NAG Library Routine Document

Integer, Intent (In)	::	n, nrhs, lda, ldaf, ldb, ldx
Integer, Intent (Inout)	::	ipiv(*)
Integer, Intent (Out)	::	iwork(n), info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), af(ldaf,), r(), c(), b(ldb,), x(ldx,)
Real (Kind=nag_wp), Intent (Out)	::	rcond, ferr(nrhs), berr(nrhs), work(max(1,4*n))
Character (1), Intent (In)	::	fact, trans
Character (1), Intent (Inout)	::	equed

C Header Interface

#include <nagmk26.h>

void

f07abf_ (const char *fact, const char *trans, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, double af[], const Integer *ldaf, Integer ipiv[], char *equed, double r[], double c[], double b[], const Integer *ldb, double x[], const Integer *ldx, double *rcond, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_fact, const Charlen length_trans, const Charlen length_equed)

The routine may be called by its LAPACK name dgesvx.

3

Description

f07abf (dgesvx) 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 f07abf (dgesvx) 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

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'

'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

Arguments

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.

$fact ='F'$: af 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. a, af and ipiv are not modified.
$fact ='N'$: The matrix $A$ will be copied to af and factorized.
$fact ='E'$: The matrix $A$ will be equilibrated if necessary, then copied to af 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: $nrhs$ – 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: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, n)

On entry: the

n

n

matrix

A

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'

, a is not modified.

fact ='E'

equed \neq'N'

A

is scaled as follows:

if $equed ='R'$ , $A = D_{R} A$ ;
if $equed ='C'$ , $A = A D_{C}$ ;
if $equed ='B'$ , $A = D_{R} A D_{C}$ .

6: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f07abf (dgesvx) is called.

Constraint:

lda \geq \max (1, n)

7: $af (ldaf *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, n)

On entry: if

fact ='F'

, af contains the factors

L

and

U

from the factorization

A = P L U

as computed by f07adf (dgetrf). If

equed \neq'N'

, af is the factorized form of the equilibrated matrix

A

fact ='N'

'E'

, af need not be set.

On exit: if

fact ='N'

, af returns the factors

L

and

U

from the factorization

A = P L U

of the original matrix

A

fact ='E'

, af returns the factors

L

and

U

from the factorization

A = P L U

of the equilibrated matrix

A

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

fact ='F'

, af is unchanged from entry.

8: $ldaf$ – IntegerInput

On entry: the first dimension of the array af as declared in the (sub)program from which f07abf (dgesvx) is called.

Constraint:

ldaf \geq \max (1, n)

9: $ipiv (*)$ – Integer arrayInput/Output

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

\max (1, n)

On entry: if

fact ='F'

, ipiv contains the pivot indices from the factorization

A = P L U

as computed by f07adf (dgetrf); 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'

'E'

, ipiv need not be set.

On exit: if

fact ='N'

, ipiv contains the pivot indices from the factorization

A = P L U

of the original matrix

A

fact ='E'

, ipiv contains the pivot indices from the factorization

A = P L U

of the equilibrated matrix

A

fact ='F'

, ipiv is unchanged from entry.

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 $equed ='R'$ , 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'

11: $r (*)$ – 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.

12: $c (*)$ – 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'

and

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.

13: $b (ldb *)$ – 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

14: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f07abf (dgesvx) is called.

Constraint:

ldb \geq \max (1, n)

15: $x (ldx *)$ – 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'

16: $ldx$ – IntegerInput

On entry: the first dimension of the array x as declared in the (sub)program from which f07abf (dgesvx) is called.

Constraint:

ldx \geq \max (1, n)

17: $rcond$ – 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})

18: $ferr (nrhs)$ – 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.

19: $berr (nrhs)$ – 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).

20: $work (\max (1, 4 \times n))$ – Real (Kind=nag_wp) arrayOutput

On exit:

work (1)

contains the reciprocal pivot growth factor

‖A‖ / ‖U‖

. The ‘max absolute element’ norm is used. 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 estimate rcond, and forward error bound ferr could be unreliable. If the factorization fails with

info > 0 and info \leq n

, then

work (1)

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

A

21: $iwork (n)$ – Integer arrayWorkspace

22: $info$ – 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.

$info > 0 and info \leq n$: Element $〈value〉$ of the diagonal 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.

$info = n + 1$: $U$ 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

Parallelism and Performance

f07abf (dgesvx) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07abf (dgesvx) 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

The factorization of

A

requires approximately

\frac{2}{3} n^{3}

floating-point operations.

Estimating the forward error involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n^{2}

operations.

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 f07apf (zgesvx).

10

Example

This example solves the equations

A X = B,

where

A

is the general matrix

A = (\begin{matrix} 1.80 & 2.88 & 2.05 & - 0.89 \\ 525.00 & - 295.00 & - 95.00 & - 380.00 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & - 0.80 \end{matrix})

and

B = (\begin{matrix} 9.52 & 18.47 \\ 2435.00 & 225.00 \\ 0.77 & - 13.28 \\ - 6.22 & - 6.21 \end{matrix}) .

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

A

and an estimate of the reciprocal of the pivot growth factor for the factorization of

A

are also output.

NAG Library Routine Document

f07abf (dgesvx)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

2

Specification

3

Description

4

References

5

Arguments

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