void	f04bgc (Nag_OrderType order, Integer n, Integer nrhs, double d[], double e[], double b[], Integer pdb, double rcond, double errbnd, NagError *fail)

The function may be called by the names: f04bgc, nag_linsys_real_posdef_tridiag_solve or nag_real_sym_posdef_tridiag_lin_solve.

3 Description

A

is factorized as

A = L D L^{T}

, where

L

is a unit lower bidiagonal matrix and

D

is diagonal, and the factored form of

A

is then used to solve the system of equations.

4 References

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

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

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: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – Integer Input

On entry: the number of right-hand sides

r

, 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: must contain the

n

diagonal elements of the tridiagonal matrix

A

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, d is overwritten by the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{T}

factorization of

A

5: $e [\dim]$ – double Input/Output

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

subdiagonal elements of the tridiagonal matrix

A

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, e is overwritten by the

(n - 1)

subdiagonal elements of the unit lower bidiagonal matrix

L

from the

L D L^{T}

factorization of

A

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

U

from the

U^{T} D U

factorization of

A

6: $b [\dim]$ – 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

matrix of right-hand sides

B

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, 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: $rcond$ – double * Output

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

rcond = 1 / ({‖ A ‖}_{1} {‖ A^{- 1} ‖}_{1})

9: $errbnd$ – double * Output

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution

\hat{x}

, such that

{‖ \hat{x} - x ‖}_{1} / {‖ x ‖}_{1} \leq errbnd

, where

\hat{x}

is a column of the computed solution returned in the array b and

x

is the corresponding column of the exact solution

X

. If rcond is less than machine precision, errbnd is returned as unity.

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.
The double allocatable memory required is n. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.
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_POS_DEF: The principal minor of order $⟨ value ⟩$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.
NE_RCOND: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

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. f04bgc uses the approximation

{‖ E ‖}_{1} = ε {‖ A ‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

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

f04bgc 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 required to solve the equations

A X = B

is proportional to

n r

. The condition number estimation requires

O (n)

floating-point operations.

See Section 15.3 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.

The complex analogue of f04bgc is f04cgc.

10 Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite tridiagonal matrix

A = (\begin{matrix} 4.0 & - 2.0 & 0 & 0 & 0 \\ - 2.0 & 10.0 & - 6.0 & 0 & 0 \\ 0 & - 6.0 & 29.0 & 15.0 & 0 \\ 0 & 0 & 15.0 & 25.0 & 8.0 \\ 0 & 0 & 0 & 8.0 & 5.0 \end{matrix}) and B = (\begin{matrix} 6.0 & 10.0 \\ 9.0 & 4.0 \\ 2.0 & 9.0 \\ 14.0 & 65.0 \\ 7.0 & 23.0 \end{matrix}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

f04bg: FL CL CPP AD PY MB

NAG CL Interfacef04bgc (real_​posdef_​tridiag_​solve)

▸▿ 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
f04bgc (real_posdef_tridiag_solve)