is the AD Library version of the primal routine
Based (in the C++ interface) on overload resolution,
f08kd can be used for primal, tangent and adjoint
evaluation. It supports tangents and adjoints of first order.
The parameter ad_handle can be used to choose whether adjoints are computed using a symbolic adjoint or straightforward algorithmic differentiation.
Corresponding to the overloaded C++ function, the Fortran interface provides five routines with names reflecting the type used for active real arguments. The actual subroutine and type names are formed by replacing AD and ADTYPE in the above as follows:
f08kdf (dgesdd) computes the singular value decomposition (SVD) of a real matrix , optionally computing the left and/or right singular vectors, by using a divide-and-conquer method.
For further information see Section 3 in the documentation for f08kdf (dgesdd).
f08kd can provide symbolic adjoints by setting the symbolic strategy as described in Section 3.3.3 in the Introduction to the NAG AD Library.
The symbolic adjoint allows you to compute the adjoints of the output arguments:
The symbolic adjoint assumes that the primal routine has successfully converged. Moreover for considering the adjoints of s the first columns of u and the first rows of vt are required. To consider the adjoints of the first columns of u and/or the first rows of vt the algorithm requires the computation of all entries of the matrices and .
Hence (to compute the desired adjoint) if the routine is run with the SVD decomposition is performed by calling f08kd with (you must ensure that all arrays are allocated as specified for ). The results are stored according to the value jobz you provided.
For all other settings of jobz the SVD decomposition is performed by calling the f08kdf with (you must ensure that all arrays are allocated as specified for ). The results are stored according to the value jobz you provided.
The symbolic adjoint uses the SVD decomposition computed by the primal routine to obtain the adjoints. To compute the adjoints it is required that
(i) for all , ;
(ii)if then for all ,
where denotes the th singular value of matrix . Please see Giles (2017) for more details.
You can set or access the adjoints of the output arguments a if , s, u if and , and vt if and . The adjoints of all other output arguments are ignored.
f08kd increments the adjoints of input argument a according to the first order adjoint model.
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
In addition to the arguments present in the interface of the primal routine,
f08kd includes some arguments specific to AD.
A brief summary of the AD specific arguments is given below. For the remainder, links are provided to the corresponding argument from the primal routine.
A tooltip popup for all arguments can be found by hovering over the argument name in Section 2 and in this section.
On entry: a configuration object that holds information on the differentiation strategy. Details on setting the AD strategy are described in AD handle object and AD Strategies in the NAG AD Library Introduction.