# NAG AD Library Routine Document

## f07ca_a1w_f (dgtsv_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 routine name and corresponding argument types is described in the NAG AD Library Introduction.

## 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 (Out) :: info Type (nagad_a1w_w_rtype), Intent (Inout) :: dl(*), d(*), du(*), b(ldb,*) Type (c_ptr), Intent (In) :: ad_handle
 void f07ca_a1w_f_ ( void *&ad_handle, const Integer &n, const Integer &nrhs, nagad_a1w_w_rtype dl[], nagad_a1w_w_rtype d[], nagad_a1w_w_rtype du[], nagad_a1w_w_rtype b[], const Integer &ldb, Integer &ifail)

## 3Description

f07caf (dgtsv) computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ 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 NAG AD Library Introduction 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
 $AT· B i,1 = X i,1 ,$ (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
 $A 1 = ∑ i=1 r - B i,1 · XiT ,$ (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.

## 4References

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

## 5Arguments

f07ca_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.
• n$n$, the number of linear equations, i.e., the order of the matrix $A$.
• nrhs$r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
• dl – on entry: must contain the $\left(n-\mathrm{1}\right)$ subdiagonal elements of the matrix $A$. on exit: if no constraints are violated, dl is overwritten by the ($n-\mathrm{2}$) elements of the second superdiagonal of the upper triangular matrix $U$ from the $LU$ factorization of $A$, in $\mathbf{dl}\left(\mathrm{1}\right),\mathbf{dl}\left(\mathrm{2}\right),\dots ,\mathbf{dl}\left(n-\mathrm{2}\right)$.
• d – on entry: must contain the $n$ diagonal elements of the matrix $A$. on exit: if no constraints are violated, this argument is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
• du – on entry: must contain the $\left(n-\mathrm{1}\right)$ superdiagonal elements of the matrix $A$. on exit: if no constraints are violated, du is overwritten by the $\left(n-\mathrm{1}\right)$ elements of the first superdiagonal of $U$.
• b – on entry: the $n$ by $r$ right-hand side matrix $B$. on exit: if the function exits successfully, the $n$ by $r$ solution matrix $X$.
• ldb – the first dimension of the array b.
• ifail – on exit: $\mathbf{ifail}=\mathrm{0}$ unless the routine detects an error (see 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 5.2 in the NAG AD Library Introduction for further information.
$\mathbf{ifail}=-899$
Dynamic memory allocation failed for AD.
See Section 5.1 in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

f07ca_a1w_f is not threaded in any implementation.