar00msstaff000000 000000 (* VL: Rechnernutzung in der Physik, WS23/24 rpol.m p1(x,d)/p2(x,d): Ausmultiplizieren und ggT finden *) Timing[num = Expand[ d* ((x + d + 2)^250 - 23*d + x^2) ];] Timing[den = Expand[ (d - 1)*((x + d + 2)^250 - 23*d + x^2) ];] Timing[Together[num/den]] Timing[Factor[num/den]] Timing[Simplify[num/den]]