Note: a1w denotes that first order adjoints are computed in working precision; this has the corresponding argument type nagad_a1w_w_rtype. Also available is the t1w (first order tangent linear) mode, the interface of which is implied by replacing a1w by t1w throughout this document. Additionally, the p0w (passive interface, as alternative to the FL interface) mode is available and can be inferred by replacing of active types by the corresponding passive types. The method of codifying AD implementations in the routine name and corresponding argument types is described in the NAG AD Library Introduction.

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## 1Purpose

f07ca_a1w_f is the adjoint version of the primal routine f07caf (dgtsv). Depending on the value of ad_handle, f07ca_a1w_f uses algorithmic differentiation or symbolic adjoints to compute adjoints of the primal.

## 2Specification

Fortran Interface
 Subroutine f07ca_a1w_f ( ad_handle, n, nrhs, dl, d, du, b, ldb, ifail)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Inout) :: ifail Type (nagad_a1w_w_rtype), Intent (Inout) :: dl(*), d(*), du(*), b(ldb,*) Type (c_ptr), Intent (Inout) :: ad_handle
The routine may be called by the names f07ca_a1w_f or nagf_lapacklin_dgtsv_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

f07ca_a1w_f is the adjoint 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).

f07ca_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.

#### 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_a1w_f increments the adjoints of input arguments b, d, du and dl 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
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_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/Output
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: n – Integer Input
3: nrhs – Integer Input
4: dl($*$) – Type (nagad_a1w_w_rtype) array Input/Output
5: d($*$) – Type (nagad_a1w_w_rtype) array Input/Output
6: du($*$) – Type (nagad_a1w_w_rtype) array Input/Output
7: b(ldb, $*$) – Type (nagad_a1w_w_rtype) 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_a1w_f 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 Section 4.8.2 in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

f07ca_a1w_f is not threaded in any implementation.

None.

## 10Example

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