NAG AD Library
e04gb_a1w_f (lsq_uncon_quasi_deriv_comp_a1w)
Note: _a1w_ denotes that first order adjoints are computed in working precision; this has the corresponding argument type
nagad_a1w_w_rtype. Further implementations, for example for higher order differentiation or using the tangent linear approach, may become available at later marks of the NAG AD Library.
The method of codifying AD implementations in the routine name and corresponding argument types is described in the
NAG AD Library Introduction.
1
Purpose
e04gb_a1w_f
is the adjoint version of the primal routine
e04gbf.
Depending on the value of
ad_handle,
e04gb_a1w_f uses algorithmic differentiation or symbolic adjoints to compute adjoints of the primal.
2
Specification
Fortran Interface
Subroutine e04gb_a1w_f ( |
ad_handle, m, n, selct, lsqfun, lsqmon, iprint, maxcal, eta, xtol, stepmx, x, fsumsq, fvec, fjac, ldfjac, s, v, ldv, niter, nf, iuser, ruser, ifail) |
Integer, Intent (In) |
:: |
m, n, selct, iprint, maxcal, ldfjac, ldv |
Integer, Intent (Inout) |
:: |
iuser(*), ifail |
Integer, Intent (Out) |
:: |
niter, nf |
Type (nagad_a1w_w_rtype), Intent (In) |
:: |
eta, xtol, stepmx |
Type (nagad_a1w_w_rtype), Intent (Inout) |
:: |
x(n), fjac(ldfjac,n), v(ldv,n), ruser(*) |
Type (nagad_a1w_w_rtype), Intent (Out) |
:: |
fsumsq, fvec(m), s(n) |
Type (c_ptr), Intent (In) |
:: |
ad_handle |
External |
:: |
lsqfun, lsqmon |
|
C++ Header Interface
#include <nagad.h>
void e04gb_a1w_f_ ( |
void *&ad_handle, const Integer &m, const Integer &n, const Integer &selct, void (NAG_CALL lsqfun)(void *&ad_handle, Integer &iflag, const Integer &m, const Integer &n, nagad_a1w_w_rtype xc[], nagad_a1w_w_rtype fvec[], nagad_a1w_w_rtype fjac[], const Integer &ldfjac, Integer iuser[], nagad_a1w_w_rtype ruser[]), void (NAG_CALL lsqmon)(void *&ad_handle, const Integer &m, const Integer &n, nagad_a1w_w_rtype xc[], nagad_a1w_w_rtype fvec[], nagad_a1w_w_rtype fjac[], const Integer &ldfjac, nagad_a1w_w_rtype s[], const Integer &igrade, const Integer &niter, const Integer &nf, Integer iuser[], nagad_a1w_w_rtype ruser[]), const Integer &iprint, const Integer &maxcal, const nagad_a1w_w_rtype &eta, const nagad_a1w_w_rtype &xtol, const nagad_a1w_w_rtype &stepmx, nagad_a1w_w_rtype x[], nagad_a1w_w_rtype &fsumsq, nagad_a1w_w_rtype fvec[], nagad_a1w_w_rtype fjac[], const Integer &ldfjac, nagad_a1w_w_rtype s[], nagad_a1w_w_rtype v[], const Integer &ldv, Integer &niter, Integer &nf, Integer iuser[], nagad_a1w_w_rtype ruser[], Integer &ifail) |
|
The routine may be called by the names e04gb_a1w_f or nagf_opt_lsq_uncon_quasi_deriv_comp_a1w.
3
Description
e04gb_a1w_f
is the adjoint version of the primal routine
e04gbf.
e04gbf is a comprehensive quasi-Newton algorithm for finding an unconstrained minimum of a sum of squares of
nonlinear functions in
variables
. First derivatives are required. The routine is intended for functions which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).
For further information see
Section 3 in the documentation for
e04gbf.
3.1
Symbolic Adjoint
e04gb_a1w_f can provide symbolic adjoints by setting the symbolic mode as described in
Section 3.2.2 in the
X10 Chapter introduction. Please see
Section 4 in the Introduction to the NAG AD Library for API description on how to use symbolic adjoints.
