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+///////////////////////////////////////////////////////////////////////////////
+// Copyright 2018 John Maddock. Distributed under the Boost
+// Software License, Version 1.0. (See accompanying file
+// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+#include <boost/multiprecision/eigen.hpp>
+#include <iostream>
+#include <Eigen/Dense>
+#include "test.hpp"
+
+using namespace Eigen;
+
+template <class T>
+struct related_number
+{
+ typedef T type;
+};
+
+template <class Num>
+void example1()
+{
+ // expected results first:
+ Matrix<Num, 2, 2> r1, r2;
+ r1 << 3, 5, 4, 8;
+ r2 << -1, -1, 2, 0;
+ Matrix<Num, 3, 1> r3;
+ r3 << -1, -4, -6;
+
+ Matrix<Num, 2, 2> a;
+ a << 1, 2, 3, 4;
+ Matrix<Num, Dynamic, Dynamic> b(2, 2);
+ b << 2, 3, 1, 4;
+ std::cout << "a + b =\n"
+ << a + b << std::endl;
+ BOOST_CHECK_EQUAL(a + b, r1);
+ std::cout << "a - b =\n"
+ << a - b << std::endl;
+ BOOST_CHECK_EQUAL(a - b, r2);
+ std::cout << "Doing a += b;" << std::endl;
+ a += b;
+ std::cout << "Now a =\n"
+ << a << std::endl;
+ Matrix<Num, 3, 1> v(1, 2, 3);
+ Matrix<Num, 3, 1> w(1, 0, 0);
+ std::cout << "-v + w - v =\n"
+ << -v + w - v << std::endl;
+ BOOST_CHECK_EQUAL(-v + w - v, r3);
+}
+
+template <class Num>
+void example2()
+{
+ Matrix<Num, 2, 2> a;
+ a << 1, 2, 3, 4;
+ Matrix<Num, 3, 1> v(1, 2, 3);
+ std::cout << "a * 2.5 =\n"
+ << a * 2.5 << std::endl;
+ std::cout << "0.1 * v =\n"
+ << 0.1 * v << std::endl;
+ std::cout << "Doing v *= 2;" << std::endl;
+ v *= 2;
+ std::cout << "Now v =\n"
+ << v << std::endl;
+ Num n(4);
+ std::cout << "Doing v *= Num;" << std::endl;
+ v *= n;
+ std::cout << "Now v =\n"
+ << v << std::endl;
+ typedef typename related_number<Num>::type related_type;
+ related_type r(6);
+ std::cout << "Doing v *= RelatedType;" << std::endl;
+ v *= r;
+ std::cout << "Now v =\n"
+ << v << std::endl;
+ std::cout << "RelatedType * v =\n"
+ << r * v << std::endl;
+ std::cout << "Doing v *= RelatedType^2;" << std::endl;
+ v *= r * r;
+ std::cout << "Now v =\n"
+ << v << std::endl;
+ std::cout << "RelatedType^2 * v =\n"
+ << r * r * v << std::endl;
+}
+
+template <class Num>
+void example3()
+{
+ using namespace std;
+ Matrix<Num, Dynamic, Dynamic> a = Matrix<Num, Dynamic, Dynamic>::Random(2, 2);
+ cout << "Here is the matrix a\n"
+ << a << endl;
+ cout << "Here is the matrix a^T\n"
+ << a.transpose() << endl;
+ cout << "Here is the conjugate of a\n"
+ << a.conjugate() << endl;
+ cout << "Here is the matrix a^*\n"
+ << a.adjoint() << endl;
+}
+
+template <class Num>
+void example4()
+{
+ Matrix<Num, 2, 2> mat;
+ mat << 1, 2,
+ 3, 4;
+ Matrix<Num, 2, 1> u(-1, 1), v(2, 0);
+ std::cout << "Here is mat*mat:\n"
+ << mat * mat << std::endl;
+ std::cout << "Here is mat*u:\n"
+ << mat * u << std::endl;
+ std::cout << "Here is u^T*mat:\n"
+ << u.transpose() * mat << std::endl;
+ std::cout << "Here is u^T*v:\n"
+ << u.transpose() * v << std::endl;
+ std::cout << "Here is u*v^T:\n"
+ << u * v.transpose() << std::endl;
+ std::cout << "Let's multiply mat by itself" << std::endl;
+ mat = mat * mat;
+ std::cout << "Now mat is mat:\n"
+ << mat << std::endl;
+}
+
+template <class Num>
+void example5()
