Eigen
3.4.99 (git rev 199c5f2b47eb1f8e5a2d20e60f07e97cd95a6ba6)

In Eigen, aliasing refers to assignment statement in which the same matrix (or array or vector) appears on the left and on the right of the assignment operators. Statements like mat = 2 * mat;
or mat = mat.transpose();
exhibit aliasing. The aliasing in the first example is harmless, but the aliasing in the second example leads to unexpected results. This page explains what aliasing is, when it is harmful, and what to do about it.
Here is a simple example exhibiting aliasing:
Example  Output 

MatrixXi mat(3,3);
mat << 1, 2, 3, 4, 5, 6, 7, 8, 9;
cout << "Here is the matrix mat:\n" << mat << endl;
// This assignment shows the aliasing problem
mat.bottomRightCorner(2,2) = mat.topLeftCorner(2,2);
cout << "After the assignment, mat = \n" << mat << endl;
 Here is the matrix mat: 1 2 3 4 5 6 7 8 9 After the assignment, mat = 1 2 3 4 1 2 7 4 1 
The output is not what one would expect. The problem is the assignment
This assignment exhibits aliasing: the coefficient mat(1,1)
appears both in the block mat.bottomRightCorner(2,2)
on the lefthand side of the assignment and the block mat.topLeftCorner(2,2)
on the righthand side. After the assignment, the (2,2) entry in the bottom right corner should have the value of mat(1,1)
before the assignment, which is 5. However, the output shows that mat(2,2)
is actually 1. The problem is that Eigen uses lazy evaluation (see Expression templates in Eigen) for mat.topLeftCorner(2,2)
. The result is similar to
Thus, mat(2,2)
is assigned the new value of mat(1,1)
instead of the old value. The next section explains how to solve this problem by calling eval().
Aliasing occurs more naturally when trying to shrink a matrix. For example, the expressions vec = vec.head(n)
and mat = mat.block(i,j,r,c)
exhibit aliasing.
In general, aliasing cannot be detected at compile time: if mat
in the first example were a bit bigger, then the blocks would not overlap, and there would be no aliasing problem. However, Eigen does detect some instances of aliasing, albeit at run time. The following example exhibiting aliasing was mentioned in Matrix and vector arithmetic :
Example  Output 

Matrix2i a; a << 1, 2, 3, 4;
cout << "Here is the matrix a:\n" << a << endl;
a = a.transpose(); // !!! do NOT do this !!!
cout << "and the result of the aliasing effect:\n" << a << endl;
 Here is the matrix a: 1 2 3 4 and the result of the aliasing effect: 1 2 2 4 
Again, the output shows the aliasing issue. However, by default Eigen uses a runtime assertion to detect this and exits with a message like
void Eigen::DenseBase<Derived>::checkTransposeAliasing(const OtherDerived&) const [with OtherDerived = Eigen::Transpose<Eigen::Matrix<int, 2, 2, 0, 2, 2> >, Derived = Eigen::Matrix<int, 2, 2, 0, 2, 2>]: Assertion `(!internal::check_transpose_aliasing_selector<Scalar,internal::blas_traits<Derived>::IsTransposed,OtherDerived>::run(internal::extract_data(derived()), other)) && "aliasing detected during transposition, use transposeInPlace() or evaluate the rhs into a temporary using .eval()"' failed.
The user can turn Eigen's runtime assertions like the one to detect this aliasing problem off by defining the EIGEN_NO_DEBUG macro, and the above program was compiled with this macro turned off in order to illustrate the aliasing problem. See Assertions for more information about Eigen's runtime assertions.
If you understand the cause of the aliasing issue, then it is obvious what must happen to solve it: Eigen has to evaluate the righthand side fully into a temporary matrix/array and then assign it to the lefthand side. The function eval() does precisely that.
For example, here is the corrected version of the first example above:
Example  Output 

MatrixXi mat(3,3);
mat << 1, 2, 3, 4, 5, 6, 7, 8, 9;
cout << "Here is the matrix mat:\n" << mat << endl;
// The eval() solves the aliasing problem
mat.bottomRightCorner(2,2) = mat.topLeftCorner(2,2).eval();
cout << "After the assignment, mat = \n" << mat << endl;
 Here is the matrix mat: 1 2 3 4 5 6 7 8 9 After the assignment, mat = 1 2 3 4 1 2 7 4 5 
Now, mat(2,2)
equals 5 after the assignment, as it should be.
The same solution also works for the second example, with the transpose: simply replace the line a = a.transpose();
with a = a.transpose().eval();
. However, in this common case there is a better solution. Eigen provides the specialpurpose function transposeInPlace() which replaces a matrix by its transpose. This is shown below:
Example  Output 

MatrixXf a(2,3); a << 1, 2, 3, 4, 5, 6;
cout << "Here is the initial matrix a:\n" << a << endl;
a.transposeInPlace();
cout << "and after being transposed:\n" << a << endl;
 Here is the initial matrix a: 1 2 3 4 5 6 and after being transposed: 1 4 2 5 3 6 
If an xxxInPlace() function is available, then it is best to use it, because it indicates more clearly what you are doing. This may also allow Eigen to optimize more aggressively. These are some of the xxxInPlace() functions provided:
Original function  Inplace function 

