--- /dev/null
+/* -*- C++ -*- ------------------------------------------------------------
+
+Copyright (c) 2007 Jesse Anders and Demian Nave http://cmldev.net/
+
+The Configurable Math Library (CML) is distributed under the terms of the
+Boost Software License, v1.0 (see cml/LICENSE for details).
+
+ *-----------------------------------------------------------------------*/
+/** @file
+ * @brief Compute the inverse of a matrix by LU factorization.
+ */
+
+#ifndef matrix_inverse_h
+#define matrix_inverse_h
+
+#include <vector>
+#include <cml/matrix/lu.h>
+
+namespace cml {
+namespace detail {
+
+/* Need to use a functional, since template functions cannot be
+ * specialized. _tag is used to specialize based upon dimension:
+ */
+template<typename MatT, int _tag> struct inverse_f;
+
+/* @todo: Reciprocal optimization for division by determinant.
+ */
+
+/* 2x2 inverse. Despite being marked for fixed_size matrices, this can
+ * be used for dynamic-sized ones also:
+ */
+template<typename MatT>
+struct inverse_f<MatT,2>
+{
+ typename MatT::temporary_type operator()(const MatT& M) const
+ {
+ typedef typename MatT::temporary_type temporary_type;
+ typedef typename temporary_type::value_type value_type;
+
+ /* Matrix containing the inverse: */
+ temporary_type Z;
+ cml::et::detail::Resize(Z,2,2);
+
+ /* Compute determinant and inverse: */
+ value_type D = value_type(1) / (M(0,0)*M(1,1) - M(0,1)*M(1,0));
+ Z(0,0) = M(1,1)*D; Z(0,1) = - M(0,1)*D;
+ Z(1,0) = - M(1,0)*D; Z(1,1) = M(0,0)*D;
+
+ return Z;
+ }
+};
+
+/* 3x3 inverse. Despite being marked for fixed_size matrices, this can
+ * be used for dynamic-sized ones also:
+ */
+template<typename MatT>
+struct inverse_f<MatT,3>
+{
+ /* [00 01 02]
+ * M = [10 11 12]
+ * [20 21 22]
+ */
+ typename MatT::temporary_type operator()(const MatT& M) const
+ {
+ /* Shorthand. */
+ typedef typename MatT::value_type value_type;
+
+ /* Compute cofactors for each entry: */
+ value_type m_00 = M(1,1)*M(2,2) - M(1,2)*M(2,1);
+ value_type m_01 = M(1,2)*M(2,0) - M(1,0)*M(2,2);
+ value_type m_02 = M(1,0)*M(2,1) - M(1,1)*M(2,0);
+
+ value_type m_10 = M(0,2)*M(2,1) - M(0,1)*M(2,2);
+ value_type m_11 = M(0,0)*M(2,2) - M(0,2)*M(2,0);
+ value_type m_12 = M(0,1)*M(2,0) - M(0,0)*M(2,1);
+
+ value_type m_20 = M(0,1)*M(1,2) - M(0,2)*M(1,1);
+ value_type m_21 = M(0,2)*M(1,0) - M(0,0)*M(1,2);
+ value_type m_22 = M(0,0)*M(1,1) - M(0,1)*M(1,0);
+
+ /* Compute determinant from the minors: */
+ value_type D =
+ value_type(1) / (M(0,0)*m_00 + M(0,1)*m_01 + M(0,2)*m_02);
+
+ /* Matrix containing the inverse: */
+ typename MatT::temporary_type Z;
+ cml::et::detail::Resize(Z,3,3);
+
+ /* Assign the inverse as (1/D) * (cofactor matrix)^T: */
+ Z(0,0) = m_00*D; Z(0,1) = m_10*D; Z(0,2) = m_20*D;
+ Z(1,0) = m_01*D; Z(1,1) = m_11*D; Z(1,2) = m_21*D;
+ Z(2,0) = m_02*D; Z(2,1) = m_12*D; Z(2,2) = m_22*D;
+
+ return Z;
+ }
+};
+
+/* 4x4 inverse. Despite being marked for fixed_size matrices, this can
+ * be used for dynamic-sized ones also:
+ */
+template<typename MatT>
