NAG Library Routine Document

F07HBF (DPBSVX)

INTEGER	N, KD, NRHS, LDAB, LDAFB, LDB, LDX, IWORK(N), INFO
REAL (KIND=nag_wp)	AB(LDAB,), AFB(LDAFB,), S(), B(LDB,), X(LDX,), RCOND, FERR(NRHS), BERR(NRHS), WORK(3N)
CHARACTER(1)	FACT, UPLO, EQUED

The routine may be called by its LAPACK name dpbsvx.

3 Description

F07HBF (DPBSVX) performs the following steps:

FACT ='E'

, real diagonal scaling factors,

D_{S}

, are computed to equilibrate the system:

(D_{S} A D_{S}) (D_{S}^{- 1} X) = D_{S} B .

Whether or not the system will be equilibrated depends on the scaling of the matrix

A

, but if equilibration is used,

A

is overwritten by

D_{S} A D_{S}

and

B

D_{S} B

FACT ='N'

'E'

, the Cholesky decomposition is used to factor the matrix

A

(after equilibration if

FACT ='E'

) as

A = U^{T} U

UPLO ='U'

A = L L^{T}

UPLO ='L'

, where

U

is an upper triangular matrix and

L

is a lower triangular matrix.

If the leading

i

i

principal minor of

A

is not positive definite, then the routine returns with

INFO = i

. Otherwise, the factored form of

A

is used to estimate the condition number of the matrix

A

. If the reciprocal of the condition number is less than machine precision,

INFO = N + 1

is returned as a warning, but the routine still goes on to solve for

X

and compute error bounds as described below.

The system of equations is solved for

X

using the factored form of

A

Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

If equilibration was used, the matrix

X

is premultiplied by

D_{S}

so that it solves the original system before equilibration.

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

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Parameters

1: $FACT$ – CHARACTER(1)Input

On entry: specifies whether or not the factorized form of the matrix

A

is supplied on entry, and if not, whether the matrix

A

should be equilibrated before it is factorized.

$FACT ='F'$: AFB contains the factorized form of $A$ . If $EQUED ='Y'$ , the matrix $A$ has been equilibrated with scaling factors given by S. AB and AFB will not be modified.
$FACT ='N'$: The matrix $A$ will be copied to AFB and factorized.
$FACT ='E'$: The matrix $A$ will be equilibrated if necessary, then copied to AFB and factorized.

Constraint:

FACT ='F'

'N'

'E'

2: $UPLO$ – 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'

3: $N$ – INTEGERInput

On entry:

n

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

A

Constraint:

N \geq 0

4: $KD$ – 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

5: $NRHS$ – INTEGERInput

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

NRHS \geq 0

6: $AB (LDAB *)$ – 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

, except if

FACT ='F'

and

EQUED ='Y'

, in which case AB must contain the equilibrated matrix

D_{S} A D_{S}

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

FACT ='E'

and

EQUED ='Y'

, AB is overwritten by

D_{S} A D_{S}

7: $LDAB$ – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07HBF (DPBSVX) is called.

Constraint:

LDAB \geq KD + 1

8: $AFB (LDAFB *)$ – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: if

FACT ='F'

, AFB contains 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

. If

EQUED ='Y'

, AFB is the factorized form of the equilibrated matrix

A

On exit: if

FACT ='N'

, AFB returns the triangular factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

FACT ='E'

, AFB returns the triangular factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

of the equilibrated matrix

A

(see the description of AB for the form of the equilibrated matrix).

9: $LDAFB$ – INTEGERInput

On entry: the first dimension of the array AFB as declared in the (sub)program from which F07HBF (DPBSVX) is called.

Constraint:

LDAFB \geq KD + 1

10: $EQUED$ – CHARACTER(1)Input/Output

On entry: if

FACT ='N'

'E'

, EQUED need not be set.

FACT ='F'

, EQUED must specify the form of the equilibration that was performed as follows:

if $EQUED ='N'$ , no equilibration;
if $EQUED ='Y'$ , equilibration was performed, i.e., $A$ has been replaced by $D_{S} A D_{S}$ .

On exit: if

FACT ='F'

, EQUED is unchanged from entry.

Otherwise, if no constraints are violated, EQUED specifies the form of the equilibration that was performed as specified above.

