NAG Library Routine Document

F07BBF (DGBSVX)

INTEGER	N, KL, KU, NRHS, LDAB, LDAFB, IPIV(*), LDB, LDX, IWORK(N), INFO
REAL (KIND=nag_wp)	AB(LDAB,), AFB(LDAFB,), R(), C(), B(LDB,), X(LDX,), RCOND, FERR(NRHS), BERR(NRHS), WORK(max(1,3*N))
CHARACTER(1)	FACT, TRANS, EQUED

The routine may be called by its LAPACK name dgbsvx.

3 Description

F07BBF (DGBSVX) performs the following steps:

Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting $FACT ='E'$ . In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems $A X = B$ and $A^{T} X = B$ are
$(D_{R} A D_{C}) (D_{C}^{- 1} X) = D_{R} B$
and
${(D_{R} A D_{C})}^{T} (D_{R}^{- 1} X) = D_{C} B,$
respectively, where $D_{R}$ and $D_{C}$ are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.

When equilibration is used, $A$ will be overwritten by $D_{R} A D_{C}$ and $B$ will be overwritten by $D_{R} B$ (or $D_{C} B$ when the solution of $A^{T} X = B$ is sought).
Factorization
The matrix $A$ , or its scaled form, is copied and factored using the $L U$ decomposition
$A = P L U,$
where $P$ is a permutation matrix, $L$ is a unit lower triangular matrix, and $U$ is upper triangular.

This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to F07BBF (DGBSVX) with the same matrix $A$ .
Condition Number Estimation
The $L U$ factorization of $A$ determines whether a solution to the linear system exists. If some diagonal element of $U$ is zero, then $U$ is exactly singular, no solution exists and the routine returns with a failure. Otherwise the factorized 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 then a warning code is returned on final exit.
Solution
The (equilibrated) system is solved for $X$ ( $D_{C}^{- 1} X$ or $D_{R}^{- 1} X$ ) using the factored form of $A$ ( $D_{R} A D_{C}$ ).
Iterative Refinement
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
Construct Solution Matrix $X$
If equilibration was used, the matrix $X$ is premultiplied by $D_{C}$ (if $TRANS ='N'$ ) or $D_{R}$ (if $TRANS ='T'$ or $'C'$ ) 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 and IPIV contain the factorized form of $A$ . If $EQUED \neq'N'$ , the matrix $A$ has been equilibrated with scaling factors given by R and C. AB, AFB and IPIV are not 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: TRANS – CHARACTER(1)Input

On entry: specifies the form of the system of equations.

$TRANS ='N'$: $A X = B$ (No transpose).
$TRANS ='T'$ or $'C'$: $A^{T} X = B$ (Transpose).

Constraint:

TRANS ='N'

'T'

'C'

3: N – INTEGERInput

On entry:

n

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

A

Constraint:

N \geq 0

4: KL – INTEGERInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

KL \geq 0

5: KU – INTEGERInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

KU \geq 0

6: 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

7: 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

1

k_{l} + k_{u} + 1

, more precisely, the element

A_{i j}

must be stored in

AB (k_{u} + 1 + i - j, j) for ​ \max (1, j - k_{u}) \leq i \leq \min (n, j + k_{l}) .

See Section 8 for further details.

FACT ='F'

and

EQUED \neq'N'

A

must have been equilibrated by the scaling factors in R and/or C.

On exit: if

FACT ='F'

'N'

, or if

FACT ='E'

and

EQUED ='N'

, AB is not modified.

EQUED \neq'N'

then, if no constraints are violated,

A

is scaled as follows:

if $EQUED ='R'$ , $A = D_{r} A$ ;
if $EQUED ='C'$ , $A = A D_{c}$ ;
if $EQUED ='B'$ , $A = D_{r} A D_{c}$ .

8: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07BBF (DGBSVX) is called.

Constraint:

LDAB \geq KL + KU + 1

9: 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 ='N'

'E'

, AFB need not be set.

FACT ='F'

, details of the

L U

factorization of the

n

n

band matrix

A

, as computed by F07BDF (DGBTRF).

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

EQUED \neq'N'

, AFB is the factorized form of the equilibrated matrix

A

On exit: if

FACT ='F'

, AFB is unchanged from entry.

Otherwise, if no constraints are violated, then if

FACT ='N'

, AFB returns details of the

L U

factorization of the band matrix

A

, and if

FACT ='E'

, AFB returns details of the

L U

factorization of the equilibrated band matrix

A

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

10: LDAFB – INTEGERInput

On entry: the first dimension of the array AFB as declared in the (sub)program from which F07BBF (DGBSVX) is called.