+{
+ using namespace std;
+ Matrix<Num, 3, 1> v(1, 2, 3);
+ Matrix<Num, 3, 1> w(0, 1, 2);
+ cout << "Dot product: " << v.dot(w) << endl;
+ Num dp = v.adjoint() * w; // automatic conversion of the inner product to a scalar
+ cout << "Dot product via a matrix product: " << dp << endl;
+ cout << "Cross product:\n"
+ << v.cross(w) << endl;
+}
+
+template <class Num>
+void example6()
+{
+ using namespace std;
+ Matrix<Num, 2, 2> mat;
+ mat << 1, 2,
+ 3, 4;
+ cout << "Here is mat.sum(): " << mat.sum() << endl;
+ cout << "Here is mat.prod(): " << mat.prod() << endl;
+ cout << "Here is mat.mean(): " << mat.mean() << endl;
+ cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;
+ cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;
+ cout << "Here is mat.trace(): " << mat.trace() << endl;
+}
+
+template <class Num>
+void example7()
+{
+ using namespace std;
+
+ Array<Num, Dynamic, Dynamic> m(2, 2);
+
+ // assign some values coefficient by coefficient
+ m(0, 0) = 1.0;
+ m(0, 1) = 2.0;
+ m(1, 0) = 3.0;
+ m(1, 1) = m(0, 1) + m(1, 0);
+
+ // print values to standard output
+ cout << m << endl
+ << endl;
+
+ // using the comma-initializer is also allowed
+ m << 1.0, 2.0,
+ 3.0, 4.0;
+
+ // print values to standard output
+ cout << m << endl;
+}
+
+template <class Num>
+void example8()
+{
+ using namespace std;
+ Array<Num, Dynamic, Dynamic> a(3, 3);
+ Array<Num, Dynamic, Dynamic> b(3, 3);
+ a << 1, 2, 3,
+ 4, 5, 6,
+ 7, 8, 9;
+ b << 1, 2, 3,
+ 1, 2, 3,
+ 1, 2, 3;
+
+ // Adding two arrays
+ cout << "a + b = " << endl
+ << a + b << endl
+ << endl;
+ // Subtracting a scalar from an array
+ cout << "a - 2 = " << endl
+ << a - 2 << endl;
+}
+
+template <class Num>
+void example9()
+{
+ using namespace std;
+ Array<Num, Dynamic, Dynamic> a(2, 2);
+ Array<Num, Dynamic, Dynamic> b(2, 2);
+ a << 1, 2,
+ 3, 4;
+ b << 5, 6,
+ 7, 8;
+ cout << "a * b = " << endl
+ << a * b << endl;
+}
+
+template <class Num>
+void example10()
+{
+ using namespace std;
+ Array<Num, Dynamic, 1> a = Array<Num, Dynamic, 1>::Random(5);
+ a *= 2;
+ cout << "a =" << endl
+ << a << endl;
+ cout << "a.abs() =" << endl
+ << a.abs() << endl;
+ cout << "a.abs().sqrt() =" << endl
+ << a.abs().sqrt() << endl;
+ cout << "a.min(a.abs().sqrt()) =" << endl
+ << (a.min)(a.abs().sqrt()) << endl;
+}
+
+template <class Num>
+void example11()
+{
+ using namespace std;
+ Matrix<Num, Dynamic, Dynamic> m(2, 2);
+ Matrix<Num, Dynamic, Dynamic> n(2, 2);
+ Matrix<Num, Dynamic, Dynamic> result(2, 2);
+ m << 1, 2,
+ 3, 4;
+ n << 5, 6,
+ 7, 8;
+ result = m * n;
+ cout << "-- Matrix m*n: --" << endl
+ << result << endl
+ << endl;
+ result = m.array() * n.array();
+ cout << "-- Array m*n: --" << endl
+ << result << endl
+ << endl;
+ result = m.cwiseProduct(n);
+ cout << "-- With cwiseProduct: --" << endl
+ << result << endl
+ << endl;
+ result = m.array() + 4;
+ cout << "-- Array m + 4: --" << endl
+ << result << endl
+ << endl;
+}
+
+template <class Num>
+void example12()
+{
+ using namespace std;
+ Matrix<Num, Dynamic, Dynamic> m(2, 2);
+ Matrix<Num, Dynamic, Dynamic> n(2, 2);
+ Matrix<Num, Dynamic, Dynamic> result(2, 2);
+ m << 1, 2,
+ 3, 4;
+ n << 5, 6,
+ 7, 8;
+
+ result = (m.array() + 4).matrix() * m;