Constraint: if

FACT ='F'

EQUED ='N'

'Y'

11: $S (*)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array S must be at least

\max (1, N)

On entry: if

FACT ='N'

'E'

, S need not be set.

FACT ='F'

and

EQUED ='Y'

, S must contain the scale factors,

D_{S}

, for

A

; each element of S must be positive.

On exit: if

FACT ='F'

, S is unchanged from entry.

Otherwise, if no constraints are violated and

EQUED ='Y'

, S contains the scale factors,

D_{S}

, for

A

; each element of S is positive.

12: $B (LDB *)$ – 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

EQUED ='N'

, B is not modified.

EQUED ='Y'

, B is overwritten by

D_{S} B

13: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07HBF (DPBSVX) is called.

Constraint:

LDB \geq \max (1, N)

14: $X (LDX *)$ – REAL (KIND=nag_wp) arrayOutput

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

\max (1, NRHS)

On exit: if

INFO = 0

N + 1

, the

n

r

solution matrix

X

to the original system of equations. Note that the arrays

A

and

B

are modified on exit if

EQUED ='Y'

, and the solution to the equilibrated system is

D_{S}^{- 1} X

15: $LDX$ – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07HBF (DPBSVX) is called.

Constraint:

LDX \geq \max (1, N)

16: $RCOND$ – REAL (KIND=nag_wp)Output

On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix

A

(after equilibration if that is performed), computed as

RCOND = 1.0 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

17: $FERR (NRHS)$ – REAL (KIND=nag_wp) arrayOutput

On exit: if

INFO = 0

N + 1

, an estimate of the forward error bound for each computed solution vector, such that

{‖{\hat{x}}_{j} - x_{j}‖}_{\infty} / {‖x_{j}‖}_{\infty} \leq FERR (j)

where

{\hat{x}}_{j}

is the

j

th column of the computed solution returned in the array X and

x_{j}

is the corresponding column of the exact solution

X

. The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.

18: $BERR (NRHS)$ – REAL (KIND=nag_wp) arrayOutput

On exit: if

INFO = 0

N + 1

, an estimate of the component-wise relative backward error of each computed solution vector

{\hat{x}}_{j}

(i.e., the smallest relative change in any element of

A

B

that makes

{\hat{x}}_{j}

an exact solution).

19: $WORK (3 \times N)$ – REAL (KIND=nag_wp) arrayWorkspace

20: $IWORK (N)$ – INTEGER arrayWorkspace

21: $INFO$ – 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.

$INFO > 0 and INFO \leq N$: 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. $RCOND = 0.0$ is returned.

$INFO = N + 1$: $U$ (or $L$ ) is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

7 Accuracy

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

if $UPLO ='U'$ , $|E| \leq c (n) ε |U^{T}| |U|$ ;
if $UPLO ='L'$ , $|E| \leq c (n) ε |L| |L^{T}|$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 10.1 of Higham (2002) for further details.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖\hat{x}‖}_{\infty}} \leq w_{c} cond (A, \hat{x}, b)

where

cond (A, \hat{x}, b) = {‖|A^{- 1}| (|A| |\hat{x}| + |b|)‖}_{\infty} / {‖\hat{x}‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

. If

\hat{x}

is the

j

th column of

X

, then

w_{c}

is returned in

BERR (j)

and a bound on

{‖x - \hat{x}‖}_{\infty} / {‖\hat{x}‖}_{\infty}

is returned in

FERR (j)

. See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

F07HBF (DPBSVX) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F07HBF (DPBSVX) 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 factorization of

A

requires approximately

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

floating-point operations, where

k

is the number of superdiagonals.

For each right-hand side, computation of the backward error involves a minimum of

8 n k

floating-point operations. Each step of iterative refinement involves an additional

12 n k

operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form

A x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

4 n k

operations.

The complex analogue of this routine is F07HPF (ZPBSVX).

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 & 5.10 \\ 9.31 & 30.81 \\ - 5.24 & - 25.82 \\ 11.83 & 22.90 \end{matrix}) .

Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix

A

are also output.

NAG Library Routine DocumentF07HBF (DPBSVX)

▸▿ 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

F07HBF (DPBSVX)