void	f04bbc (Nag_OrderType order, Integer n, Integer kl, Integer ku, Integer nrhs, double ab[], Integer pdab, Integer ipiv[], double b[], Integer pdb, double rcond, double errbnd, NagError *fail)

The function may be called by the names: f04bbc, nag_linsys_real_band_solve or nag_real_band_lin_solve.

3 Description

The

L U

decomposition with partial pivoting and row interchanges is used to factor

A

A = P L U

, where

P

is a permutation matrix,

L

is the product of permutation matrices and unit lower triangular matrices with

k_{l}

subdiagonals, and

U

is upper triangular with

(k_{l} + k_{u})

superdiagonals. The factored form of

A

is then used to solve the system of equations

A X = B

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: $kl$ – Integer Input

On entry: the number of subdiagonals

k_{l}

, within the band of

A

Constraint:

kl \geq 0

4: $ku$ – Integer Input

On entry: the number of superdiagonals

k_{u}

, within the band of

A

Constraint:

ku \geq 0

5: $nrhs$ – 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

6: $ab [\dim]$ – double Input/Output

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

\max (1, pdab \times n)

On entry: the

n \times n

matrix

A

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements

A_{i j}

, for row

i = 1, \dots, n

and column

j = \max (1, i - k_{l}), \dots, \min (n, i + k_{u})

, depends on the order argument as follows:

if $order = Nag_ColMajor$ , $A_{i j}$ is stored as $ab [(j - 1) \times pdab + kl + ku + i - j]$ ;
if $order = Nag_RowMajor$ , $A_{i j}$ is stored as $ab [(i - 1) \times pdab + kl + j - i]$ .

See Section 9 for further details.

On exit: ab is overwritten by details of the factorization.

The elements,

u_{i j}

, of the upper triangular band factor

U

with

k_{l} + k_{u}

super-diagonals, and the multipliers,

l_{i j}

, used to form the lower triangular factor

L

are stored. The elements

u_{i j}

, for

i = 1, \dots, n

and

j = i, \dots, \min (n, i + k_{l} + k_{u})

, and

l_{i j}

, for

i = 1, \dots, n

and

j = \max (1, i - k_{l}), \dots, i

, are stored where

A_{i j}

is stored on entry.

7: $pdab$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array ab.

Constraint:

pdab \geq 2 \times kl + ku + 1

8: $ipiv [n]$ – Integer Output

On exit: if

fail . code =

NE_NOERROR, the pivot indices that define the permutation matrix

P

; at the

i

th step row

i

of the matrix was interchanged with row

ipiv [i - 1]

ipiv [i - 1] = i

indicates a row interchange was not required.

9: $b [\dim]$ – 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

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

11: $rcond$ – double * Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

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

12: $errbnd$ – 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.

13: $fail$ – 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 Integer allocatable memory required is n, and the double allocatable memory required is $3 \times n$ . 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, $kl = ⟨ value ⟩$ .
Constraint: $kl \geq 0$ .

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

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

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

On entry, $pdab = ⟨ value ⟩$ .
Constraint: $pdab > 0$ .

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_INT_3: On entry, $pdab = ⟨ value ⟩$ , $kl = ⟨ value ⟩$ and $ku = ⟨ value ⟩$ .
Constraint: $pdab \geq 2 \times kl + ku + 1$ .
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_RCOND: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
NE_SINGULAR: Diagonal element $⟨ value ⟩$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.

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. f04bbc 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.

f04bbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f04bbc 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 band storage scheme for the array ab stored in Nag_ColMajor is illustrated by the following example, when

n = 5

k_{l} = 2

, and

k_{u} = 1

. Storage of the band matrix

A

in the array ab:

Band matrix $A$	Band storage in array ab
Band matrix $A$	$order = Nag_ColMajor$	$order = Nag_RowMajor$
$\begin{array}{l} a_{11} & a_{12} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} & a_{45} \\ a_{53} & a_{54} & a_{55} \end{array}$	$\begin{array}{l} * & * & * & + & + \\ * & * & + & + & + \\ * & a_{12} & a_{23} & a_{34} & a_{45} \\ a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\ a_{21} & a_{32} & a_{43} & a_{54} & * \\ a_{31} & a_{42} & a_{53} & * & * \end{array}$	$\begin{array}{l} * & * & a_{11} & a_{12} & + & + \\ * & a_{21} & a_{22} & a_{23} & + & + \\ a_{31} & a_{32} & a_{33} & a_{34} & + & * \\ a_{42} & a_{43} & a_{44} & a_{45} & * & * \\ a_{53} & a_{54} & a_{55} & * & * & * \end{array}$

Array elements marked

*

need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements

u_{13}

u_{14}

u_{24}

u_{25}

and

u_{35}

. In this example when

order = Nag_ColMajor

the first referenced element of ab is

ab [3] = a_{11}

; while for

order = Nag_RowMajor

the first referenced element is

ab [2] = a_{11}

In general, elements

a_{i j}

are stored as follows:

if $order = Nag_ColMajor$ , $a_{i j}$ are stored in $ab [(j - 1) \times pdab + kl + ku + i - j]$
if $order = Nag_RowMajor$ , $a_{i j}$ are stored in $ab [(i - 1) \times pdab + kl + j - i]$

where

\max (1, i - kl) \leq j \leq \min (n, i + ku)

The total number of floating-point operations required to solve the equations

A X = B

depends upon the pivoting required, but if

n ≫ k_{l} + k_{u}

then it is approximately bounded by

O (n k_{l} (k_{l} + k_{u}))

for the factorization and

O (n (2 k_{l} + k_{u}) r)

for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of f04bbc is f04cbc.

10 Example

This example solves the equations

A X = B,

where

A

is the band matrix

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0 & 2.56 & 2.46 & 4.07 \\ 0 & 0 & - 4.78 & - 3.82 \end{matrix}) and B = (\begin{matrix} 4.42 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

An estimate of the condition number of

A

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

f04bb: FL CL CPP AD PY MB

NAG CL Interfacef04bbc (real_​band_​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
f04bbc (real_band_solve)