+struct inverse_f<MatT,4>
+{
+ /* [00 01 02 03]
+ * M = [10 11 12 13]
+ * [20 21 22 23]
+ * [30 31 32 33]
+ *
+ * |11 12 13| |10 12 13|
+ * C00 = |21 22 23| C01 = |20 22 23|
+ * |31 32 33| |30 32 33|
+ *
+ * |10 11 13| |10 11 12|
+ * C02 = |20 21 23| C03 = |20 21 22|
+ * |30 31 33| |30 31 32|
+ */
+ typename MatT::temporary_type operator()(const MatT& M) const
+ {
+ /* Shorthand. */
+ typedef typename MatT::value_type value_type;
+
+ /* Common cofactors, rows 0,1: */
+ value_type m_22_33_23_32 = M(2,2)*M(3,3) - M(2,3)*M(3,2);
+ value_type m_23_30_20_33 = M(2,3)*M(3,0) - M(2,0)*M(3,3);
+ value_type m_20_31_21_30 = M(2,0)*M(3,1) - M(2,1)*M(3,0);
+ value_type m_21_32_22_31 = M(2,1)*M(3,2) - M(2,2)*M(3,1);
+ value_type m_23_31_21_33 = M(2,3)*M(3,1) - M(2,1)*M(3,3);
+ value_type m_20_32_22_30 = M(2,0)*M(3,2) - M(2,2)*M(3,0);
+
+ /* Compute minors: */
+ value_type d00
+ = M(1,1)*m_22_33_23_32+M(1,2)*m_23_31_21_33+M(1,3)*m_21_32_22_31;
+
+ value_type d01
+ = M(1,0)*m_22_33_23_32+M(1,2)*m_23_30_20_33+M(1,3)*m_20_32_22_30;
+
+ value_type d02
+ = M(1,0)*-m_23_31_21_33+M(1,1)*m_23_30_20_33+M(1,3)*m_20_31_21_30;
+
+ value_type d03
+ = M(1,0)*m_21_32_22_31+M(1,1)*-m_20_32_22_30+M(1,2)*m_20_31_21_30;
+
+ /* Compute minors: */
+ value_type d10
+ = M(0,1)*m_22_33_23_32+M(0,2)*m_23_31_21_33+M(0,3)*m_21_32_22_31;
+
+ value_type d11
+ = M(0,0)*m_22_33_23_32+M(0,2)*m_23_30_20_33+M(0,3)*m_20_32_22_30;
+
+ value_type d12
+ = M(0,0)*-m_23_31_21_33+M(0,1)*m_23_30_20_33+M(0,3)*m_20_31_21_30;
+
+ value_type d13
+ = M(0,0)*m_21_32_22_31+M(0,1)*-m_20_32_22_30+M(0,2)*m_20_31_21_30;
+
+ /* Common cofactors, rows 2,3: */
+ value_type m_02_13_03_12 = M(0,2)*M(1,3) - M(0,3)*M(1,2);
+ value_type m_03_10_00_13 = M(0,3)*M(1,0) - M(0,0)*M(1,3);
+ value_type m_00_11_01_10 = M(0,0)*M(1,1) - M(0,1)*M(1,0);
+ value_type m_01_12_02_11 = M(0,1)*M(1,2) - M(0,2)*M(1,1);
+ value_type m_03_11_01_13 = M(0,3)*M(1,1) - M(0,1)*M(1,3);
+ value_type m_00_12_02_10 = M(0,0)*M(1,2) - M(0,2)*M(1,0);
+
+ /* Compute minors (uses row 3 as the multipliers instead of row 0,
+ * which uses the same signs as row 0):
+ */
+ value_type d20
+ = M(3,1)*m_02_13_03_12+M(3,2)*m_03_11_01_13+M(3,3)*m_01_12_02_11;
+
+ value_type d21
+ = M(3,0)*m_02_13_03_12+M(3,2)*m_03_10_00_13+M(3,3)*m_00_12_02_10;
+
+ value_type d22
+ = M(3,0)*-m_03_11_01_13+M(3,1)*m_03_10_00_13+M(3,3)*m_00_11_01_10;
+
+ value_type d23
+ = M(3,0)*m_01_12_02_11+M(3,1)*-m_00_12_02_10+M(3,2)*m_00_11_01_10;
+
+ /* Compute minors: */
+ value_type d30
+ = M(2,1)*m_02_13_03_12+M(2,2)*m_03_11_01_13+M(2,3)*m_01_12_02_11;
+
+ value_type d31
+ = M(2,0)*m_02_13_03_12+M(2,2)*m_03_10_00_13+M(2,3)*m_00_12_02_10;
+
+ value_type d32
+ = M(2,0)*-m_03_11_01_13+M(2,1)*m_03_10_00_13+M(2,3)*m_00_11_01_10;
+
+ value_type d33
+ = M(2,0)*m_01_12_02_11+M(2,1)*-m_00_12_02_10+M(2,2)*m_00_11_01_10;
+
+ /* Finally, compute determinant from the minors, and assign the
+ * inverse as (1/D) * (cofactor matrix)^T:
+ */
+ typename MatT::temporary_type Z;
+ cml::et::detail::Resize(Z,4,4);
+
+ value_type D = value_type(1) /