MatrixBase::adjoint()  MatrixBase::adjointInPlace() 
DenseBase::reverse()  DenseBase::reverseInPlace() 
LDLT::solve()  LDLT::solveInPlace() 
LLT::solve()  LLT::solveInPlace() 
TriangularView::solve()  TriangularView::solveInPlace() 
DenseBase::transpose()  DenseBase::transposeInPlace() 
In the special case where a matrix or vector is shrunk using an expression like vec = vec.head(n)
, you can use conservativeResize() .
As explained above, it may be dangerous if the same matrix or array occurs on both the lefthand side and the righthand side of an assignment operator, and it is then often necessary to evaluate the righthand side explicitly. However, applying componentwise operations (such as matrix addition, scalar multiplication and array multiplication) is safe.
The following example has only componentwise operations. Thus, there is no need for eval() even though the same matrix appears on both sides of the assignments.
Example  Output 

MatrixXf mat(2,2);
mat << 1, 2, 4, 7;
cout << "Here is the matrix mat:\n" << mat << endl << endl;
mat = 2 * mat;
cout << "After 'mat = 2 * mat', mat = \n" << mat << endl << endl;
mat = mat  MatrixXf::Identity(2,2);
cout << "After the subtraction, it becomes\n" << mat << endl << endl;
ArrayXXf arr = mat;
arr = arr.square();
cout << "After squaring, it becomes\n" << arr << endl << endl;
// Combining all operations in one statement:
mat << 1, 2, 4, 7;
mat = (2 * mat  MatrixXf::Identity(2,2)).array().square();
cout << "Doing everything at once yields\n" << mat << endl << endl;
static const IdentityReturnType Identity() Definition: CwiseNullaryOp.h:799  Here is the matrix mat: 1 2 4 7 After 'mat = 2 * mat', mat = 2 4 8 14 After the subtraction, it becomes 1 4 8 13 After squaring, it becomes 1 16 64 169 Doing everything at once yields 1 16 64 169 
In general, an assignment is safe if the (i,j) entry of the expression on the righthand side depends only on the (i,j) entry of the matrix or array on the lefthand side and not on any other entries. In that case it is not necessary to evaluate the righthand side explicitly.
Matrix multiplication is the only operation in Eigen that assumes aliasing by default, under the condition that the destination matrix is not resized. Thus, if matA
is a squared matrix, then the statement matA = matA * matA;
is safe. All other operations in Eigen assume that there are no aliasing problems, either because the result is assigned to a different matrix or because it is a componentwise operation.
Example  Output 

MatrixXf matA(2,2);
matA << 2, 0, 0, 2;
matA = matA * matA;
cout << matA;
 4 0 0 4 
However, this comes at a price. When executing the expression matA = matA * matA
, Eigen evaluates the product in a temporary matrix which is assigned to matA
after the computation. This is fine. But Eigen does the same when the product is assigned to a different matrix (e.g., matB = matA * matA
). In that case, it is more efficient to evaluate the product directly into matB
instead of evaluating it first into a temporary matrix and copying that matrix to matB
.
The user can indicate with the noalias() function that there is no aliasing, as follows: matB.noalias() = matA * matA
. This allows Eigen to evaluate the matrix product matA * matA
directly into matB
.
Example  Output 

MatrixXf matA(2,2), matB(2,2);
matA << 2, 0, 0, 2;
// Simple but not quite as efficient
matB = matA * matA;
cout << matB << endl << endl;
// More complicated but also more efficient
matB.noalias() = matA * matA;
cout << matB;
 4 0 0 4 4 0 0 4 
Of course, you should not use noalias()
when there is in fact aliasing taking place. If you do, then you may get wrong results:
Example  Output 

MatrixXf matA(2,2);
matA << 2, 0, 0, 2;
matA.noalias() = matA * matA;
cout << matA;
 4 0 0 4 
Moreover, starting in Eigen 3.3, aliasing is not assumed if the destination matrix is resized and the product is not directly assigned to the destination. Therefore, the following example is also wrong:
Example  Output 

MatrixXf A(2,2), B(3,2);
B << 2, 0, 0, 3, 1, 1;
A << 2, 0, 0, 2;
A = (B * A).cwiseAbs();
cout << A;
 4 0 0 6 2 2 
As for any aliasing issue, you can resolve it by explicitly evaluating the expression prior to assignment:
Example  Output 

MatrixXf A(2,2), B(3,2);
B << 2, 0, 0, 3, 1, 1;
A << 2, 0, 0, 2;
A = (B * A).eval().cwiseAbs();
cout << A;
 4 0 0 6 2 2 
Aliasing occurs when the same matrix or array coefficients appear both on the left and the righthand side of an assignment operator.