f07ca is the AD Library version of the primal routine f07caf (dgtsv). Based (in the C++ interface) on overload resolution, f07ca can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first and second order. The parameter ad_handle can be used to choose whether adjoints are computed using a symbolic adjoint or straightforward algorithmic differentiation.

2 Specification

Fortran Interface

Subroutine f07ca_AD_f (

ad_handle, n, nrhs, dl, d, du, b, ldb, ifail)

Integer, Intent (In)	::	n, nrhs, ldb
Integer, Intent (Inout)	::	ifail
ADTYPE, Intent (Inout)	::	dl(), d(), du(), b(ldb,)
Type (c_ptr), Intent (Inout)	::	ad_handle

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:

when ADTYPE is	Real(kind=nag_wp)	then AD is p0w
when ADTYPE is	Type(nagad_a1w_w_rtype)	then AD is a1w
when ADTYPE is	Type(nagad_t1w_w_rtype)	then AD is t1w
when ADTYPE is	Type(nagad_a1t1w_w_rtype)	then AD is a1t1w
when ADTYPE is	Type(nagad_t2w_w_rtype)	then AD is t2w

C++ Interface

#include <dco.hpp>
#include <nagad.h>
namespace nag {
namespace ad {

void f07ca (

handle_t &ad_handle, const Integer &n, const Integer &nrhs, ADTYPE dl[], ADTYPE d[], ADTYPE du[], ADTYPE b[], const Integer &ldb, Integer &ifail)

}
}

The function is overloaded on ADTYPE which represents the type of active arguments. ADTYPE may be any of the following types:
double,
dco::ga1s<double>::type,
dco::gt1s<double>::type,
dco::gt1s<dco::gt1s<double>::type>::type,
dco::ga1s<dco::gt1s<double>::type>::type

Note: this function can be used with AD tools other than dco/c++. For details, please contact NAG.

3 Description

f07ca is the AD Library version of the primal routine f07caf (dgtsv).

f07caf (dgtsv) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n \times n

tridiagonal matrix and

X

and

B

are

n \times r

matrices. For further information see Section 3 in the documentation for f07caf (dgtsv).

3.1 Symbolic Adjoint

Symbolic strategy may be selected by calling ad_handle.set_strategy(nag::ad::symbolic) prior to calling f07ca. No further changes are needed compared to using the algorithmic strategy.

3.1.1 Mathematical Background

The symbolic adjoint uses the

L U

decomposition computed by the primal routine to obtain the adjoint of the right-hand side

B

by solving

A^{T} \cdot B_{i, (1)} = X_{i, (1)},

(1)

where

B_{i, (1)}

and

X_{i, (1)}

denote the

i

th column of the matrices

B_{(1)}

and

X_{(1)}

respectively. The adjoint of the matrix

A

is then computed according to

A_{(1)} = \sum_{i = 1}^{r} - B_{i, (1)} \cdot X_{i}^{T},

(2)

where

B_{i, (1)}

and

X_{i}

denote the

i

th column of the matrices

B_{(1)}

and

X

respectively.

Please see Du Toit and Naumann (2017).

3.1.2 Usable Adjoints

You can set or access the adjoints of output argument b. The adjoints of all other output arguments are ignored.

f07ca increments the adjoints of input arguments b, d, du and dl according to the first order adjoint model.

4 References

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

Du Toit J, Naumann U (2017) Adjoint Algorithmic Differentiation Tool Support for Typical Numerical Patterns in Computational Finance

5 Arguments

In addition to the arguments present in the interface of the primal routine, f07ca 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 – nag::ad::handle_t Input/Output: 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.
2: n – Integer Input
3: nrhs – Integer Input
4: dl( $*$ ) – ADTYPE array Input/Output: Please consult Overwriting of Inputs in the NAG AD Library Introduction.
5: d( $*$ ) – ADTYPE array Input/Output: Please consult Overwriting of Inputs in the NAG AD Library Introduction.
6: du( $*$ ) – ADTYPE array Input/Output: Please consult Overwriting of Inputs in the NAG AD Library Introduction.
7: b(ldb, $*$ ) – ADTYPE array Input/Output: Please consult Overwriting of Inputs in the NAG AD Library Introduction.
8: ldb – Integer Input
9: ifail – Integer Input/Output: On entry: must be set to $0$ , $- 1 or 1$ .
On exit: any errors are indicated as described in Section 6.

6 Error Indicators and Warnings

f07ca uses the standard NAG ifail mechanism. Any errors indicated via info values returned by f07caf may be indicated with the same value returned by ifail. In addition, this routine may return:

$ifail = - 89$: An unexpected AD error has been triggered by this routine. Please contact NAG.
See Error Handling in the NAG AD Library Introduction for further information.

$ifail = - 199$: The routine was called using a strategy that has not yet been implemented.
See AD Strategies in the NAG AD Library Introduction for further information.

$ifail = - 444$: A C++ exception was thrown.
The error message will show the details of the C++ exception text.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

f07ca is not threaded in any implementation.

9 Further Comments

Since b is not a pure output and there is overwriting of variables, accessing adjoints later may result in wrong values, so a copy of the active input/output is used to obtain correct derivative values. See the example f07ca_a1_algo_dcoe.cpp for details.

10 Example

The following examples are variants of the example for f07caf (dgtsv), modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example solves the equations

A x = b,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) and b = (\begin{matrix} 2.7 \\ - 0.5 \\ 2.6 \\ 0.6 \\ 2.7 \end{matrix}) .

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	f07ca_a1t1w_fe.f90	f07ca_a1t1w_fe.d	f07ca_a1t1w_fe.r
Fortran	f07ca_a1w_fe.f90	f07ca_a1w_fe.d	f07ca_a1w_fe.r
C++	f07ca_a1_algo_dcoe.cpp	None	f07ca_a1_algo_dcoe.r
C++	f07ca_a1_sym_dcoe.cpp	None	f07ca_a1_sym_dcoe.r
C++	f07ca_a1t1_algo_dcoe.cpp	None	f07ca_a1t1_algo_dcoe.r
C++	f07ca_a1t1_sym_dcoe.cpp	None	f07ca_a1t1_sym_dcoe.r

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	f07ca_t1w_fe.f90	f07ca_t1w_fe.d	f07ca_t1w_fe.r
Fortran	f07ca_t2w_fe.f90	f07ca_t2w_fe.d	f07ca_t2w_fe.r
C++	f07ca_t1_dcoe.cpp	None	f07ca_t1_dcoe.r
C++	f07ca_t2_dcoe.cpp	None	f07ca_t2_dcoe.r