In comparison to the algorithmic adjoint, the user-supplied primal and adjoint callbacks need specific implementation to support symbolic adjoint computation. Please see
Section 4.2.3 in the Introduction to the NAG AD Library and recall what primal and adjoint callbacks need to calculate in the case of an algorithmic adjoint.
Assuming the original user-supplied function evaluates
where
is given by the
w or by use of COMMON globals. The variables
,
and
correspond to
xc,
fvec and
fjac of
lsqfun. The symbolic adjoint of
e04gbf then also requires the following capability / modes:
-
(a)Function value evaluation only.
-
(b)Function value evaluation and adjoint computation w.r.t. xc only (corresponds to in the following equation), i.e.,
-
(c)Function value evaluation and adjoint computation w.r.t. only, i.e.,
Here is a placeholder for any user variable either passed via the user segment of w or via COMMON global variables.
3.1.1
Mathematical Background
To be more specific, the symbolic adjoint solves
followed by an adjoint projection through the user-supplied adjoint routine
The Hessian as well as the mixed derivative tensor is computed using the user-supplied adjoint routine.
3.1.2
Usable Adjoints
You can set or access the adjoints of output arguments x, fvec, fjac and fsumsq. The adjoints of all other output arguments are ignored.
e04gb_a1w_f increments the adjoints of the variable
, where
is given by the argument
w or by use of COMMON globals (see
(1)).
4
References
Du Toit J, Naumann U (2017) Adjoint Algorithmic Differentiation Tool Support for Typical Numerical Patterns in Computational Finance
Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation
Naumann U, Lotz J, Leppkes K and Towara M (2017) Algorithmic Differentiation of Numerical Methods: Tangent and Adjoint Solvers for Parameterized Systems of Nonlinear Equations
5
Arguments
In addition to the arguments present in the interface of the primal routine,
e04gb_a1w_f 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.
-
1:
ad_handle – Type (c_ptr)
Input
-
On entry: a handle to the AD configuration data object, as created by x10aa_a1w_f. Symbolic adjoint mode may be selected by calling x10aa_a1w_f with this handle.
-
2:
m – Integer
Input
-
3:
n – Integer
Input
-
4:
selct – Integer
Input
-
this argument enables you to specify whether the linear minimizations (i.e., minimizations of with respect to ) are to be performed by a routine which just requires the evaluation of the (), or by a routine which also requires the first derivatives of the ().
-
5:
lsqfun – Subroutine
External Procedure
-
The specification of lsqfun is:
Fortran Interface
Integer, Intent (In) |
:: |
m, n, ldfjac |
Integer, Intent (Inout) |
:: |
iflag, iuser(*) |
Type (nagad_a1w_w_rtype), Intent (In) |
:: |
xc(n) |
Type (nagad_a1w_w_rtype), Intent (Inout) |
:: |
fjac(ldfjac,n), ruser(*) |
Type (nagad_a1w_w_rtype), Intent (Out) |
:: |
fvec(m) |
Type (c_ptr), Intent (In) |
:: |
ad_handle |
|
C++ Interface
#include <nagad.h>
void lsqfun ( |
void *&ad_handle, Integer &iflag, const Integer &m, const Integer &n, nagad_a1w_w_rtype xc[], nagad_a1w_w_rtype fvec[], nagad_a1w_w_rtype fjac[], const Integer &ldfjac, Integer iuser[], nagad_a1w_w_rtype ruser[]) |
|
-
1:
ad_handle – Type (c_ptr)
Input
-
On entry: a handle to the AD configuration data object.
