The dot-product test is a valuable checkpoint, which can tell us whether the implementation of the adjoint operator is wrong (however it cannot guarantee that it is indeed correct). The concept is the following: Assuming that we have coded an operator
and its adjoint
. Then for any two vectors or functions
denotes the dot product. Remember that the dot product of two functions
while the dot product of two vectors
. Notice that for vectors eq. (1) becomes
which is obviously true.
The lhs of eq. (1) is computed using
, while the rhs is computed using the adjoint
. For the dot-product test, one just needs to load the vectors x and y with random numbers and perform the two computations. If the two results are not equal (within machine precision), then the computation of either
is erroneous. Note that truncation errors have identical effects on both operators, so the two results should be almost equal. The dot-product test (for real operators only) is implemented by [sec:sf_dot_test]sf_dot_test.