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

3

Description

nag_dgbsv (f07bac) uses the

L U

decomposition with partial pivoting and row interchanges to factor

A

A = P L U

, where

P

is a permutation matrix,

L

is a product of permutation 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 http://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_OrderTypeInput

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $n$ – IntegerInput

On entry:

n

, the number of linear equations, i.e., the order of the matrix

A

Constraint:

n \geq 0

3: $kl$ – IntegerInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

4: $ku$ – IntegerInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

5: $nrhs$ – IntegerInput

On entry:

r

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

B

Constraint:

nrhs \geq 0

6: $ab [\dim]$ – doubleInput/Output

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

\max (1, pdab \times n)

On entry: the

n

n

coefficient 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: $pdab$ – IntegerInput

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: $ipiv [n]$ – IntegerOutput

On exit: if no constraints are violated, 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: $b [\dim]$ – doubleInput/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

r

right-hand side matrix

B

On exit: if

fail . code =

NE_NOERROR, the

n

r

solution matrix

X

10: $pdb$ – IntegerInput

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: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR: Element $〈value〉$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so 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. See Section 4.4 of Anderson et al. (1999) for further details.

Following the use of nag_dgbsv (f07bac), nag_dgbcon (f07bgc) can be used to estimate the condition number of

A

and nag_dgbrfs (f07bhc) can be used to obtain approximate error bounds. Alternatives to nag_dgbsv (f07bac), which return condition and error estimates directly are nag_real_band_lin_solve (f04bbc) and nag_dgbsvx (f07bbc).

8

Parallelism and Performance

nag_dgbsv (f07bac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgbsv (f07bac) 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 is illustrated by the following example, when

n = 6

k_{l} = 1

, and

k_{u} = 2

. Storage of the band matrix

A

in the array ab:

\begin{matrix} order = Nag_ColMajor \\ \begin{matrix} * & * & * & + & + & + \\ * & * & a_{13} & a_{24} & a_{35} & a_{46} \\ * & a_{12} & a_{23} & a_{34} & a_{45} & a_{56} \\ a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \\ a_{21} & a_{32} & a_{43} & a_{54} & a_{65} & * \end{matrix} \end{matrix} \begin{matrix} order = Nag_RowMajor \\ \begin{matrix} * & a_{11} & a_{12} & a_{13} & + \\ a_{21} & a_{22} & a_{23} & a_{24} & + \\ a_{32} & a_{33} & a_{34} & a_{35} & + \\ a_{43} & a_{44} & a_{45} & a_{46} & * \\ a_{54} & a_{55} & a_{56} & * & * \\ a_{65} & a_{66} & * & * & * \end{matrix} \end{matrix}

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_{14}

u_{25}

and

u_{36}

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 complex analogue of this function is nag_zgbsv (f07bnc).

10

Example

This example solves the equations

A x = b,

where

A

is the band matrix

A = (\begin{array}{r} - 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{array}) and b = (\begin{matrix} 4.42 \\ 27.13 \\ - 6.14 \\ 10.50 \end{matrix}) .

Details of the

L U

factorization of

A

are also output.

NAG Library Function Document

nag_dgbsv (f07bac)

▸▿ 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

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