+ (M(0,0)*d00 - M(0,1)*d01 + M(0,2)*d02 - M(0,3)*d03);
+ Z(0,0) = +d00*D; Z(0,1) = -d10*D; Z(0,2) = +d20*D; Z(0,3) = -d30*D;
+ Z(1,0) = -d01*D; Z(1,1) = +d11*D; Z(1,2) = -d21*D; Z(1,3) = +d31*D;
+ Z(2,0) = +d02*D; Z(2,1) = -d12*D; Z(2,2) = +d22*D; Z(2,3) = -d32*D;
+ Z(3,0) = -d03*D; Z(3,1) = +d13*D; Z(3,2) = -d23*D; Z(3,3) = +d33*D;
+
+ return Z;
+ }
+};
+
+/* If more extensive general linear algebra functionality is offered in
+ * future versions it may be useful to make the elementary row and column
+ * operations separate functions. For now they're simply performed in place,
+ * but the commented-out lines of code show where the calls to these functions
+ * should go if and when they become available.
+ */
+
+/* @todo: In-place version, and address memory allocation for pivot vector.
+ */
+
+/* General NxN inverse by Gauss-Jordan elimination with full pivoting: */
+template<typename MatT, int _tag>
+struct inverse_f
+{
+ typename MatT::temporary_type operator()(const MatT& M) const
+ {
+ /* Shorthand. */
+ typedef typename MatT::value_type value_type;
+
+ /* Size of matrix */
+ size_t N = M.rows();
+
+ /* Matrix containing the inverse: */
+ typename MatT::temporary_type Z;
+ cml::et::detail::Resize(Z,N,N);
+ Z = M;
+
+ /* For tracking pivots */
+ std::vector<size_t> row_index(N);
+ std::vector<size_t> col_index(N);
+ std::vector<size_t> pivoted(N,0);
+
+ /* For each column */
+ for (size_t i = 0; i < N; ++i) {
+
+ /* Find the pivot */
+ size_t row = 0, col = 0;
+ value_type max = value_type(0);
+ for (size_t j = 0; j < N; ++j) {
+ if (!pivoted[j]) {
+ for (size_t k = 0; k < N; ++k) {
+ if (!pivoted[k]) {
+ value_type mag = std::fabs(Z(j,k));
+ if (mag > max) {
+ max = mag;
+ row = j;
+ col = k;
+ }
+ }
+ }
+ }
+ }
+
+ /* TODO: Check max against epsilon here to catch singularity */
+
+ row_index[i] = row;
+ col_index[i] = col;
+
+ /* Swap rows if necessary */
+ if (row != col) {
+ /*Z.row_op_swap(row,col);*/
+ for (size_t j = 0; j < Z.cols(); ++j) {
+ std::swap(Z(row,j),Z(col,j));
+ }
+ }
+
+ /* Process pivot row */
+ pivoted[col] = true;
+ value_type pivot = Z(col,col);
+ Z(col,col) = value_type(1);
+ /*Z.row_op_mult(col,value_type(1)/pivot);*/
+ value_type k = value_type(1)/pivot;
+ for (size_t j = 0; j < Z.cols(); ++j) {
+ Z(col,j) *= k;
+ }
+
+ /* Process other rows */
+ for (size_t j = 0; j < N; ++j) {
+ if (j != col) {
+ value_type mult = -Z(j,col);
+ Z(j,col) = value_type(0);
+ /*Z.row_op_add_mult(col,j,mult);*/
+ for (size_t k = 0; k < Z.cols(); ++k) {
+ Z(j,k) += mult * Z(col,k);
+ }
+ }
+ }
+ }
+
+ /* Swap columns if necessary */
+ for (int i = N-1; i >= 0; --i) {
+ if (row_index[i] != col_index[i]) {
+ /*Z.col_op_swap(row_index[i],col_index[i]);*/
+ for (size_t j = 0; j < Z.rows(); ++j) {
+ std::swap(Z(j,row_index[i]),Z(j,col_index[i]));
+ }
+ }
+ }
+
+ /* Return result */
+ return Z;
+ }
+};
+
+/* Inversion by LU factorization is turned off for now due to lack of
+ * pivoting in the implementation, but we may switch back to it at some future
+ * time.
