NAG Library Function Document

nag_dgbtrf (f07bdc)

+− 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

nag_dgbtrf (f07bdc) computes the

L U

factorization of a real

m

n

band matrix.

2 Specification

#include <nag.h>

#include <nagf07.h>

void	nag_dgbtrf (Nag_OrderType order, Integer m, Integer n, Integer kl, Integer ku, double ab[], Integer pdab, Integer ipiv[], NagError *fail)

3 Description

nag_dgbtrf (f07bdc) forms the

L U

factorization of a real

m

n

band matrix

A

using partial pivoting, with row interchanges. Usually

m = n

, and then, if

A

has

k_{l}

nonzero subdiagonals and

k_{u}

nonzero superdiagonals, the factorization has the form

A = P L U

, where

P

is a permutation matrix,

L

is a lower triangular matrix with unit diagonal elements and at most

k_{l}

nonzero elements in each column, and

U

is an upper triangular band matrix with

k_{l} + k_{u}

superdiagonals.

Note that

L

is not a band matrix, but the nonzero elements of

L

can be stored in the same space as the subdiagonal elements of

A

U

is a band matrix but with

k_{l}

additional superdiagonals compared with

A

. These additional superdiagonals are created by the row interchanges.

4 References

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: m – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: n – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: kl – IntegerInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

5: ku – IntegerInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

6: ab[ $\dim$ ] – doubleInput/Output

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

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

On entry: the

m

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, m

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 in nag_dgbsv (f07bac) 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, m

and

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

, and

l_{i j}

, for

i = 1, \dots, m

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[ $\min (m, n)$ ] – IntegerOutput

On exit: the pivot indices that define the permutation matrix. At the

i

th step, if

ipiv [i - 1] > i

then row

i

of the matrix

A

was interchanged with row

ipiv [i - 1]

, for

i = 1, 2, \dots, \min (m, n)

ipiv [i - 1] \leq i

indicates that, at the

i

th step, a row interchange was not required.

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pdab = ⟨value⟩$ .
Constraint: $pdab > 0$ .
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.
NE_SINGULAR: $U (⟨value⟩, ⟨value⟩)$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7 Accuracy

The computed factors

L

and

U

are the exact factors of a perturbed matrix

A + E

, where

|E| \leq c (k) ε P |L| |U|,

c (k)

is a modest linear function of

k = k_{l} + k_{u} + 1

, and

ε

is the machine precision. This assumes

k ≪ \min (m, n)

8 Parallelism and Performance

nag_dgbtrf (f07bdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgbtrf (f07bdc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations varies between approximately

2 n k_{l} (k_{u} + 1)

and

2 n k_{l} (k_{l} + k_{u} + 1)

, depending on the interchanges, assuming

m = n ≫ k_{l}

and

n ≫ k_{u}

A call to nag_dgbtrf (f07bdc) may be followed by calls to the functions:

nag_dgbtrs (f07bec) to solve $A X = B$ or $A^{T} X = B$ ;
nag_dgbcon (f07bgc) to estimate the condition number of $A$ .

The complex analogue of this function is nag_zgbtrf (f07brc).

10 Example

This example computes the

L U

factorization of the matrix

A

, where

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0.00 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0.00 & 2.56 & 2.46 & 4.07 \\ 0.00 & 0.00 & - 4.78 & - 3.82 \end{matrix}) .

Here

A

is treated as a band matrix with one subdiagonal and two superdiagonals.

NAG Library Function Documentnag_dgbtrf (f07bdc)

+− 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 Library Function Document

nag_dgbtrf (f07bdc)