NAG Library Routine Document

F07BAF (DGBSV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07BAF (DGBSV) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

band matrix, with

k_{l}

subdiagonals and

k_{u}

superdiagonals, and

X

and

B

are

n

r

matrices.

2 Specification

SUBROUTINE F07BAF (

N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)

INTEGER	N, KL, KU, NRHS, LDAB, IPIV(N), LDB, INFO
REAL (KIND=nag_wp)	AB(LDAB,), B(LDB,)

The routine may be called by its LAPACK name dgbsv.

3 Description

F07BAF (DGBSV) 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 Parameters

1: N – INTEGERInput

On entry:

n

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

A

Constraint:

N \geq 0

2: KL – INTEGERInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

KL \geq 0

3: KU – INTEGERInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

KU \geq 0

4: 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

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

n

n

coefficient 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 (n, j + k_{l}) .

See Section 8 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 F07BAF (DGBSV) is called.

Constraint:

LDAB \geq 2 \times KL + KU + 1

7: IPIV(N) – INTEGER arrayOutput

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)

IPIV (i) = i

indicates a row interchange was not required.

8: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, NRHS)

On entry: the

n

r

right-hand side matrix

B

On exit: if

INFO = 0

, the

n

r

solution matrix

X

9: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07BAF (DGBSV) is called.

Constraint:

LDB \geq \max (1, N)

10: INFO – INTEGEROutput

On exit:

INFO = 0

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

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

$INFO > 0$: If $INFO = i$ , $u_{i i}$ 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 F07BAF (DGBSV), F07BGF (DGBCON) can be used to estimate the condition number of

A

and F07BHF (DGBRFS) can be used to obtain approximate error bounds. Alternatives to F07BAF (DGBSV), which return condition and error estimates directly are F04BBF and F07BBF (DGBSVX).

8 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} * & * & * & + & + & + \\ * & * & 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}

Array elements marked

*

need not be set and are not referenced by the routine. Array elements marked

+

need not be set, but are defined on exit from the routine 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 routine is F07BNF (ZGBSV).

9 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 Routine DocumentF07BAF (DGBSV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07BAF (DGBSV)