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- To: eigen <eigen@xxxxxxxxxxxxxxxxxxx>
- Subject: [eigen] ASCII quick reference for Eigen2
- From: Keir Mierle <mierle@xxxxxxxxx>
- Date: Sun, 1 Feb 2009 17:25:39 -0800
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This probably has some errors and is incomplete (no geometry or sparse or triangular stuff) but does include some Matlab translations. Comments and corrections welcome.
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.
double s;
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.
A.setRandom(); // Fill A with uniform random numbers in (-1, 1).
// Requires #include <Eigen/Array>.
A.setIdentity(); // Fill A with the identity.
// 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.start(n) // x(1:n)
x.start<n>() // x(1:n)
x.end(n) // N = rows(x); x(N - n: N)
x.end<n>() // N = rows(x); x(N - n: N)
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.corner(TopLeft, rows, cols) // P(1:rows, 1:cols)
P.corner(TopRight, rows, cols) // [m n]=size(P); P(1:rows, n-cols+1:n)
P.corner(BottomLeft, rows, cols) // [m n]=size(P); P(m-rows+1:m, 1:cols)
P.corner(BottomRight, rows, cols) // [m n]=size(P); P(m-rows+1:m, n-cols+1:n)
P.corner<rows,cols>(TopLeft) // P(1:rows, 1:cols)
P.corner<rows,cols>(TopRight) // [m n]=size(P); P(1:rows, n-cols+1:n)
P.corner<rows,cols>(BottomLeft) // [m n]=size(P); P(m-rows+1:m, 1:cols)
P.corner<rows,cols>(BottomRight) // [m n]=size(P); P(m-rows+1:m, n-cols+1:n)
P.minor(i, j) // Something nasty.
// 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() // conj(R')
R.transpose() // R'
R.diagonal() // diag(R)
x.asDiagonal() // diag(x)
// 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.cwise() * Q; // R = P .* Q
R = P.cwise() / Q; // R = P ./ Q
R = P.cwise() + s; // R = P + s
R = P.cwise() - s; // R = P - s
R.cwise() += s; // R = R + s
R.cwise() -= s; // R = R - s
R.cwise() *= s; // R = R * s
R.cwise() /= s; // R = R / s
R.cwise() < Q; // R < Q
R.cwise() <= Q; // R <= Q
R.cwise().inverse(); // 1 ./ P
R.cwise().sin() // sin(P)
R.cwise().cos() // cos(P)
R.cwise().pow(s) // P .^ s
R.cwise().square() // P .^ 2
R.cwise().cube() // P .^ 3
R.cwise().sqrt() // sqrt(P)
R.cwise().exp() // exp(P)
R.cwise().log() // log(P)
R.cwise().max(P) // max(R, P)
R.cwise().min(P) // min(R, P)
R.cwise().abs() // abs(P)
R.cwise().abs2() // abs(P.^2)
// Reductions.
int r, c;
// Eigen // Matlab
R.minCoeff() // min(R(:))
R.maxCoeff() // max(R(:))
s = R.minCoeff(&r, &c) // [aa, bb] = min(R); [cc, dd] = min(aa);
// r = bb(dd); c = dd; s = cc
s = R.maxCoeff(&r, &c) // [aa, bb] = max(R); [cc, dd] = max(aa);
// row = bb(dd); col = dd; s = cc
R..sum() // sum(R(:))
R.colwise.sum() // sum(R)
R.rowwise.sum() // sum(R, 2) or sum(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.norm2() // dot(x, x)
x.squaredNorm() // dot(x, x)
x.cross(y) // cross(x, y)
// Eigen can map existing memory into Eigen matrices.
float array[3];
Map<Vector3f>(array, 3).fill(10);
int data[4] = 1, 2, 3, 4;
Matrix2i mat2x2(data);
MatrixXi mat2x2 = Map<Matrix2i>(data);
MatrixXi mat2x2 = Map<MatrixXi>(data, 2, 2);
// Solve Ax = b. Result stored in x. Matlab: x = A \ b.
bool solved;
solved = A.ldlt().solve(b, &x)); // A symmetric p.s.d.
solved = A.llt() .solve(b, &x)); // A symmetric p.d.
solved = A.lu() .solve(b, &x)); // Stable and fast.
solved = A.qr() .solve(b, &x)); // No pivoting.
solved = A.svd() .solve(b, &x)); // Most stable, slowest.
// .ldlt() -> .matrixL() and .matrixD()
// .llt() -> .matrixL()
// .lu() -> .matrixL() and .matrixU()
// .qr() -> .matrixQ() and .matrixR()
// ..svd() -> .matrixU(), .singularValues(), and .matrixV()