Dense matrix and array manipulation » Aliasing module

Examples

Here is a simple example exhibiting aliasing:

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 mat.bottomRightCorner(2,2) = mat.topLeftCorner(2,2); This assignment exhibits aliasing: the coefficient mat(1,1) appears both in the block mat.bottomRightCorner(2,2) on the left-hand side of the assignment and the block mat.topLeftCorner(2,2) on the right-hand 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

mat(1,1) = mat(0,0);
mat(1,2) = mat(0,1);
mat(2,1) = mat(1,0);
mat(2,2) = mat(1,1);

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 :

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 run-time 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 run-time 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 run-time assertions.

Resolving aliasing issues

If you understand the cause of the aliasing issue, then it is obvious what must happen to solve it: Eigen has to evaluate the right-hand side fully into a temporary matrix/array and then assign it to the left-hand side. The function eval() does precisely that.

For example, here is the corrected version of the first example above:

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 special-purpose function transposeInPlace() which replaces a matrix by its transpose. This is shown below:

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:

In the special case where a matrix or vector is shrunk using an expression like vec = vec.head(n), you can use conservativeResize().

Aliasing and component-wise operations

As explained above, it may be dangerous if the same matrix or array occurs on both the left-hand side and the right-hand side of an assignment operator, and it is then often necessary to evaluate the right-hand side explicitly. However, applying component-wise operations (such as matrix addition, scalar multiplication and array multiplication) is safe.

The following example has only component-wise operations. Thus, there is no need for eval() even though the same matrix appears on both sides of the assignments.

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;

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 right-hand side depends only on the (i,j) entry of the matrix or array on the left-hand side and not on any other entries. In that case it is not necessary to evaluate the right-hand side explicitly.

Aliasing and matrix multiplication

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 component-wise operation.

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.

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:

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:

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:

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

Summary

Aliasing occurs when the same matrix or array coefficients appear both on the left- and the right-hand side of an assignment operator.

• Aliasing is harmless with coefficient-wise computations; this includes scalar multiplication and matrix or array addition.
• When you multiply two matrices, Eigen assumes that aliasing occurs. If you know that there is no aliasing, then you can use noalias().
• In all other situations, Eigen assumes that there is no aliasing issue and thus gives the wrong result if aliasing does in fact occur. To prevent this, you have to use eval() or one of the xxxInPlace() functions.