+ cout << "-- Combination 1: --" << endl
+ << result << endl
+ << endl;
+ result = (m.array() * n.array()).matrix() * m;
+ cout << "-- Combination 2: --" << endl
+ << result << endl
+ << endl;
+}
+
+template <class Num>
+void example13()
+{
+ using namespace std;
+ Matrix<Num, Dynamic, Dynamic> m(4, 4);
+ m << 1, 2, 3, 4,
+ 5, 6, 7, 8,
+ 9, 10, 11, 12,
+ 13, 14, 15, 16;
+ cout << "Block in the middle" << endl;
+ cout << m.template block<2, 2>(1, 1) << endl
+ << endl;
+ for (int i = 1; i <= 3; ++i)
+ {
+ cout << "Block of size " << i << "x" << i << endl;
+ cout << m.block(0, 0, i, i) << endl
+ << endl;
+ }
+}
+
+template <class Num>
+void example14()
+{
+ using namespace std;
+ Array<Num, 2, 2> m;
+ m << 1, 2,
+ 3, 4;
+ Array<Num, 4, 4> a = Array<Num, 4, 4>::Constant(0.6);
+ cout << "Here is the array a:" << endl
+ << a << endl
+ << endl;
+ a.template block<2, 2>(1, 1) = m;
+ cout << "Here is now a with m copied into its central 2x2 block:" << endl
+ << a << endl
+ << endl;
+ a.block(0, 0, 2, 3) = a.block(2, 1, 2, 3);
+ cout << "Here is now a with bottom-right 2x3 block copied into top-left 2x2 block:" << endl
+ << a << endl
+ << endl;
+}
+
+template <class Num>
+void example15()
+{
+ using namespace std;
+ Eigen::Matrix<Num, Dynamic, Dynamic> m(3, 3);
+ m << 1, 2, 3,
+ 4, 5, 6,
+ 7, 8, 9;
+ cout << "Here is the matrix m:" << endl
+ << m << endl;
+ cout << "2nd Row: " << m.row(1) << endl;
+ m.col(2) += 3 * m.col(0);
+ cout << "After adding 3 times the first column into the third column, the matrix m is:\n";
+ cout << m << endl;
+}
+
+template <class Num>
+void example16()
+{
+ using namespace std;
+ Matrix<Num, 4, 4> m;
+ m << 1, 2, 3, 4,
+ 5, 6, 7, 8,
+ 9, 10, 11, 12,
+ 13, 14, 15, 16;
+ cout << "m.leftCols(2) =" << endl
+ << m.leftCols(2) << endl
+ << endl;
+ cout << "m.bottomRows<2>() =" << endl
+ << m.template bottomRows<2>() << endl
+ << endl;
+ m.topLeftCorner(1, 3) = m.bottomRightCorner(3, 1).transpose();
+ cout << "After assignment, m = " << endl
+ << m << endl;
+}
+
+template <class Num>
+void example17()
+{
+ using namespace std;
+ Array<Num, Dynamic, 1> v(6);
+ v << 1, 2, 3, 4, 5, 6;
+ cout << "v.head(3) =" << endl
+ << v.head(3) << endl
+ << endl;
+ cout << "v.tail<3>() = " << endl
+ << v.template tail<3>() << endl
+ << endl;
+ v.segment(1, 4) *= 2;
+ cout << "after 'v.segment(1,4) *= 2', v =" << endl
+ << v << endl;
+}
+
+template <class Num>
+void example18()
+{
+ using namespace std;
+ Matrix<Num, 2, 2> mat;
+ mat << 1, 2,
+ 3, 4;
+ cout << "Here is mat.sum(): " << mat.sum() << endl;
+ cout << "Here is mat.prod(): " << mat.prod() << endl;
+ cout << "Here is mat.mean(): " << mat.mean() << endl;
+ cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;
+ cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;
+ cout << "Here is mat.trace(): " << mat.trace() << endl;
+
+ BOOST_CHECK_EQUAL(mat.sum(), 10);
+ BOOST_CHECK_EQUAL(mat.prod(), 24);
+ BOOST_CHECK_EQUAL(mat.mean(), Num(5) / 2);
+ BOOST_CHECK_EQUAL(mat.minCoeff(), 1);
+ BOOST_CHECK_EQUAL(mat.maxCoeff(), 4);
+ BOOST_CHECK_EQUAL(mat.trace(), 5);
+}
+
+template <class Num>
+void example18a()
+{
+ using namespace std;
+ Matrix<Num, 2, 2> mat;
+ mat << 1, 2,
+ 3, 4;