+ */
+
+#if 0
+
+/* General NxN inverse by LU factorization: */
+template<typename MatT, int _tag>
+struct inverse_f
+{
+ typename MatT::temporary_type operator()(const MatT& M) const
+ {
+ /* Shorthand. */
+ typedef typename MatT::value_type value_type;
+
+ /* Compute LU factorization: */
+ size_t N = M.rows();
+ typename MatT::temporary_type LU;
+ cml::et::detail::Resize(LU,N,N);
+ LU = lu(M);
+
+ /* Matrix containing the inverse: */
+ typename MatT::temporary_type Z;
+ cml::et::detail::Resize(Z,N,N);
+
+ typename MatT::col_vector_type v, x;
+ cml::et::detail::Resize(v,N);
+ cml::et::detail::Resize(x,N);
+ for(size_t i = 0; i < N; ++i)
+ v[i] = value_type(0);
+ /* XXX Need a fill() function here. */
+
+ /* Use lu_solve to solve M*x = v for x, where v = [0 ... 1 ... 0]^T: */
+ for(size_t i = 0; i < N; ++i) {
+ v[i] = 1.;
+ x = lu_solve(LU,v);
+
+ /* x is column i of the inverse of LU: */
+ for(size_t k = 0; k < N; ++ k) {
+ Z(k,i) = x[k];
+ }
+ v[i] = 0.;
+ }
+
+ return Z;
+ }
+
+};
+
+#endif
+
+/* Note: force_NxN is for checking general NxN inversion against the special-
+ * case 2x2, 3x3 and 4x4 code. I'm leaving it in for now since we may need to
+ * test the NxN code further if the implementation changes. At some future
+ * time when the implementation is stable, everything related to force_NxN can
+ * be taken out.
+ */
+
+/* Note: Commenting the force_NxN stuff out, but leaving the code here in
+ * case we need to do more testing in the future.
+ */
+
+/* Generator for the inverse functional for fixed-size matrices: */
+template<typename MatT> typename MatT::temporary_type
+inverse(const MatT& M, fixed_size_tag/*, bool force_NxN*/)
+{
+ /* Require a square matrix: */
+ cml::et::CheckedSquare(M, fixed_size_tag());
+
+ /*
+ if (force_NxN) {
+ return inverse_f<MatT,0>()(M);
+ } else {
+ */
+ return inverse_f<MatT,MatT::array_rows>()(M);
+ /*
+ }
+ */
+}
+
+/* Generator for the inverse functional for dynamic-size matrices: */
+template<typename MatT> typename MatT::temporary_type
+inverse(const MatT& M, dynamic_size_tag/*, bool force_NxN*/)
+{
+ /* Require a square matrix: */
+ cml::et::CheckedSquare(M, dynamic_size_tag());
+
+ /*
+ if (force_NxN) {
+ return inverse_f<MatT,0>()(M);
+ } else {
+ */
+ /* Dispatch based upon the matrix dimension: */
+ switch(M.rows()) {
+ case 2: return inverse_f<MatT,2>()(M); // 2x2
+ case 3: return inverse_f<MatT,3>()(M); // 3x3
+ case 4: return inverse_f<MatT,4>()(M); // 4x4
+ default: return inverse_f<MatT,0>()(M); // > 4x4 (or 1x1)
+ }
+ /*
+ }
+ */
+}
+
+} // namespace detail
+
+/** Inverse of a matrix. */
+template<typename E, class AT, typename BO, typename L> inline
+typename matrix<E,AT,BO,L>::temporary_type
+inverse(const matrix<E,AT,BO,L>& M/*, bool force_NxN = false*/)
+{
+ typedef typename matrix<E,AT,BO,L>::size_tag size_tag;
+ return detail::inverse(M,size_tag()/*,force_NxN*/);
+}
+
+/** Inverse of a matrix expression. */
+template<typename XprT> inline
+typename et::MatrixXpr<XprT>::temporary_type
+inverse(const et::MatrixXpr<XprT>& e/*, bool force_NxN = false*/)
+{
+ typedef typename et::MatrixXpr<XprT>::size_tag size_tag;
+ return detail::inverse(e,size_tag()/*,force_NxN*/);
+}
+
+} // namespace cml
+
+#endif
+
+// -------------------------------------------------------------------------
+// vim:ft=cpp
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