namespace gismo {

/** \defgroup Matrix Matrix module
\ingroup Modules

    This module provides dense and sparse matrix data structures. It
    is based on the <a href="http://eigen.tuxfamily.org"
    target="_blank">Eigen</a> library and ports all the linear algebra
    related infrastructure to \gismo. The main objects are gsMatrix
    for dense matrices, gsVector for dense vectors, gsSparseMatrix for
    sparse matrices and gsSparseVector for sparse vectors. Typical
    matrix decompositions such as LU, QR, SVD, and so on, are
    available.  Furthermore, the user has also access to iterative
    solvers like conjugate gradient methods with different
    preconditioners. Finally, one can use popular high-end linear
    solver packages like PARADISO and SuperLU through a common
    interface.

    <a href="http://eigen.tuxfamily.org/dox" target="_blank">Eigen Documentation</a>


\section matop Main matrix operations

~~~~~
// A simple quickref for Eigen. Add anything that's missing.
// Main author: Keir Mierle

#include <Eigen/Dense>

gsMatrix<double, 3, 3> A;               // Fixed rows and cols. Same as Matrix3d.
Matrix<double, 3, Dynamic> B;         // Fixed rows, dynamic cols.
Matrix<double, Dynamic, Dynamic> C;   // Full dynamic. Same as MatrixXd.
Matrix<double, 3, 3, RowMajor> E;     // Row major; default is column-major.
Matrix3f P, Q, R;                     // 3x3 float matrix.
Vector3f x, y, z;                     // 3x1 float matrix.
RowVector3f a, b, c;                  // 1x3 float matrix.
VectorXd v;                           // Dynamic column vector of doubles
double s;                            

// Basic usage
// Eigen          // Matlab           // comments
x.size()          // length(x)        // vector size
C.rows()          // size(C,1)        // number of rows
C.cols()          // size(C,2)        // number of columns
x(i)              // x(i+1)           // Matlab is 1-based
C(i,j)            // C(i+1,j+1)       //

A.resize(4, 4);   // Runtime error if assertions are on.
B.resize(4, 9);   // Runtime error if assertions are on.
A.resize(3, 3);   // Ok; size didn't change.
B.resize(3, 9);   // Ok; only dynamic cols changed.
                  
A << 1, 2, 3,     // Initialize A. The elements can also be
     4, 5, 6,     // matrices, which are stacked along cols
     7, 8, 9;     // and then the rows are stacked.
B << A, A, A;     // B is three horizontally stacked A's.
A.fill(10);       // Fill A with all 10's.

// Eigen                            // Matlab
MatrixXd::Identity(rows,cols)       // eye(rows,cols)
C.setIdentity(rows,cols)            // C = eye(rows,cols)
MatrixXd::Zero(rows,cols)           // zeros(rows,cols)
C.setZero(rows,cols)                // C = ones(rows,cols)
MatrixXd::Ones(rows,cols)           // ones(rows,cols)
C.setOnes(rows,cols)                // C = ones(rows,cols)
MatrixXd::Random(rows,cols)         // rand(rows,cols)*2-1        // MatrixXd::Random returns uniform random numbers in (-1, 1).
C.setRandom(rows,cols)              // C = rand(rows,cols)*2-1
VectorXd::LinSpaced(size,low,high)   // linspace(low,high,size)'
v.setLinSpaced(size,low,high)        // v = linspace(low,high,size)'


// Matrix slicing and blocks. All expressions listed here are read/write.
// Templated size versions are faster. Note that Matlab is 1-based (a size N
// vector is x(1)...x(N)).
// Eigen                           // Matlab
x.head(n)                          // x(1:n)
x.head<n>()                        // x(1:n)
x.tail(n)                          // x(end - n + 1: end)
