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## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine f07ca_AD_f ( 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:
C++ Interface
#include <dco.hpp>
namespace nag {
}
}
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,

## 3Description

f07ca is the AD Library version of the primal routine f07caf (dgtsv).
f07caf (dgtsv) computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ tridiagonal matrix and $X$ and $B$ are $n×r$ matrices. For further information see Section 3 in the documentation for f07caf (dgtsv).

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.

#### 3.1.1Mathematical Background

The symbolic adjoint uses the $LU$ decomposition computed by the primal routine to obtain the adjoint of the right-hand side $B$ by solving
 $AT·Bi,(1)=Xi,(1),$ (1)
where ${B}_{i,\left(1\right)}$ and ${X}_{i,\left(1\right)}$ denote the $i$th column of the matrices ${B}_{\left(1\right)}$ and ${X}_{\left(1\right)}$ respectively. The adjoint of the matrix $A$ is then computed according to
 $A(1)=∑ i=1 r -Bi,(1)·XiT,$ (2)
where ${B}_{i,\left(1\right)}$ and ${X}_{i}$ denote the $i$th column of the matrices ${B}_{\left(1\right)}$ and $X$ respectively.
Please see Du Toit and Naumann (2017).

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.

## 4References

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

## 5Arguments

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.
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
5: d($*$) – ADTYPE array Input/Output
6: du($*$) – ADTYPE array Input/Output
7: b(ldb, $*$) – ADTYPE array Input/Output
8: ldb – Integer Input
9: ifail – Integer Input/Output
On entry: must be set to $0$, .
On exit: any errors are indicated as described in Section 6.

## 6Error 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:
${\mathbf{ifail}}=-89$
See Error Handling in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-199$
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.
${\mathbf{ifail}}=-444$
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

f07ca is not threaded in any implementation.

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.

## 10Example

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
 $Ax=b ,$
where $A$ is the tridiagonal matrix
 $A = ( 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 ) and b = ( 2.7 -0.5 2.6 0.6 2.7 ) .$

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

### 10.2Tangent 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

### 10.3Passive mode

Language Source File Data Results
Fortran f07ca_p0w_fe.f90 f07ca_p0w_fe.d f07ca_p0w_fe.r
C++ f07ca_passive_dcoe.cpp None f07ca_passive_dcoe.r