1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
|
/// @ref gtx_matrix_factorisation
namespace glm
{
template <length_t C, length_t R, typename T, qualifier Q>
GLM_FUNC_QUALIFIER mat<C, R, T, Q> flipud(mat<C, R, T, Q> const& in)
{
mat<R, C, T, Q> tin = transpose(in);
tin = fliplr(tin);
mat<C, R, T, Q> out = transpose(tin);
return out;
}
template <length_t C, length_t R, typename T, qualifier Q>
GLM_FUNC_QUALIFIER mat<C, R, T, Q> fliplr(mat<C, R, T, Q> const& in)
{
mat<C, R, T, Q> out;
for (length_t i = 0; i < C; i++)
{
out[i] = in[(C - i) - 1];
}
return out;
}
template <length_t C, length_t R, typename T, qualifier Q>
GLM_FUNC_QUALIFIER void qr_decompose(mat<C, R, T, Q> const& in, mat<(C < R ? C : R), R, T, Q>& q, mat<C, (C < R ? C : R), T, Q>& r)
{
// Uses modified Gram-Schmidt method
// Source: https://en.wikipedia.org/wiki/Gram–Schmidt_process
// And https://en.wikipedia.org/wiki/QR_decomposition
//For all the linearly independs columns of the input...
// (there can be no more linearly independents columns than there are rows.)
for (length_t i = 0; i < (C < R ? C : R); i++)
{
//Copy in Q the input's i-th column.
q[i] = in[i];
//j = [0,i[
// Make that column orthogonal to all the previous ones by substracting to it the non-orthogonal projection of all the previous columns.
// Also: Fill the zero elements of R
for (length_t j = 0; j < i; j++)
{
q[i] -= dot(q[i], q[j])*q[j];
r[j][i] = 0;
}
//Now, Q i-th column is orthogonal to all the previous columns. Normalize it.
q[i] = normalize(q[i]);
//j = [i,C[
//Finally, compute the corresponding coefficients of R by computing the projection of the resulting column on the other columns of the input.
for (length_t j = i; j < C; j++)
{
r[j][i] = dot(in[j], q[i]);
}
}
}
template <length_t C, length_t R, typename T, qualifier Q>
GLM_FUNC_QUALIFIER void rq_decompose(mat<C, R, T, Q> const& in, mat<(C < R ? C : R), R, T, Q>& r, mat<C, (C < R ? C : R), T, Q>& q)
{
// From https://en.wikipedia.org/wiki/QR_decomposition:
// The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices.
// QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column.
// RQ decomposition is Gram–Schmidt orthogonalization of rows of A, started from the last row.
mat<R, C, T, Q> tin = transpose(in);
tin = fliplr(tin);
mat<R, (C < R ? C : R), T, Q> tr;
mat<(C < R ? C : R), C, T, Q> tq;
qr_decompose(tin, tq, tr);
tr = fliplr(tr);
r = transpose(tr);
r = fliplr(r);
tq = fliplr(tq);
q = transpose(tq);
}
} //namespace glm
|
