NAG Library Routine Document

F07HAF (DPBSV)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F07HAF (DPBSV) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

symmetric positive definite band matrix of bandwidth

(2 k_{d} + 1)

and

X

and

B

are

n

r

matrices.

2 Specification

SUBROUTINE F07HAF (

UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)

INTEGER	N, KD, NRHS, LDAB, LDB, INFO
REAL (KIND=nag_wp)	AB(LDAB,), B(LDB,)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dpbsv.

3 Description

F07HAF (DPBSV) uses the Cholesky decomposition to factor

A

A = U^{T} U

UPLO ='U'

A = L L^{T}

UPLO ='L'

, where

U

is an upper triangular band matrix, and

L

is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as

A

. 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: $UPLO$ – CHARACTER(1)Input

On entry: if

UPLO ='U'

, the upper triangle of

A

is stored.

UPLO ='L'

, the lower triangle of

A

is stored.

Constraint:

UPLO ='U'

'L'

2: $N$ – INTEGERInput

On entry:

n

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

A

Constraint:

N \geq 0

3: $KD$ – INTEGERInput

On entry:

k_{d}

, the number of superdiagonals of the matrix

A

UPLO ='U'

, or the number of subdiagonals if

UPLO ='L'

Constraint:

KD \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 upper or lower triangle of the symmetric band matrix

A

The matrix is stored in rows

1

k_{d} + 1

, more precisely,

if $UPLO ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $UPLO ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

On exit: if

INFO = 0

, the triangular factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

of the band matrix

A

, in the same storage format as

A

6: $LDAB$ – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07HAF (DPBSV) is called.

Constraint:

LDAB \geq KD + 1

7: $B (LDB *)$ – 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

8: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07HAF (DPBSV) is called.

Constraint:

LDB \geq \max (1, N)

9: $INFO$ – 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. An explanatory message is output, and execution of the program is terminated.

$INFO > 0$: The leading minor of order $〈value〉$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been 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.

F07HBF (DPBSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04BFF solves

A x = b

and returns a forward error bound and condition estimate. F04BFF calls F07HAF (DPBSV) to solve the equations.

8 Parallelism and Performance

F07HAF (DPBSV) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F07HAF (DPBSV) 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

When

n ≫ k

, the total number of floating-point operations is approximately

n {(k + 1)}^{2} + 4 n k r

, where

k

is the number of superdiagonals and

r

is the number of right-hand sides.

The complex analogue of this routine is F07HNF (ZPBSV).

10 Example

This example solves the equations

A x = b,

where

A

is the symmetric positive definite band matrix

A = (\begin{array}{r} 5.49 & 2.68 & 0 & 0 \\ 2.68 & 5.63 & - 2.39 & 0 \\ 0 & - 2.39 & 2.60 & - 2.22 \\ 0 & 0 & - 2.22 & 5.17 \end{array}) and b = (\begin{matrix} 22.09 \\ 9.31 \\ - 5.24 \\ 11.83 \end{matrix}) .

Details of the Cholesky factorization of

A

are also output.

NAG Library Routine DocumentF07HAF (DPBSV)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

F07HAF (DPBSV)