NAG Library Routine Document

void	f07bdf_ (const Integer m, const Integer n, const Integer kl, const Integer ku, double ab[], const Integer ldab, Integer ipiv[], Integer info)

The routine may be called by its LAPACK name dgbtrf.

3

Description

f07bdf (dgbtrf) 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: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

2: $n$ – IntegerInput

On entry:

n

, the number of columns 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: $ab (ldab *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array ab must be at least

\max (1, n)

On entry: the

m

n

matrix

A

The matrix is stored in rows

k_{l} + 1

2 k_{l} + k_{u} + 1

; the first

k_{l}

rows need not be set, more precisely, the element

A_{i j}

must be stored in

ab (k_{l} + k_{u} + 1 + i - j, j) = A_{i j} for ​ \max (1, j - k_{u}) \leq i \leq \min (m, j + k_{l}) .

See Section 9 in f07baf (dgbsv) for further details.

On exit: if

info \geq 0

, ab is overwritten by details of the factorization.

The upper triangular band matrix

U

, with

k_{l} + k_{u}

superdiagonals, is stored in rows

1

k_{l} + k_{u} + 1

of the array, and the multipliers used to form the matrix

L

are stored in rows

k_{l} + k_{u} + 2

2 k_{l} + k_{u} + 1

6: $ldab$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f07bdf (dgbtrf) is called.

Constraint:

ldab \geq 2 \times kl + ku + 1

7: $ipiv (\min (m, n))$ – Integer arrayOutput

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

i

th step, if

ipiv (i) > i

then row

i

of the matrix

A

was interchanged with row

ipiv (i)

, for

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

ipiv (i) \leq i

indicates that, at the

i

th step, a row interchange was not required.

8: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value.

If $info = - 999$ , dynamic memory allocation failed. See Section 3.7 in How to Use the NAG Library and its Documentation for further information. An explanatory message is output, and execution of the program is terminated.

$info > 0$: Element $〈value〉$ of the diagonal 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

f07bdf (dgbtrf) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07bdf (dgbtrf) 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 routine. 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 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 f07bdf (dgbtrf) may be followed by calls to the routines:

f07bef (dgbtrs) to solve $A X = B$ or $A^{T} X = B$ ;
f07bgf (dgbcon) to estimate the condition number of $A$ .

The complex analogue of this routine is f07brf (zgbtrf).

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 Routine Document

f07bdf (dgbtrf)

▸▿ 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