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+/// @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