-
2:
iflag – Integer
Input/Output
-
3:
m – Integer
Input
-
4:
n – Integer
Input
-
5:
xc – Type (nagad_a1w_w_rtype) array
Input
-
6:
fvec – Type (nagad_a1w_w_rtype) array
Output
-
7:
fjac – Type (nagad_a1w_w_rtype) array
Output
-
8:
ldfjac – Integer
Input
-
9:
iuser() – Integer array
User Workspace
-
10:
ruser() – Type (nagad_a1w_w_rtype) array
User Workspace
-
6:
lsqmon – Subroutine
External Procedure
-
The specification of lsqmon is:
Fortran Interface
Subroutine lsqmon ( |
ad_handle, m, n, xc, fvec, fjac, ldfjac, s, igrade, niter, nf, iuser, ruser) |
Integer, Intent (In) |
:: |
m, n, ldfjac, igrade, niter, nf |
Integer, Intent (Inout) |
:: |
iuser(*) |
Type (nagad_a1w_w_rtype), Intent (In) |
:: |
xc(n), fvec(m), fjac(ldfjac,n), s(n) |
Type (nagad_a1w_w_rtype), Intent (Inout) |
:: |
ruser(*) |
Type (c_ptr), Intent (In) |
:: |
ad_handle |
|
C++ Interface
#include <nagad.h>
void lsqmon ( |
void *&ad_handle, const Integer &m, const Integer &n, nagad_a1w_w_rtype xc[], nagad_a1w_w_rtype fvec[], nagad_a1w_w_rtype fjac[], const Integer &ldfjac, nagad_a1w_w_rtype s[], const Integer &igrade, const Integer &niter, const Integer &nf, Integer iuser[], nagad_a1w_w_rtype ruser[]) |
|
-
1:
ad_handle – Type (c_ptr)
Input
-
On entry: a handle to the AD configuration data object.
-
2:
m – Integer
Input
-
3:
n – Integer
Input
-
4:
xc – Type (nagad_a1w_w_rtype) array
Input
-
5:
fvec – Type (nagad_a1w_w_rtype) array
Input
-
6:
fjac – Type (nagad_a1w_w_rtype) array
Input
-
7:
ldfjac – Integer
Input
-
8:
s – Type (nagad_a1w_w_rtype) array
Input
-
9:
igrade – Integer
Input
-
10:
niter – Integer
Input
-
11:
nf – Integer
Input
-
12:
iuser() – Integer array
User Workspace
-
13:
ruser() – Type (nagad_a1w_w_rtype) array
User Workspace
-
7:
iprint – Integer
Input
-
8:
maxcal – Integer
Input
-
9:
eta – Type (nagad_a1w_w_rtype)
Input
-
10:
xtol – Type (nagad_a1w_w_rtype)
Input
-
11:
stepmx – Type (nagad_a1w_w_rtype)
Input
-
12:
x(n) – Type (nagad_a1w_w_rtype) array
Input/Output
-
13:
fsumsq – Type (nagad_a1w_w_rtype)
Output
-
14:
fvec(m) – Type (nagad_a1w_w_rtype) array
Output
-
15:
fjac(ldfjac, n) – Type (nagad_a1w_w_rtype) array
Output
-
16:
ldfjac – Integer
Input
-
17:
s(n) – Type (nagad_a1w_w_rtype) array
Output
-
18:
v(ldv, n) – Type (nagad_a1w_w_rtype) array
Output
-
19:
ldv – Integer
Input
-
20:
niter – Integer
Output
-
21:
nf – Integer
Output
-
22:
iuser() – Integer array
User Workspace
-
User workspace.
-
23:
ruser() – Type (nagad_a1w_w_rtype) array
User Workspace
-
User workspace.
-
24:
ifail – Integer
Input/Output
6
Error Indicators and Warnings
e04gb_a1w_f preserves all error codes from
e04gbf and in addition can return:
An unexpected AD error has been triggered by this routine. Please
contact
NAG.
See
Section 4.5.2 in the NAG AD Library Introduction for further information.
Dynamic memory allocation failed for AD.
See
Section 4.5.1 in the NAG AD Library Introduction for further information.
In symbolic mode the following may be returned:
-
In attempting to compute the symbolic adjoint a singular Hessian was encountered
and the computation could not proceed.
-
In attempting to compute the symbolic adjoint a Hessian was encountered with reciprocal
condition number less than machine precision;
the computation did therefore not proceed.
7
Accuracy
Not applicable.
8
Parallelism and Performance
e04gb_a1w_f
is not threaded in any implementation.
None.
10
Example
The following examples are variants of the example for
e04gbf,
modified to demonstrate calling the NAG AD Library.