some tests with unitary matrices
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135
t.cc
135
t.cc
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@ -2592,15 +2592,140 @@ cin>>x;
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cout <<mantissa(x,20)<<endl;
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}
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if(1)
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if(0)
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{
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double t,t2;
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cin>>t;
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bitvector m = mantissa(t,64);
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int s;
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cin>>t>>s;
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bitvector m = mantissa(t,64,s);
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cout <<m<<std::endl;
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bitvector_decimal(t2,m);
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cout <<t2;
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bitvector_decimal(t2,m,-s);
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cout <<t2<<endl;
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}
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if(0)
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{
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int n;
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cin >>n;
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NRMat<std::complex<double> > a(n,n);
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a.randomize(1);
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NRMat<std::complex<double> > ah=a.transpose(true);
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NRMat<std::complex<double> > b=a-ah;
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//general non-symmetric unitary matrix
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NRMat<std::complex<double> > u=exp(b);
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NRMat<std::complex<double> > bsym(b),banti(b);
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bsym.copyonwrite();
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banti.copyonwrite();
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for(int i=0; i<n; ++i) for(int j=0; j<n; ++j)
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{
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bsym(i,j).real(0);
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banti(i,j).imag(0);
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}
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NRMat<std::complex<double> > usym=exp(bsym);
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NRMat<std::complex<double> > uanti=exp(banti);
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cout <<b<<u;
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cout<<bsym<<usym;
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//usym is unitary symmetric complex
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cout<<banti<<uanti;
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//uanti is unitary real
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NRMat<std::complex<double> > uh=u.transpose(true);
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NRMat<std::complex<double> > usymh=usym.transpose(true);
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NRMat<std::complex<double> > uantih=uanti.transpose(true);
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cout <<"error of unitarity of u*uh = "<<(u*uh).norm(1.)<<endl;
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cout <<"error of unitarity of usym*usymh = "<<(usym*usymh).norm(1.)<<endl;
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cout <<"error of unitarity of uanti*uantih = "<<(uanti*uantih).norm(1.)<<endl;
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cout <<"expected nonzero norm of commutator of sym and anti parts = "<<commutator(bsym,banti).norm()<<endl;
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cout <<"expected nonzero norm of difference between u and usym*uanti = "<<(u-usym*uanti).norm()<<endl;
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NRSMat<double> hs(n);
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hs.randomize(1);
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NRMat<double> h(hs);
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NRMat<std::complex<double> > ih=imagmatrix(h);
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NRMat<std::complex<double> > c=commutator(bsym,ih);
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//cout <<"commutator of two sym parts = "<<c<<endl; //is antisymmetric
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NRMat<std::complex<double> > uc=exp(c);
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NRMat<std::complex<double> > usym2=exp(ih);
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cout <<"error of unitarity of uc = "<<(uc*uc.transpose(true)).norm(1.)<<endl;
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cout <<"error of unitarity of usym2 = "<<(usym2*usym2.transpose(true)).norm(1.)<<endl;
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NRMat<std::complex<double> > up = exp(bsym+ih);
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cout <<"error of unitarity of up = "<<(up*up.transpose(true)).norm(1.)<<endl;
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cout <<"expected nonzero norm of difference between up and usym*usym2 = "<<(up-usym*usym2).norm()<<endl;
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NRMat<std::complex<double> > ubch1 = exp(bsym+ih+c*0.5);
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cout <<"error of unitarity of ubch1 = "<<(ubch1*ubch1.transpose(true)).norm(1.)<<endl;
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cout <<"expected nonzero norm of difference between ubch1 and usym*usym2 = "<<(ubch1-usym*usym2).norm()<<endl;
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}
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if(0)
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{
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int n;
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cin >>n;
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NRSMat<double> hs(n);
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hs.randomize(1);
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NRMat<double> h(hs);
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//since H is real, the unitary U will be symmetric and its complex conjugation thus equal to hermitian conjugation
