void	f07uec (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer n, Integer nrhs, const double ap[], double b[], Integer pdb, NagError *fail)

The function may be called by the names: f07uec, nag_lapacklin_dtptrs or nag_dtptrs.

3 Description

f07uec solves a real triangular system of linear equations

A X = B

A^{T} X = B

, using packed storage.

4 References

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

Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

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: $uplo$ – Nag_UploType Input

On entry: specifies whether

A

is upper or lower triangular.

$uplo = Nag_Upper$: $A$ is upper triangular.
$uplo = Nag_Lower$: $A$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $trans$ – Nag_TransType Input

On entry: indicates the form of the equations.

$trans = Nag_NoTrans$: The equations are of the form $A X = B$ .
$trans = Nag_Trans$ or $Nag_ConjTrans$: The equations are of the form $A^{T} X = B$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

4: $diag$ – Nag_DiagType Input

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$diag = Nag_NonUnitDiag$: $A$ is a nonunit triangular matrix.
$diag = Nag_UnitDiag$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag = Nag_NonUnitDiag

Nag_UnitDiag

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $nrhs$ – Integer Input

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

7: $ap [\dim]$ – const double Input

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

\max (1, n \times (n + 1) / 2)

On entry: the

n \times n

triangular matrix

A

, packed by rows or columns.

The storage of elements

A_{i j}

depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ap [(j - 1) \times j / 2 + i - 1]$ , for $i \leq j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ap [(2 n - j) \times (j - 1) / 2 + i - 1]$ , for $i \geq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ap [(2 n - i) \times (i - 1) / 2 + j - 1]$ , for $i \leq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ap [(i - 1) \times i / 2 + j - 1]$ , for $i \geq j$ .

diag = Nag_UnitDiag

, the diagonal elements of

AP

are assumed to be

1

, and are not referenced; the same storage scheme is used whether

diag = Nag_NonUnitDiag

diag = Nag_UnitDiag

8: $b [\dim]$ – double 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: the

n \times r

solution matrix

X

9: $pdb$ – 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 $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

10: $fail$ – 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_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.
NE_SINGULAR: Element $⟨ value ⟩$ of the diagonal is exactly zero. $A$ is singular and the solution has not been computed.

7 Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

| E | \leq c (n) ε | A |,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖ x - \hat{x} ‖}_{\infty}}{{‖ x ‖}_{\infty}} \leq c (n) cond (A, x) ε,   provided c (n) cond (A, x) ε < 1,

where

cond (A, x) = {‖ | A^{- 1} | | A | | x | ‖}_{\infty} / {‖ x ‖}_{\infty}

Note that

cond (A, x) \leq cond (A) = {‖ | A^{- 1} | | A | ‖}_{\infty} \leq κ_{\infty} (A)

;

cond (A, x)

can be much smaller than

cond (A)

and it is also possible for

cond (A^{T})

to be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling f07uhc, and an estimate for

κ_{\infty} (A)

can be obtained by calling f07ugc with

norm = Nag_InfNorm

8 Parallelism and Performance

f07uec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07uec 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 total number of floating-point operations is approximately

n^{2} r

The complex analogue of this function is f07usc.

10 Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 4.30 & 0.00 & 0.00 & 0.00 \\ - 3.96 & - 4.87 & 0.00 & 0.00 \\ 0.40 & 0.31 & - 8.02 & 0.00 \\ - 0.27 & 0.07 & - 5.95 & 0.12 \end{matrix}) and B = (\begin{matrix} - 12.90 & - 21.50 \\ 16.75 & 14.93 \\ - 17.55 & 6.33 \\ - 11.04 & 8.09 \end{matrix}),

using packed storage for

A

f07ue: FL CL CPP AD

NAG CL Interfacef07uec (dtptrs)

▸▿ 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
f07uec (dtptrs)