n=9,h=1/(n+1),K=2*eye(n)-diag(ones(n-1,1),1)-diag(ones(n-1,1),-1); K=(1/h)*K, F=h*ones(n,1), U=K\F n = 9 h = 0.10000000000000 K = 20 -10 0 0 0 0 0 0 0 -10 20 -10 0 0 0 0 0 0 0 -10 20 -10 0 0 0 0 0 0 0 -10 20 -10 0 0 0 0 0 0 0 -10 20 -10 0 0 0 0 0 0 0 -10 20 -10 0 0 0 0 0 0 0 -10 20 -10 0 0 0 0 0 0 0 -10 20 -10 0 0 0 0 0 0 0 -10 20 F = 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 0.10000000000000 U = 0.04500000000000 0.08000000000000 0.10500000000000 0.12000000000000 0.12500000000000 0.12000000000000 0.10500000000000 0.08000000000000 0.04500000000000 n=7,h=1/(n+1),K=2*eye(n)-diag(ones(n-1,1),1)-diag(ones(n-1,1),-1); K=(1/h)*K, F=h*ones(n,1), U=K\F n = 7 h = 0.12500000000000 K = 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 F = 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 U = 0.05468750000000 0.09375000000000 0.11718750000000 0.12500000000000 0.11718750000000 0.09375000000000 0.05468750000000 n=7,h=1/(n+1),K=2*eye(n)-diag(ones(n-1,1),1)-diag(ones(n-1,1),-1); K=(1/h)*K, F=h*ones(n,1), U=K\F, x=h:h:1-h; n = 7 h = 0.12500000000000 K = 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 -8 0 0 0 0 0 -8 16 F = 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 0.12500000000000 U = 0.05468750000000 0.09375000000000 0.11718750000000 0.12500000000000 0.11718750000000 0.09375000000000 0.05468750000000 plot(x,U) n=3,h=1/(n+1),K=2*eye(n)-diag(ones(n-1,1),1)-diag(ones(n-1,1),-1); K=(1/h)*K, F=h*ones(n,1), U=K\F, x=h:h:1-h; n = 3 h = 0.25000000000000 K = 8 -4 0 -4 8 -4 0 -4 8 F = 0.25000000000000 0.25000000000000 0.25000000000000 U = 0.09375000000000 0.12500000000000 0.09375000000000 hold on plot(x,U,'r') n=51,h=1/(n+1),K=2*eye(n)-diag(ones(n-1,1),1)-diag(ones(n-1,1),-1); K=(1/h)*K, F=h*ones(n,1), U=K\F, x=h:h:1-h; n = 51 h = 0.01923076923077 K = Columns 1 through 20 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 21 through 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 104 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 41 through 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -52 0 0 0 0 0 0 0 0 0 0 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 -52 0 0 0 0 0 0 0 0 0 -52 104 F = 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 0.01923076923077 U = 0.00943047337278 0.01849112426035 0.02718195266272 0.03550295857988 0.04345414201183 0.05103550295858 0.05824704142012 0.06508875739645 0.07156065088757 0.07766272189349 0.08339497041420 0.08875739644970 0.09375000000000 0.09837278106509 0.10262573964497 0.10650887573964 0.11002218934911 0.11316568047337 0.11593934911243 0.11834319526627 0.12037721893491 0.12204142011834 0.12333579881657 0.12426035502958 0.12481508875740 0.12500000000000 0.12481508875740 0.12426035502958 0.12333579881657 0.12204142011834 0.12037721893491 0.11834319526627 0.11593934911242 0.11316568047337 0.11002218934911 0.10650887573964 0.10262573964497 0.09837278106509 0.09375000000000 0.08875739644970 0.08339497041420 0.07766272189349 0.07156065088757 0.06508875739645 0.05824704142012 0.05103550295858 0.04345414201183 0.03550295857988 0.02718195266272 0.01849112426035 0.00943047337278 plot(x,U,'g') exact=(-0.5*x.*(x-1))' exact = 0.00943047337278 0.01849112426036 0.02718195266272 0.03550295857988 0.04345414201183 0.05103550295858 0.05824704142012 0.06508875739645 0.07156065088757 0.07766272189349 0.08339497041420 0.08875739644970 0.09375000000000 0.09837278106509 0.10262573964497 0.10650887573964 0.11002218934911 0.11316568047337 0.11593934911243 0.11834319526627 0.12037721893491 0.12204142011834 0.12333579881657 0.12426035502959 0.12481508875740 0.12500000000000 0.12481508875740 0.12426035502959 0.12333579881657 0.12204142011834 0.12037721893491 0.11834319526627 0.11593934911243 0.11316568047337 0.11002218934911 0.10650887573964 0.10262573964497 0.09837278106509 0.09375000000000 0.08875739644970 0.08339497041420 0.07766272189349 0.07156065088757 0.06508875739645 0.05824704142012 0.05103550295858 0.04345414201183 0.03550295857988 0.02718195266272 0.01849112426036 0.00943047337278 norm(U-exact,1) ans = 4.064804048908854e-014 n=3,h=1/(n+1),K=2*eye(n)-diag(ones(n-1,1),1)-diag(ones(n-1,1),-1); K=(1/h)*K, F=h*ones(n,1), U=K\F, x=h:h:1-h; n = 3 h = 0.25000000000000 K = 8 -4 0 -4 8 -4 0 -4 8 F = 0.25000000000000 0.25000000000000 0.25000000000000 U = 0.09375000000000 0.12500000000000 0.09375000000000 exact=(-0.5*x.*(x-1))' exact = 0.09375000000000 0.12500000000000 0.09375000000000 diary off