Constraint:

LDAFB \geq 2 \times KL + KU + 1

11: IPIV( $*$ ) – INTEGER arrayInput/Output

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

\max (1, N)

On entry: if

FACT ='N'

'E'

, IPIV need not be set.

FACT ='F'

, IPIV contains the pivot indices from the factorization

A = L U

, as computed by F07BDF (DGBTRF); row

i

of the matrix was interchanged with row

IPIV (i)

On exit: if

FACT ='F'

, IPIV is unchanged from entry.

Otherwise, if no constraints are violated, IPIV contains 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.

FACT ='N'

, the pivot indices are those corresponding to the factorization

A = L U

of the original matrix

A

FACT ='E'

, the pivot indices are those corresponding to the factorization of

A = L U

of the equilibrated matrix

A

12: 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 $EQUED ='R'$ , row equilibration, i.e., $A$ has been premultiplied by $D_{R}$ ;
if $EQUED ='C'$ , column equilibration, i.e., $A$ has been postmultiplied by $D_{C}$ ;
if $EQUED ='B'$ , both row and column equilibration, i.e., $A$ has been replaced by $D_{R} A D_{C}$ .

On exit: if

FACT ='F'

, EQUED is unchanged from entry.

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

Constraint: if

FACT ='F'

EQUED ='N'

'R'

'C'

'B'

13: R( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: if

FACT ='N'

'E'

, R need not be set.

FACT ='F'

and

EQUED ='R'

'B'

, R must contain the row scale factors for

A

D_{R}

; each element of R must be positive.

On exit: if

FACT ='F'

, R is unchanged from entry.

Otherwise, if no constraints are violated and

EQUED ='R'

'B'

, R contains the row scale factors for

A

D_{R}

, such that

A

is multiplied on the left by

D_{R}

; each element of R is positive.

14: C( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: if

FACT ='N'

'E'

, C need not be set.

FACT ='F'

EQUED ='C'

'B'

, C must contain the column scale factors for

A

D_{C}

; each element of C must be positive.

On exit: if

FACT ='F'

, C is unchanged from entry.

Otherwise, if no constraints are violated and

EQUED ='C'

'B'

, C contains the row scale factors for

A

D_{C}

; each element of C is positive.

15: 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.

TRANS ='N'

and

EQUED ='R'

'B'

, B is overwritten by

D_{R} B

TRANS ='T'

'C'

and

EQUED ='C'

'B'

, B is overwritten by

D_{C} B

16: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07BBF (DGBSVX) is called.

Constraint:

LDB \geq \max (1, N)

17: 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 \neq'N'

, and the solution to the equilibrated system is

D_{C}^{- 1} X

TRANS ='N'

and

EQUED ='C'

'B'

, or

D_{R}^{- 1} X

TRANS ='T'

'C'

and

EQUED ='R'

'B'

18: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07BBF (DGBSVX) is called.

Constraint:

LDX \geq \max (1, N)

19: 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})

20: 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.

21: 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).

22: WORK( $\max (1, 3 \times N)$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: if

INFO = 0

WORK (1)

contains the reciprocal pivot growth factor

\max |a_{i j}| / \max |u_{i j}|

. If

WORK (1)

is much less than

1

, then the stability of the

L U

factorization of the (equilibrated) matrix

A

could be poor. This also means that the solution

X

, condition estimator RCOND, and forward error bound FERR could be unreliable. If the factorization fails with

INFO > 0 and INFO \leq N

WORK (1)

contains the reciprocal pivot growth factor for the leading INFO columns of

A

23: IWORK(N) – INTEGER arrayWorkspace

24: 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.

$INFO > 0 and INFO \leq N$: If $INFO = i$ , $u_{i i}$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $RCOND = 0.0$ is returned.

$INFO = N + 1$: The triangular matrix $U$ 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

\hat{x}

is the exact solution of a perturbed system of equations

(A + E) \hat{x} = b

, where

|E| \leq c (n) ε P |L| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 9.3 of Higham (2002) for further details.

x

is the true solution, then the computed solution

\hat{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 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}

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 condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement; see F07BHF (DGBRFS) for information on the floating point operations required.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of this routine is F07BPF (ZGBSVX).

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 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

Estimates for the backward errors, forward errors, condition number and pivot growth are also output, together with information on the equilibration of

A

NAG Library Routine DocumentF07BBF (DGBSVX)

+− 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

F07BBF (DGBSVX)