e04gb_ad_f : NAG AD Library Routine Document, Mark 26

Fortran Interface

Subroutine e04gb_a1w_f (

ad_handle, m, n, lsqlin, 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, 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	::	lsqlin, lsqfun, lsqmon

Function lsqlin (

ad_handle, selct)

Integer, Intent (Out)	::	selct
Type (c_ptr), Intent (In)	::	ad_handle

Subroutine lsqfun (

ad_handle, iflag, m, n, xc, fvec, fjac, ldfjac, iuser, ruser)

Integer, Intent (In)	::	m, n, ldfjac
Integer, Intent (Inout)	::	iflag, iuser(*)
Type (nagad_a1w_w_rtype), Intent (Inout)	::	xc(n), fjac(ldfjac,n), ruser(*), fvec(m)
Type (c_ptr), Intent (In)	::	ad_handle

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 (Inout)	::	xc(n), fvec(m), fjac(ldfjac,n), s(n), ruser(*)
Type (c_ptr), Intent (In)	::	ad_handle

C++ Header Interface

#include <nagad.h>

void e04gb_a1w_f_ (

void *&ad_handle, const Integer &m, const Integer &n,
void (NAG_CALL lsqlin)(void *&ad_handle, 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)

Description

e04gbf is a comprehensive quasi-Newton algorithm for finding an unconstrained minimum of a sum of squares of

m

nonlinear functions in

n

variables

(m \geq n)

. 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 NAG AD Library Introduction 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 NAG AD Library Introduction and recall what primal and adjoint callbacks need to calculate in the case of an algorithmic adjoint.

Assuming the original user-supplied function evaluates

(z, g) = (f (x, p), \nabla_{x} f (x, p)),

(1)

where

p

is given by the w or by use of COMMON globals. The variables

x

z

and

g

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

x

in the following equation), i.e.,

x_{(1)} + = {[\nabla_{x} f (x, p)]}^{T} z_{(1)} + {[\nabla_{x}^{2} f (x, p)]}^{T} g_{(1)}

(2)

(c)

Function value evaluation and adjoint computation w.r.t.

p

only, i.e.,

p_{(1)} + = {[\nabla_{p} f (x, p)]}^{T} z_{(1)} + {[\nabla_{x, p}^{2} f (x, p)]}^{T} g_{(1)} .

(3)

Here

p

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

[\nabla_{x}^{2} F (x, p)] z = - x_{(1)}

(4)

followed by an adjoint projection through the user-supplied adjoint routine

p_{(1) k} = \sum_{j = 1}^{n} \frac{\partial^{2} F (x, p)}{\partial x_{j} \partial p_{k}} z_{j} = 2 \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\frac{\partial f_{i}}{\partial p_{k}} \frac{\partial f_{i}}{\partial x_{j}} z_{j} + f_{i} \frac{\partial^{2} f_{i}}{\partial x_{j} \partial p_{k}} z_{j}] .

(5)

The Hessian

\nabla_{x}^{2} F (x, p)

as well as the mixed derivative tensor

\frac{d^{2} F (x, p)}{d x_{j} d p_{k}}

is computed using the user-supplied adjoint routine.

Please see Du Toit and Naumann (2017), Naumann et al. (2017) and Giles (2017) for reference.

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

p

, where

p

is given by the argument w or by use of COMMON globals (see (1)).

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

Arguments

e04gb_a1w_f provides access to all the arguments available in the primal routine. There are also additional arguments specific to AD. A tooltip popup for each argument can be found by hovering over the argument name in Section 2 and a summary of the arguments are provided below:

ad_handle – a handle to the AD configuration data object, as created by x10aa_a1w_f. Symbolic adjoint mode may be selected by calling x10ac_a1w_f with this handle.
m – the number $m$ of residuals, $f_{i} (x)$ , and the number $n$ of variables, $x_{j}$ .
n – the number $m$ of residuals, $f_{i} (x)$ , and the number $n$ of variables, $x_{j}$ .
lsqlin – this argument enables you to specify whether the linear minimizations (i.e., minimizations of $F (x^{(k)} + α^{(k)} p^{(k)})$ with respect to $α^{(k)}$ ) are to be performed by a routine which just requires the evaluation of the $f_{i} (x)$ (routine), or by a routine which also requires the first derivatives of the $f_{i} (x)$ (routine).
lsqfun – lsqfun must calculate the vector of values $f_{i} (x)$ and Jacobian matrix of first derivatives $\frac{\partial f_{i}}{\partial x_{j}}$ at any point $x$ .
lsqmon – If $iprint \geq 0$ , you must supply lsqmon which is suitable for monitoring the minimization process.
iprint – the frequency with which lsqmon is to be called.
maxcal – enables you to limit the number of times that lsqfun is called by routine.
eta – every iteration of routine involves a linear minimization (i.e., minimization of $F (x^{(k)} + α^{(k)} p^{(k)})$ with respect to $α^{(k)}$ ).
xtol – the accuracy in $x$ to which the solution is required.
stepmx – an estimate of the Euclidean distance between the solution and the starting point supplied by you.
x – on entry: $x (j)$ must be set to a guess at the $j$ th component of the position of the minimum, for $j = 1, 2, \dots, n$ . on exit: the final point $x^{(k)}$ .
fsumsq – on exit: the value of $F (x)$ , the sum of squares of the residuals $f_{i} (x)$ , at the final point given in x.
fvec – on exit: the value of the residual $f_{i} (x)$ at the final point given in x, for $i = 1, 2, \dots, m$ .
fjac – on exit: the value of the first derivative $\frac{\partial f_{i}}{\partial x_{j}}$ evaluated at the final point given in x, for $j = 1, 2, \dots, n$ , for $i = 1, 2, \dots, m$ .
ldfjac – the first dimension of the array fjac.
s – on exit: the singular values of the Jacobian matrix at the final point.
v – on exit: the matrix $V$ associated with the singular value decomposition. $J = U S V^{T}$ . of the Jacobian matrix at the final point, stored by columns.
ldv – the first dimension of the array v.
niter – on exit: the number of iterations which have been performed in routine.
nf – on exit: the number of times that the residuals have been evaluated (i.e., the number of calls of lsqfun).
iuser – may be used to pass information to user-supplied argument(s).
ruser – may be used to pass information to user-supplied argument(s).
ifail – on entry: ifail must be set to $0$ , $- 1 or 1$ . on exit: ifail = 0 unless the routine detects an error or a warning has been flagged (see Section 6).

Language	Source File	Data	Results
Fortan	e04gb_a1w_fe.f90	e04gb_a1w_fe.d	e04gb_a1w_fe.r
C++	e04gb_a1w_hcppe.cpp	e04gb_a1w_hcppe.d	e04gb_a1w_hcppe.r

NAG AD Library Routine Document

e04gb_a1w_f (lsq_uncon_quasi_deriv_comp_a1w)

▸▿ Contents

1

Purpose

2

Specification

3

Description

3.1

Symbolic Adjoint

3.1.1

Mathematical Background

3.1.2

Usable Adjoints

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

NAG AD Library Routine Document

e04gb_a1w_f (lsq_uncon_quasi_deriv_comp_a1w)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Symbolic Adjoint

3.1.1 Mathematical Background

3.1.2 Usable Adjoints

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example