f07ca_ad_f : NAG AD Library Routine Document, Mark 26

Specification

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

C++ Header Interface

#include <nagad.h>

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)

Description

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

A X = 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).

3.1

Symbolic Adjoint

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.

3.1.1

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_a1w_f increments the adjoints of input arguments b, d, du and dl according to the first order adjoint model.

References

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

Arguments

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 $(n - 1)$ subdiagonal elements of the matrix $A$ . on exit: if no constraints are violated, dl is overwritten by the ( $n - 2$ ) elements of the second superdiagonal of the upper triangular matrix $U$ from the $L U$ factorization of $A$ , in $dl (1), dl (2), \dots, dl (n - 2)$ .
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 $L U$ factorization of $A$ .
du – on entry: must contain the $(n - 1)$ superdiagonal elements of the matrix $A$ . on exit: if no constraints are violated, du is overwritten by the $(n - 1)$ 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: $ifail = 0$ unless the routine detects an error (see Section 6).

Error 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:

$ifail = - 89$: 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.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Section 5.1 in the NAG AD Library Introduction for further information.

Language	Source File	Data	Results
Fortan	f07ca_a1w_fe.f90	f07ca_a1w_fe.d	f07ca_a1w_fe.r
C++	f07ca_a1w_hcppe.cpp	f07ca_a1w_hcppe.d	f07ca_a1w_hcppe.r

NAG AD Library Routine Document

f07ca_a1w_f (dgtsv_a1w)

▸▿ Contents

1

Purpose

2

Specification

3

Description

3.1

Symbolic Adjoint

3.1.1

Mathematical Background

3.1.2

Usable Adjoints

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

NAG AD Library Routine Document

f07ca_a1w_f (dgtsv_a1w)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Symbolic Adjoint

3.1.1 Mathematical Background

3.1.2 Usable Adjoints

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

1

Purpose

2

Specification

3

Description

3.1

Symbolic Adjoint

3.1.1

Mathematical Background

3.1.2

Usable Adjoints

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example