+ cout << "Here is mat.sum(): " << mat.sum() << endl;
+ cout << "Here is mat.prod(): " << mat.prod() << endl;
+ cout << "Here is mat.mean(): " << mat.mean() << endl;
+ //cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;
+ //cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;
+ cout << "Here is mat.trace(): " << mat.trace() << endl;
+}
+
+template <class Num>
+void example19()
+{
+ using namespace std;
+ Matrix<Num, Dynamic, 1> v(2);
+ Matrix<Num, Dynamic, Dynamic> m(2, 2), n(2, 2);
+
+ v << -1,
+ 2;
+
+ m << 1, -2,
+ -3, 4;
+ cout << "v.squaredNorm() = " << v.squaredNorm() << endl;
+ cout << "v.norm() = " << v.norm() << endl;
+ cout << "v.lpNorm<1>() = " << v.template lpNorm<1>() << endl;
+ cout << "v.lpNorm<Infinity>() = " << v.template lpNorm<Infinity>() << endl;
+ cout << endl;
+ cout << "m.squaredNorm() = " << m.squaredNorm() << endl;
+ cout << "m.norm() = " << m.norm() << endl;
+ cout << "m.lpNorm<1>() = " << m.template lpNorm<1>() << endl;
+ cout << "m.lpNorm<Infinity>() = " << m.template lpNorm<Infinity>() << endl;
+}
+
+template <class Num>
+void example20()
+{
+ using namespace std;
+ Matrix<Num, 3, 3> A;
+ Matrix<Num, 3, 1> b;
+ A << 1, 2, 3, 4, 5, 6, 7, 8, 10;
+ b << 3, 3, 4;
+ cout << "Here is the matrix A:\n"
+ << A << endl;
+ cout << "Here is the vector b:\n"
+ << b << endl;
+ Matrix<Num, 3, 1> x = A.colPivHouseholderQr().solve(b);
+ cout << "The solution is:\n"
+ << x << endl;
+}
+
+template <class Num>
+void example21()
+{
+ using namespace std;
+ Matrix<Num, 2, 2> A, b;
+ A << 2, -1, -1, 3;
+ b << 1, 2, 3, 1;
+ cout << "Here is the matrix A:\n"
+ << A << endl;
+ cout << "Here is the right hand side b:\n"
+ << b << endl;
+ Matrix<Num, 2, 2> x = A.ldlt().solve(b);
+ cout << "The solution is:\n"
+ << x << endl;
+}
+
+template <class Num>
+void example22()
+{
+ using namespace std;
+ Matrix<Num, Dynamic, Dynamic> A = Matrix<Num, Dynamic, Dynamic>::Random(100, 100);
+ Matrix<Num, Dynamic, Dynamic> b = Matrix<Num, Dynamic, Dynamic>::Random(100, 50);
+ Matrix<Num, Dynamic, Dynamic> x = A.fullPivLu().solve(b);
+ Matrix<Num, Dynamic, Dynamic> axmb = A * x - b;
+ double relative_error = static_cast<double>(abs(axmb.norm() / b.norm())); // norm() is L2 norm
+ cout << "norm1 = " << axmb.norm() << endl;
+ cout << "norm2 = " << b.norm() << endl;
+ cout << "The relative error is:\n"
+ << relative_error << endl;
+}
+
+template <class Num>
+void example23()
+{
+ using namespace std;
+ Matrix<Num, 2, 2> A;
+ A << 1, 2, 2, 3;
+ cout << "Here is the matrix A:\n"
+ << A << endl;
+ SelfAdjointEigenSolver<Matrix<Num, 2, 2> > eigensolver(A);
+ if (eigensolver.info() != Success)
+ {
+ std::cout << "Eigenvalue solver failed!" << endl;
+ }
+ else
+ {
+ cout << "The eigenvalues of A are:\n"
+ << eigensolver.eigenvalues() << endl;
+ cout << "Here's a matrix whose columns are eigenvectors of A \n"
+ << "corresponding to these eigenvalues:\n"
+ << eigensolver.eigenvectors() << endl;
+ }
+}
+
+template <class Num>
+void example24()
+{
+ using namespace std;
+ Matrix<Num, 3, 3> A;
+ A << 1, 2, 1,
+ 2, 1, 0,
+ -1, 1, 2;
+ cout << "Here is the matrix A:\n"
+ << A << endl;
+ cout << "The determinant of A is " << A.determinant() << endl;
