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.
f07caf (dgtsv) computes the solution to a real system of linear equations
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.
The symbolic adjoint uses the
$LU$ decomposition computed by the primal routine to obtain the adjoint of the right-hand side
$B$ by solving
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
where
${B}_{i,\left(1\right)}$ and
${X}_{i}$ denote the
$i$th column of the matrices
${B}_{\left(1\right)}$ and
$X$ respectively.
Du Toit J, Naumann U (2017) Adjoint Algorithmic Differentiation Tool Support for Typical Numerical Patterns in Computational Finance
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 $\mathbf{dl}\left(\mathrm{1}\right),\mathbf{dl}\left(\mathrm{2}\right),\dots ,\mathbf{dl}(n-\mathrm{2})$.
- 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).
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:
- $\mathbf{ifail}=-89$
An unexpected AD error has been triggered by this routine. Please
contact
NAG.
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.
None.
The following examples are variants of the example for
f07caf (dgtsv), modified to demonstrate calling the NAG AD Library.