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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – Integer Input

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

4: $d [\dim]$ – double Input/Output

Note: the dimension, dim, of the array d must be at least

\max (1, n)

On entry: the

n

diagonal elements of the tridiagonal matrix

A

On exit: the

n

diagonal elements of the diagonal matrix

D

from the factorization

A = L D L^{H}

5: $e [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array e must be at least

\max (1, n - 1)

On entry: the

(n - 1)

subdiagonal elements of the tridiagonal matrix

A

On exit: the

(n - 1)

subdiagonal elements of the unit bidiagonal factor

L

from the

L D L^{H}

factorization of

A

. (e can also be regarded as the superdiagonal of the unit bidiagonal factor

U

from the

U^{H} D U

factorization of

A

6: $b [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n \times r

right-hand side matrix

B

On exit: if

fail . code =

NE_NOERROR, the

n \times r

solution matrix

X

7: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = ⟨ value ⟩$ .
Constraint: $nrhs \geq 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $nrhs = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_POS_DEF: The leading minor of order $⟨ value ⟩$ is not positive definite, and the solution has not been computed. The factorization has not been completed unless $n = ⟨ value ⟩$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖ E ‖}_{1} = O (ε) {‖ A ‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖ \hat{x} - x ‖}_{1}}{{‖ x ‖}_{1}} \leq κ (A) \frac{{‖ E ‖}_{1}}{{‖ A ‖}_{1}},

where

κ (A) = {‖ A^{- 1} ‖}_{1} {‖ A ‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

f07jpc is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04cgc solves

A x = b

and returns a forward error bound and condition estimate. f04cgc calls f07jnc to solve the equations.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f07jnc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The number of floating-point operations required for the factorization of

A

is proportional to

n

, and the number of floating-point operations required for the solution of the equations is proportional to

n r

, where

r

is the number of right-hand sides.

The real analogue of this function is f07jac.

10 Example

This example solves the equations

A x = b,

where

A

is the Hermitian positive definite tridiagonal matrix

A = (\begin{array}{r} 16.0 & 16.0 - 16.0 i & 0 & 0 \\ 16.0 + 16.0 i & 41.0 & 18.0 + 9.0 i & 0 \\ 0 & 18.0 - 9.0 i & 46.0 & 1.0 + 4.0 i \\ 0 & 0 & 1.0 - 4.0 i & 21.0 \end{array})

and

b = (\begin{array}{r} 64.0 + 16.0 i \\ 93.0 + 62.0 i \\ 78.0 - 80.0 i \\ 14.0 - 27.0 i \end{array}) .

Details of the

L D L^{H}

factorization of

A

are also output.

f07jn: FL CL CPP AD PY MB

NAG CL Interfacef07jnc (zptsv)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f07jnc (zptsv)