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NRMat<std::complex<double> > ih=imagmatrix(h);
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NRMat<std::complex<double> > u=expi(h);
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NRMat<std::complex<double> > uh = u.transpose(true);
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NRMat<std::complex<double> > uc = u.conjugate();
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NRMat<std::complex<double> > ut = u.transpose(false);
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cout <<"error of symmetry of exp(iH) = "<<(u-ut).norm()<<endl;
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NRMat<std::complex<double> > uct = uc.transpose(false);
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cout <<"error of uct - uh = "<<(uct-uh).norm()<<endl;
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//cout <<u << uc<<uh;
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cout <<"nonunitarity of u*uh = "<<(u*uh).norm(1.)<<endl;
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cout <<"nonunitarity of u*uc = "<<(u*uc).norm(1.)<<endl;
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NRMat<std::complex<double> > uu=exp(ih);
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NRMat<std::complex<double> > uuh = uu.transpose(true);
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cout <<"nonunitarity of uu = "<<(uu*uuh).norm(1.)<<endl;
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cout <<"error of complex exponential = "<<(u-uu).norm()<<endl;
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NRMat<double> d(h);
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NRVec<double> e(n);
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diagonalize(d,e);
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//now try similarity transformed H
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NRMat<double> a(n,n);
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a.randomize(1);
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NRMat<double> ainv = inverse(a);
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NRMat<double> atr = a.transpose();
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NRMat<double> ainvtr = ainv.transpose();
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cout <<"error of inverse = "<<(a*ainv).norm(1.)<<endl;
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NRMat<double> hbar = ainv * h * a;
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NRVec<double> er(n),ei(n);
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NRMat<double> hwork(hbar);
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gdiagonalize(hwork,er,ei);
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er.sort();
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cout <<"errors of isospectral eigenvalues = "<<(er-e).norm()<<" "<<ei.norm()<<endl;
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NRMat<std::complex<double> > uubar=expi(hbar);
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NRMat<std::complex<double> > ubar = NRMat<std::complex<double> >(ainv) * u * NRMat<std::complex<double> >(a);
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cout <<"error of similarity transformed exponential = "<<(ubar-uubar).norm()<<endl;
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NRMat<std::complex<double> > ubarh = ubar.transpose(true);
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NRMat<std::complex<double> > ubarh2 = NRMat<std::complex<double> >(atr) * u.transpose(true) * NRMat<std::complex<double> >(ainvtr);
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cout <<"error of ubar hermit conjugation = "<<(ubarh-ubarh2).norm()<<endl;
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NRMat<std::complex<double> > uubarh = uubar.transpose(true);
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//cout <<"H = "<<h;
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//cout <<"Hbar = "<<hbar;
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//cout <<"U = "<<u;
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//cout <<"Ubar = "<<ubar;
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NRMat<std::complex<double> > ubartest2=ubar*ubarh;
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cout <<"expected nonunitarity of ubar = "<<ubartest2.norm(1.)<<endl;
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NRMat<std::complex<double> > uubartest=uubarh*uubar;
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cout <<"expected nonunitarity of uubar = "<<uubartest.norm(1.)<<endl;
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NRMat<std::complex<double> > uubartest2 = NRMat<std::complex<double> >(atr) * u.transpose(true) * NRMat<std::complex<double> >(ainvtr * ainv) * u * NRMat<std::complex<double> >(a);
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cout <<"expected nonunitarity of explicit uubar = "<<uubartest2.norm(1.)<<endl;
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//pseudounitarity
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NRMat<std::complex<double> > ubarc = ubar.conjugate();
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NRMat<std::complex<double> > ubarc2 = NRMat<std::complex<double> >(ainv) * u.conjugate() * NRMat<std::complex<double> >(a);
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cout <<"error of ubar complex conjugation = "<<(ubarc-ubarc2).norm()<<endl;
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NRMat<std::complex<double> > uubarc = uubar.conjugate();
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NRMat<std::complex<double> > ubartest2c=ubar*ubarc;
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cout <<"nonpseudounitarity of ubar = "<<ubartest2c.norm(1.)<<endl;
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NRMat<std::complex<double> > uubartestc=uubarc*uubar;
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cout <<"nonpseudounitarity of uubar = "<<uubartestc.norm(1.)<<endl;
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NRMat<std::complex<double> > uubartest2c = NRMat<std::complex<double> >(ainv) * u.conjugate() * NRMat<std::complex<double> >(a * ainv) * u * NRMat<std::complex<double> >(a);
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cout <<"nonpseudounitarity of explicit uubar = "<<uubartest2c.norm(1.)<<endl;
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}
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if(1)
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{
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}
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