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### Lecture 23: Apr 2/3, 2013 ###
--------- today.txt --------
7.1 Basic Quadrature Rules
  7.1.3 Degree of exactness; order of accuracy
--------- review.txt --------
6.3 The QR factorization
  example
  H_n ... H_2 H_1 A = R  => virtual QR factorization
Chapter 7 Quadrature
  rules of the game
7.1 Basic Quadrature Rules
  7.1.1 Three simple rules: trapezoid, midpoint, Simpson
  7.1.2 Composite quadrature rules
-------------------
|| 7.1.3 ||
------ section 7.1.3 ---------
function Q = compsimp(f, a, b, N)  % compsimp.m
   w = zeros(1, 2*N + 1);
   w(1) = w(end) = 1/6;
   w(2:2:end-1) = 2/3;
   w(3:2:end-2) = 1/3;
   x = linspace(a, b, 2*N + 1)';
   h = (b - a)/N;
   Q = h*(w*f(x));

f = @(x) exp(x);
Ns = 2.^(0:10);
Qs = [];
for N = Ns; Qs = [Qs compsimp(f, -4, 2, N)]; end
Qx = exp(2) - exp(-4);
loglog(Ns, 1./abs(Qs - Qx), '+')
p = 4;
loglog(Ns, 1./abs(Qs - Qx), '+', Ns, Ns.^p, '-')

f = @(x) x >= 1/5;
Qs = [];
for N = Ns; Qs = [Qs compsimp(f, 0, 1, N)]; end
Qx = 4/5;
loglog(Ns, 1./abs(Qs - Qx), '+', Ns, Ns.^p, '-')
p = 1;
loglog(Ns, 1./abs(Qs - Qx), '+', Ns, Ns.^p, '-')

f = @(x) x.^(1/3);
Qs = [];
for N = Ns; Qs = [Qs compsimp(f, 0, 1, N)]; end
Qx = 3/4;
loglog(Ns, 1./abs(Qs - Qx), '+', Ns, Ns.^p, '-')
p = 1;
loglog(Ns, 1./abs(Qs - Qx), '+', Ns, Ns.^p, '-')
p = 4/3;
loglog(Ns, 1./abs(Qs - Qx), '+', Ns, Ns.^p, '-')
------------------------------------
      H y = - sign(y_1) ||y|| e_1  prove it!
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