x.tail<n>()                        // x(end - n + 1: end)
x.segment(i, n)                    // x(i+1 : i+n)
x.segment<n>(i)                    // x(i+1 : i+n)
P.block(i, j, rows, cols)          // P(i+1 : i+rows, j+1 : j+cols)
P.block<rows, cols>(i, j)          // P(i+1 : i+rows, j+1 : j+cols)
P.row(i)                           // P(i+1, :)
P.col(j)                           // P(:, j+1)
P.leftCols<cols>()                 // P(:, 1:cols)
P.leftCols(cols)                   // P(:, 1:cols)
P.middleCols<cols>(j)              // P(:, j+1:j+cols)
P.middleCols(j, cols)              // P(:, j+1:j+cols)
P.rightCols<cols>()                // P(:, end-cols+1:end)
P.rightCols(cols)                  // P(:, end-cols+1:end)
P.topRows<rows>()                  // P(1:rows, :)
P.topRows(rows)                    // P(1:rows, :)
P.middleRows<rows>(i)              // P(i+1:i+rows, :)
P.middleRows(i, rows)              // P(i+1:i+rows, :)
P.bottomRows<rows>()               // P(end-rows+1:end, :)
P.bottomRows(rows)                 // P(end-rows+1:end, :)
P.topLeftCorner(rows, cols)        // P(1:rows, 1:cols)
P.topRightCorner(rows, cols)       // P(1:rows, end-cols+1:end)
P.bottomLeftCorner(rows, cols)     // P(end-rows+1:end, 1:cols)
P.bottomRightCorner(rows, cols)    // P(end-rows+1:end, end-cols+1:end)
P.topLeftCorner<rows,cols>()       // P(1:rows, 1:cols)
P.topRightCorner<rows,cols>()      // P(1:rows, end-cols+1:end)
P.bottomLeftCorner<rows,cols>()    // P(end-rows+1:end, 1:cols)
P.bottomRightCorner<rows,cols>()   // P(end-rows+1:end, end-cols+1:end)

// Of particular note is Eigen's swap function which is highly optimized.
// Eigen                           // Matlab
R.row(i) = P.col(j);               // R(i, :) = P(:, i)
R.col(j1).swap(mat1.col(j2));      // R(:, [j1 j2]) = R(:, [j2, j1])

// Views, transpose, etc; all read-write except for .adjoint().
// Eigen                           // Matlab
R.adjoint()                        // R'
R.transpose()                      // R.' or conj(R')
R.diagonal()                       // diag(R)
x.asDiagonal()                     // diag(x)
R.transpose().colwise().reverse(); // rot90(R)
R.conjugate()                      // conj(R)

// All the same as Matlab, but matlab doesn't have *= style operators.
// Matrix-vector.  Matrix-matrix.   Matrix-scalar.
y  = M*x;          R  = P*Q;        R  = P*s;
a  = b*M;          R  = P - Q;      R  = s*P;
a *= M;            R  = P + Q;      R  = P/s;
                   R *= Q;          R  = s*P;
                   R += Q;          R *= s;
                   R -= Q;          R /= s;

// Vectorized operations on each element independently
// Eigen                  // Matlab
R = P.cwiseProduct(Q);    // R = P .* Q
R = P.array() * s.array();// R = P .* s
R = P.cwiseQuotient(Q);   // R = P ./ Q
R = P.array() / Q.array();// R = P ./ Q
R = P.array() + s.array();// R = P + s
R = P.array() - s.array();// R = P - s
R.array() += s;           // R = R + s
R.array() -= s;           // R = R - s
R.array() < Q.array();    // R < Q
R.array() <= Q.array();   // R <= Q
R.cwiseInverse();         // 1 ./ P
R.array().inverse();      // 1 ./ P
R.array().sin()           // sin(P)
R.array().cos()           // cos(P)
R.array().pow(s)          // P .^ s
R.array().square()        // P .^ 2
R.array().cube()          // P .^ 3
R.cwiseSqrt()             // sqrt(P)
R.array().sqrt()          // sqrt(P)
R.array().exp()           // exp(P)
R.array().log()           // log(P)
R.cwiseMax(P)             // max(R, P)
R.array().max(P.array())  // max(R, P)
