1+ function [x ] = lsquare_plot_with_qr(t , y , basis , plot_step )
2+ % LSQUARE_PLOT_WITH_QR solve Ax = y for x in the least-square sense and
3+ % plot the least-square fit using QR decomposition.
4+ %
5+ % INPUT
6+ % t & y: m paired input cordinates
7+ % basis: cell array of n basis function handles that have t as input
8+ % these are used for forming A & plotting the least-square fit
9+ % plot_step: (OPTIONAL) define the step for least-square fit plot,
10+ % defaults to 0.01
11+ % OUTPUT
12+ % a plot of t & y and a least-square fit function for the data
13+ % x: least-square fit solution of Ax = y
14+
15+ % validate and possibly default input
16+ m = length(t );
17+ assert(m > 0 , ' length of t must be > 0'
18+ assert(length(y ) == m , ' length of y should be same as t'
19+ n = length(basis );
20+ assert(n > 0 , ' length of basis must be > 0'
21+ if nargin == 3 || plot_step == 0
22+ plot_step = 0.01 ;
23+ end
24+
25+ % form matrix A
26+ A = zeros(m ,n );
27+ for i = 1 : m
28+ for j = 1 : n
29+ A(i ,j ) = basis{j }(t(i ));
30+ end
31+ end
32+
33+ % find QR decomposition and w & U
34+ [Q ,R ] = qr(A );
35+ w = Q ' *y ;
36+ w = w(1 : n , : );
37+ U = R(1 : n , : );
38+
39+ % backward subs to solve x of Ux = w because U is upper triangular
40+ x = zeros(n ,1 );
41+ for i = n : - 1 : 1
42+ other_terms = 0 ;
43+ for j = n : - 1 : i + 1
44+ other_terms = U(i ,j )*x(j ) + other_terms ;
45+ end
46+ x(i ) = (w(i ) - other_terms ) / U(i ,i );
47+ end
48+
49+ % make the curve from basis functions
50+ numbers = min(t ) : plot_step : max(t );
51+ curve = x(1 ) * basis{1 }(numbers );
52+ for i = 2 : n
53+ curve = curve + x(i ) * basis{i }(numbers );
54+ end
55+
56+ % plot the given data points and the curve
57+ scatter(t ,y );
58+ hold on
59+ plot(numbers ,curve );
60+ hold off
61+
62+ end
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