Re: [eigen] ASCII quick reference for Eigen2

[Gael Guennebaud <gael.guennebaud@xxxxxxxxx>]

FYI, I added it in eigen2/doc with the suggestions I made.

On Mon, Feb 2, 2009 at 11:18 AM, Gael Guennebaud
<gael.guennebaud@xxxxxxxxx> wrote:
> 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()
>>
>