R.cwiseMin(P)             // min(R, P)
R.array().min(P.array())  // min(R, P)
R.cwiseAbs()              // abs(P)
R.array().abs()           // abs(P)
R.cwiseAbs2()             // abs(P.^2)
R.array().abs2()          // abs(P.^2)
(R.array() < s).select(P,Q);  // (R < s ? P : Q)

// Reductions.
int r, c;
// Eigen                  // Matlab
R.minCoeff()              // min(R(:))
R.maxCoeff()              // max(R(:))
s = R.minCoeff(&r, &c)    // [s, i] = min(R(:)); [r, c] = ind2sub(size(R), i);
s = R.maxCoeff(&r, &c)    // [s, i] = max(R(:)); [r, c] = ind2sub(size(R), i);
R.sum()                   // sum(R(:))
R.colwise().sum()         // sum(R)
R.rowwise().sum()         // sum(R, 2) or sum(R')'
R.prod()                  // prod(R(:))
R.colwise().prod()        // prod(R)
R.rowwise().prod()        // prod(R, 2) or prod(R')'
R.trace()                 // trace(R)
R.all()                   // all(R(:))
R.colwise().all()         // all(R)
R.rowwise().all()         // all(R, 2)
R.any()                   // any(R(:))
R.colwise().any()         // any(R)
R.rowwise().any()         // any(R, 2)

// Dot products, norms, etc.
// Eigen                  // Matlab
x.norm()                  // norm(x).    Note that norm(R) doesn't work in Eigen.
x.squaredNorm()           // dot(x, x)   Note the equivalence is not true for complex
x.dot(y)                  // dot(x, y)
x.cross(y)                // cross(x, y) Requires #include <Eigen/Geometry>

//// Type conversion
// Eigen                           // Matlab
A.cast<double>();                  // double(A)
A.cast<float>();                   // single(A)
A.cast<int>();                     // int32(A)
A.real();                          // real(A)
A.imag();                          // imag(A)
// if the original type equals destination type, no work is done

// Note that for most operations Eigen requires all operands to have the same type:
MatrixXf F = MatrixXf::Zero(3,3);
A += F;                // illegal in Eigen. In Matlab A = A+F is allowed
A += F.cast<double>(); // F converted to double and then added (generally, conversion happens on-the-fly)

// Eigen can map existing memory into Eigen matrices.
float array[3];
Vector3f::Map(array).fill(10);            // create a temporary Map over array and sets entries to 10
int data[4] = {1, 2, 3, 4};
Matrix2i mat2x2(data);                    // copies data into mat2x2
Matrix2i::Map(data) = 2*mat2x2;           // overwrite elements of data with 2*mat2x2
MatrixXi::Map(data, 2, 2) += mat2x2;      // adds mat2x2 to elements of data (alternative syntax if size is not know at compile time)

// Solve Ax = b. Result stored in x. Matlab: x = A \ b.
x = A.ldlt().solve(b));  // A sym. p.s.d.    #include <Eigen/Cholesky>
x = A.llt() .solve(b));  // A sym. p.d.      #include <Eigen/Cholesky>
x = A.lu()  .solve(b));  // Stable and fast. #include <Eigen/LU>
x = A.qr()  .solve(b));  // No pivoting.     #include <Eigen/QR>
x = A.svd() .solve(b));  // Stable, slowest. #include <Eigen/SVD>
// .ldlt() -> .matrixL() and .matrixD()
// .llt()  -> .matrixL()
// .lu()   -> .matrixL() and .matrixU()
// .qr()   -> .matrixQ() and .matrixR()
// .svd()  -> .matrixU(), .singularValues(), and .matrixV()

// Eigenvalue problems
// Eigen                          // Matlab
A.eigenvalues();                  // eig(A);
EigenSolver<Matrix3d> eig(A);     // [vec val] = eig(A)
eig.eigenvalues();                // diag(val)
eig.eigenvectors();               // vec
// For self-adjoint matrices use SelfAdjointEigenSolver<>
~~~~~

*/

}

