1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
|
// 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)).
/******************************************************************************/
/* PLEASE HELP US IMPROVING THIS SECTION */
/* Eigen 3.4 supports a much improved API for sub-matrices, including, */
/* slicing and indexing from arrays: */
/* http://eigen.tuxfamily.org/dox-devel/group__TutorialSlicingIndexing.html */
/******************************************************************************/
// 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;
/******************************************************************************/
/* PLEASE HELP US IMPROVING THIS SECTION */
/* Eigen 3.4 supports a new API for reshaping: */
/* http://eigen.tuxfamily.org/dox-devel/group__TutorialReshape.html */
/******************************************************************************/
// 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,R) // 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<>
|
