Re: [eigen] ASCII quick reference for Eigen2

[Gael Guennebaud <gael.guennebaud@xxxxxxxxx>]

Hi,

that's wondeful ! 2/3 remarks:

- the function norm2() has been removed
- I'd add:   x.dot(y);    // dot(x,y)
- x.squaredNorm()          // dot(x, x)
 => this might be confusing since this is valid for real vector only.
      For complex, you the MatLab counterpart is:
      dot(abs(x),abs(x))
- I'd add a note stating that most R.cwise().* functions are in Eigen/Array
- what about adding select:
  (R.cwise() < s).select(P,Q);  // (R < s ? P : Q)
  (I don't know the equivalent MatLab call)
- last add:
  A.eigenvalues();                        // eig(A);
  EigenSolver<Matrix3d> eig(A);   // [vec val] = eig(A)
  eig.eigenvalues();                      // diag(val)
  eig.eigenvectors();                     // vec

that's all for now,
gael.


On Mon, Feb 2, 2009 at 2:25 AM, Keir Mierle <mierle@xxxxxxxxx> wrote:
> 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()
>