n=4;
m=40;
randn('state',0);
u = linspace(-1,1,m);
v = 1./(5+40*u.^2) + 0.1*u.^3 + 0.01*randn(1,m);
a = -1/3; b = 1/3;
u1 = u(find(u<a)); m1 = length(u1);
u2 = u(find((u >= a) & (u<b))); m2 = length(u2);
u3 = u(find((u >= b))); m3 = length(u3);
A1 = vander(u1'); A1 = fliplr(A1(:,m1-n+[1:n]));
A2 = vander(u2'); A2 = fliplr(A2(:,m2-n+[1:n]));
A3 = vander(u3'); A3 = fliplr(A3(:,m3-n+[1:n]));
fprintf(1,'Computing splines in the case of L2-norm...');
cvx_begin
variables x1(n) x2(n) x3(n)
minimize ( norm( [A1*x1;A2*x2;A3*x3] - v') )
[1 a a^2 a^3]*x1 == [1 a a^2 a^3]*x2;
[0 1 2*a 3*a^2]*x1 == [0 1 2*a 3*a^2]*x2;
[0 0 2 6*a ]*x1 == [0 0 2 6*a ]*x2;
[1 b b^2 b^3]*x2 == [1 b b^2 b^3]*x3;
[0 1 2*b 3*b^2]*x2 == [0 1 2*b 3*b^2]*x3;
[0 0 2 6*b ]*x2 == [0 0 2 6*b ]*x3;
cvx_end
fprintf(1,'Done! \n');
fprintf(1,'Computing splines in the case of Linfty-norm...');
cvx_begin
variables xl1(n) xl2(n) xl3(n)
minimize ( norm( [A1*xl1;A2*xl2;A3*xl3] - v', inf) )
[1 a a^2 a^3]*xl1 == [1 a a^2 a^3]*xl2;
[0 1 2*a 3*a^2]*xl1 == [0 1 2*a 3*a^2]*xl2;
[0 0 2 6*a ]*xl1 == [0 0 2 6*a ]*xl2;
[1 b b^2 b^3]*xl2 == [1 b b^2 b^3]*xl3;
[0 1 2*b 3*b^2]*xl2 == [0 1 2*b 3*b^2]*xl3;
[0 0 2 6*b ]*xl2 == [0 0 2 6*b ]*xl3;
cvx_end
fprintf(1,'Done! \n');
u1s = linspace(-1.0,a,1000)';
p1 = x1(1) + x1(2)*u1s + x1(3)*u1s.^2 + x1(4).*u1s.^3;
p1l1 = xl1(1) + xl1(2)*u1s + xl1(3)*u1s.^2 + xl1(4).*u1s.^3;
u2s = linspace(a,b,1000)';
p2 = x2(1) + x2(2)*u2s + x2(3)*u2s.^2 + x2(4).*u2s.^3;
p2l1 = xl2(1) + xl2(2)*u2s + xl2(3)*u2s.^2 + xl2(4).*u2s.^3;
u3s = linspace(b,1.0,1000)';
p3 = x3(1) + x3(2)*u3s + x3(3)*u3s.^2 + x3(4).*u3s.^3;
p3l1 = xl3(1) + xl3(2)*u3s + xl3(3)*u3s.^2 + xl3(4).*u3s.^3;
us = [u1s;u2s;u3s];
p = [p1;p2;p3];
pl = [p1l1;p2l1;p3l1];
d = plot(us,p,'b-',u,v,'go', us,pl,'r--',...
[-1 -1], [-0.1 0.25], 'k--', [1 1], [-0.1 0.25], 'k--', ...
[a a], [-0.1 0.25], 'k--', [b b], [-0.1 0.25], 'k--');
title('Approximation using 2 cubic splines');
xlabel('u');
ylabel('f(u)');
legend('L_2 norm','Data points','L_{\infty} norm', 'Location','Best');
Computing splines in the case of L2-norm...
Calling sedumi: 47 variables, 13 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Put 6 free variables in a quadratic cone
eqs m = 13, order n = 5, dim = 49, blocks = 3
nnz(A) = 197 + 0, nnz(ADA) = 169, nnz(L) = 91
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 4.15E+00 0.000
1 : -1.74E-01 8.17E-01 0.000 0.1969 0.9000 0.9000 1.71 1 1 1.2E+00
2 : -8.96E-02 1.72E-01 0.000 0.2111 0.9000 0.9000 1.25 1 1 3.0E-01
3 : -1.16E-01 5.70E-03 0.000 0.0331 0.9900 0.9900 0.91 1 1 4.3E-03
4 : -1.17E-01 4.68E-05 0.302 0.0082 0.9990 0.9990 1.00 1 1 4.1E-05
5 : -1.17E-01 3.23E-06 0.000 0.0691 0.9900 0.9900 1.00 1 1 4.0E-06
6 : -1.17E-01 1.30E-08 0.152 0.0040 0.9990 0.9990 1.00 1 1 1.6E-08
7 : -1.17E-01 4.75E-10 0.000 0.0366 0.9900 0.9902 1.00 1 1 1.1E-09
iter seconds digits c*x b*y
7 0.0 8.9 -1.1660335735e-01 -1.1660335750e-01
|Ax-b| = 1.5e-10, [Ay-c]_+ = 2.3E-11, |x|= 2.6e+00, |y|= 1.9e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 4.000E-02 1.000E-02
Max-norms: ||b||=1, ||c|| = 2.018994e-01,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 6.24939.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.116603
Done!
Computing splines in the case of Linfty-norm...
Calling sedumi: 126 variables, 53 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 6 free variables
eqs m = 53, order n = 133, dim = 133, blocks = 1
nnz(A) = 592 + 0, nnz(ADA) = 577, nnz(L) = 315
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 8.39E+01 0.000
1 : -4.68E-01 4.44E+01 0.000 0.5294 0.9000 0.9000 2.90 1 1 3.8E+01
2 : -3.12E-02 2.14E+01 0.000 0.4813 0.9000 0.9000 12.63 1 1 1.7E+00
3 : -2.65E-02 1.07E+01 0.000 0.4985 0.9000 0.9000 1.83 1 1 7.1E-01
4 : -3.14E-02 4.14E+00 0.000 0.3886 0.9000 0.9000 1.17 1 1 2.7E-01
5 : -3.16E-02 1.48E+00 0.000 0.3575 0.9000 0.9000 1.10 1 1 9.6E-02
6 : -3.20E-02 1.16E-01 0.000 0.0785 0.9900 0.9900 1.02 1 1 7.5E-03
7 : -3.20E-02 1.99E-05 0.000 0.0002 0.9999 0.9999 1.00 1 1
iter seconds digits c*x b*y
7 0.0 15.1 -3.2038334523e-02 -3.2038334523e-02
|Ax-b| = 1.4e-16, [Ay-c]_+ = 1.2E-16, |x|= 7.3e-01, |y|= 2.4e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 3.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 2.018994e-01,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 6.03999.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0320383
Done!