gocarlos/Eigen Cheat sheet

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

#include <Eigen/Dense>

Matrix<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 = zeros(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)'
VectorXi::LinSpaced(((hi-low)/step)+1,      // low:step:hi
                    low,low+step*(size-1))  //


// 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(:, j)
R.col(j1).swap(mat1.col(j2));      // R(:, [j1 j2]) = R(:, [j2, j1])

// Views, transpose, etc;
// Eigen                           // Matlab
R.adjoint()                        // R'
R.transpose()                      // R.' or conj(R')       // Read-write
R.diagonal()                       // diag(R)               // Read-write
x.asDiagonal()                     // diag(x)
R.transpose().colwise().reverse()  // rot90(R)              // Read-write
R.rowwise().reverse()              // fliplr(R)
R.colwise().reverse()              // flipud(R)
R.replicate(i,j)                   // repmat(P,i,j)


// 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)
R = (Q.array()==0).select(P,A) // R(Q==0) = P(Q==0)
R = P.unaryExpr(ptr_fun(func)) // R = arrayfun(func, P)   // with: scalar func(const scalar &x);


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

	#include <Eigen/Dense>

	Matrix<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 = zeros(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)'
	VectorXi::LinSpaced(((hi-low)/step)+1, // low:step:hi
	low,low+step*(size-1)) //


	// 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(:, j)
	R.col(j1).swap(mat1.col(j2)); // R(:, [j1 j2]) = R(:, [j2, j1])

	// Views, transpose, etc;
	// Eigen // Matlab
	R.adjoint() // R'
	R.transpose() // R.' or conj(R') // Read-write
	R.diagonal() // diag(R) // Read-write
	x.asDiagonal() // diag(x)
	R.transpose().colwise().reverse() // rot90(R) // Read-write
	R.rowwise().reverse() // fliplr(R)
	R.colwise().reverse() // flipud(R)
	R.replicate(i,j) // repmat(P,i,j)


	// All the same as Matlab, but matlab doesn't have *= style operators.
	// Matrix-vector. Matrix-matrix. Matrix-scalar.
	y = Mx; R = PQ; R = P*s;
	a = bM; R = P - Q; R = sP;
	a *= M; R = P + Q; R = P/s;
	R = Q; R = sP;
	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)
	R = (Q.array()==0).select(P,A) // R(Q==0) = P(Q==0)
	R = P.unaryExpr(ptr_fun(func)) // R = arrayfun(func, P) // with: scalar func(const scalar &x);


	// 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) = 2mat2x2; // overwrite elements of data with 2mat2x2
	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<>