+ cout << "The inverse of A is:\n"
+ << A.inverse() << endl;
+}
+
+template <class Num>
+void test_integer_type()
+{
+ example1<Num>();
+ //example2<Num>();
+ example18<Num>();
+}
+
+template <class Num>
+void test_float_type()
+{
+ std::cout << "Epsilon = " << Eigen::NumTraits<Num>::epsilon() << std::endl;
+ std::cout << "Dummy Prec = " << Eigen::NumTraits<Num>::dummy_precision() << std::endl;
+ std::cout << "Highest = " << Eigen::NumTraits<Num>::highest() << std::endl;
+ std::cout << "Lowest = " << Eigen::NumTraits<Num>::lowest() << std::endl;
+ std::cout << "Digits10 = " << Eigen::NumTraits<Num>::digits10() << std::endl;
+
+ example1<Num>();
+ example2<Num>();
+ example4<Num>();
+ example5<Num>();
+ example6<Num>();
+ example7<Num>();
+ example8<Num>();
+ example9<Num>();
+ example10<Num>();
+ example11<Num>();
+ example12<Num>();
+ example13<Num>();
+ example14<Num>();
+ example15<Num>();
+ example16<Num>();
+ example17<Num>();
+ /*
+ example18<Num>();
+ example19<Num>();
+ example20<Num>();
+ example21<Num>();
+ example22<Num>();
+ example23<Num>();
+ example24<Num>();
+ */
+}
+
+template <class Num>
+void test_float_type_2()
+{
+ std::cout << "Epsilon = " << Eigen::NumTraits<Num>::epsilon() << std::endl;
+ std::cout << "Dummy Prec = " << Eigen::NumTraits<Num>::dummy_precision() << std::endl;
+ std::cout << "Highest = " << Eigen::NumTraits<Num>::highest() << std::endl;
+ std::cout << "Lowest = " << Eigen::NumTraits<Num>::lowest() << std::endl;
+ std::cout << "Digits10 = " << Eigen::NumTraits<Num>::digits10() << std::endl;
+
+ example18<Num>();
+ example19<Num>();
+ example20<Num>();
+ example21<Num>();
+
+ //example22<Num>();
+ //example23<Num>();
+ //example24<Num>();
+}
+
+template <class Num>
+void test_float_type_3()
+{
+ std::cout << "Epsilon = " << Eigen::NumTraits<Num>::epsilon() << std::endl;
+ std::cout << "Dummy Prec = " << Eigen::NumTraits<Num>::dummy_precision() << std::endl;
+ std::cout << "Highest = " << Eigen::NumTraits<Num>::highest() << std::endl;
+ std::cout << "Lowest = " << Eigen::NumTraits<Num>::lowest() << std::endl;
+ std::cout << "Digits10 = " << Eigen::NumTraits<Num>::digits10() << std::endl;
+
+ example22<Num>();
+ example23<Num>();
+ example24<Num>();
+}
+
+template <class Num>
+void test_complex_type()
+{
+ std::cout << "Epsilon = " << Eigen::NumTraits<Num>::epsilon() << std::endl;
+ std::cout << "Dummy Prec = " << Eigen::NumTraits<Num>::dummy_precision() << std::endl;
+ std::cout << "Highest = " << Eigen::NumTraits<Num>::highest() << std::endl;
+ std::cout << "Lowest = " << Eigen::NumTraits<Num>::lowest() << std::endl;
+ std::cout << "Digits10 = " << Eigen::NumTraits<Num>::digits10() << std::endl;
+
+ example1<Num>();
+ example2<Num>();
+ example3<Num>();
+ example4<Num>();
+ example5<Num>();
+ example7<Num>();
+ example8<Num>();
+ example9<Num>();
+ example11<Num>();
+ example12<Num>();
+ example13<Num>();
+ example14<Num>();
+ example15<Num>();
+ example16<Num>();
+ example17<Num>();
+ example18a<Num>();
+ example19<Num>();
+ example20<Num>();
+ example21<Num>();
+ example22<Num>();
+ // example23<Num>(); //requires comparisons.
+ example24<Num>();
+}
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