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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

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 $INFO = i$ , the leading minor of order $i$ 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 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).

9 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 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07HAF (DPBSV)