N = 10;
x1 = 0:1/N:1;
x1 = x1+(1/(2*pi)-.01)*sin(2*pi*x1);
X1 = 0:1/(10*N):1;
X1 = X1+(1/(2*pi)-.01)*sin(2*pi*X1);
u = linear_elements(@f,x1);
phi = linear_building_blocks(x1,X1);
figure(1)
clf
  plot(X1,phi(1,:),'LineWidth',2);
  hold on
  plot(X1,phi(3,:),'LineWidth',2);
  plot(X1,phi(5,:),'LineWidth',2);
  plot(X1,phi(7,:),'LineWidth',2);
  plot(X1,phi(9,:),'LineWidth',2);
  xlabel('x','FontSize',16);
print -dps nonunif_basis_functions.ps
figure(1)

U1 = u(2:N+1)*phi;
U = (1/12)*X1.^4-(1/6)*X1.^3+(1/6)*X1;
figure(2)
clf
plot(x1,u,'-o')
hold on
plot(X1,U1,'x')
figure(2)

err1 = U1-U;
figure(3)
clf
plot(X1,abs(err1),'LineWidth',2);
xlabel('x','FontSize',16)
ylabel('abs(err)','FontSize',16);
print -dps ten_node_nonunif_error.ps
figure(3)

figure(4)
clf
plot(X1,log10(abs(err1)));
figure(4)

N = 2*N;
x2 = 0:1/N:1;
x2 = x2+(1/(2*pi)-.01)*sin(2*pi*x2);
X2 = 0:1/(10*N):1;
X2 = X2+(1/(2*pi)-.01)*sin(2*pi*X2);
u = linear_elements(@f,x2);
phi = linear_building_blocks(x2,X2);
U2 = u(2:N+1)*phi;
U = (1/12)*X2.^4-(1/6)*X2.^3+(1/6)*X2;
err2 = U2-U;

N = 2*N;
x3 = 0:1/N:1;
x3 = x3+(1/(2*pi)-.01)*sin(2*pi*x3);
X3 = 0:1/(10*N):1;
X3 = X3+(1/(2*pi)-.01)*sin(2*pi*X3);
u = linear_elements(@f,x3);
phi = linear_building_blocks(x3,X3);
U3 = u(2:N+1)*phi;
U = (1/12)*X3.^4-(1/6)*X3.^3+(1/6)*X3;
err3 = U3-U;

N = 2*N;
x4 = 0:1/N:1;
x4 = x4+(1/(2*pi)-.01)*sin(2*pi*x4);
X4 = 0:1/(10*N):1;
X4 = X4+(1/(2*pi)-.01)*sin(2*pi*X4);
u = linear_elements(@f,x4);
phi = linear_building_blocks(x4,X4);
U4 = u(2:N+1)*phi;
U = (1/12)*X4.^4-(1/6)*X4.^3+(1/6)*X4;
err4 = U4-U;

N = 2*N;
x5 = 0:1/N:1;
x5 = x5+(1/(2*pi)-.01)*sin(2*pi*x5);
X5 = 0:1/(10*N):1;
X5 = X5+(1/(2*pi)-.01)*sin(2*pi*X5);
u = linear_elements(@f,x5);
phi = linear_building_blocks(x5,X5);
U5 = u(2:N+1)*phi;
U = (1/12)*X5.^4-(1/6)*X5.^3+(1/6)*X5;
err5 = U5-U;

N = 2*N;
x6 = 0:1/N:1;
x6 = x6+(1/(2*pi)-.01)*sin(2*pi*x6);
X6 = 0:1/(10*N):1;
X6 = X6+(1/(2*pi)-.01)*sin(2*pi*X6);
u = linear_elements(@f,x6);
phi = linear_building_blocks(x6,X6);
U6 = u(2:N+1)*phi;
U = (1/12)*X6.^4-(1/6)*X6.^3+(1/6)*X6;
err6 = U6-U;

N = 2*N;
x7 = 0:1/N:1;
x7 = x7+(1/(2*pi)-.01)*sin(2*pi*x7);
X7 = 0:1/(10*N):1;
X7 = X7+(1/(2*pi)-.01)*sin(2*pi*X7);
u = linear_elements(@f,x7);
phi = linear_building_blocks(x7,X7);
U7 = u(2:N+1)*phi;
U = (1/12)*X7.^4-(1/6)*X7.^3+(1/6)*X7;
err7 = U7-U;


figure(1)
clf
plot(X1,U1);
hold on
plot(X2,U2);
plot(X3,U3);
plot(X4,U4);
plot(X5,U5);
plot(X6,U6);
plot(X7,U7);
plot(X7,U,'r')
figure(1)

format short e
format compact
display('L^infty errors')
max(abs(err1))
max(abs(err2))
max(abs(err3))
max(abs(err4))
max(abs(err5))
max(abs(err6))
max(abs(err7))
display(' ')
display('ratios of L^infty errors')
max(abs(err1))/max(abs(err2))
max(abs(err2))/max(abs(err3))
max(abs(err3))/max(abs(err4))
max(abs(err4))/max(abs(err5))
max(abs(err5))/max(abs(err6))
max(abs(err6))/max(abs(err7))
display(' ')

display('L^2 errors')
sqrt((X1(2)-X1(1))*sum(err1.^2))
sqrt((X2(2)-X2(1))*sum(err2.^2))
sqrt((X3(2)-X3(1))*sum(err3.^2))
sqrt((X4(2)-X4(1))*sum(err4.^2))
sqrt((X5(2)-X5(1))*sum(err5.^2))
sqrt((X6(2)-X6(1))*sum(err6.^2))
sqrt((X7(2)-X7(1))*sum(err7.^2))
display(' ')
display('ratios of L^2 errors')
sqrt((X1(2)-X1(1))*sum(err1.^2))/sqrt((X2(2)-X2(1))*sum(err2.^2))
sqrt((X2(2)-X2(1))*sum(err2.^2))/sqrt((X3(2)-X3(1))*sum(err3.^2))
sqrt((X3(2)-X3(1))*sum(err3.^2))/sqrt((X4(2)-X4(1))*sum(err4.^2))
sqrt((X4(2)-X4(1))*sum(err4.^2))/sqrt((X5(2)-X5(1))*sum(err5.^2))
sqrt((X5(2)-X5(1))*sum(err5.^2))/sqrt((X6(2)-X6(1))*sum(err6.^2))
sqrt((X6(2)-X6(1))*sum(err6.^2))/sqrt((X7(2)-X7(1))*sum(err7.^2))
display(' ')