f07abc (Nag_OrderType order, Nag_FactoredFormType fact, Nag_TransType trans, Integer n, Integer nrhs, double a[], Integer pda, double af[], Integer pdaf, Integer ipiv[], Nag_EquilibrationType *equed, double r[], double c[], double b[], Integer pdb, double x[], Integer pdx, double *rcond, double ferr[], double berr[], double *recip_growth_factor, NagError *fail)

The function may be called by the names: f07abc, nag_lapacklin_dgesvx or nag_dgesvx.

3 Description

f07abc performs the following steps:

1.Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting $fact = Nag_EquilibrateAndFactor$ . 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).
2.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 f07abc with the same matrix $A$ .
3.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 function 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.
4.Solution
The (equilibrated) system is solved for $X$ ( $D_{C}^{- 1} X$ or $D_{R}^{- 1} X$ ) using the factored form of $A$ ( $D_{R} A D_{C}$ ).
5.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.
6.Construct Solution Matrix $X$
If equilibration was used, the matrix $X$ is premultiplied by $D_{C}$ (if $trans = Nag_NoTrans$ ) or $D_{R}$ (if $trans = Nag_Trans$ or $Nag_ConjTrans$ ) 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 https://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: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $fact$ – Nag_FactoredFormType 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.

$fact = Nag_Factored$: af and ipiv contain the factorized form of $A$ . If $equed \neq Nag_NoEquilibration$ , the matrix $A$ has been equilibrated with scaling factors given by r and c. a, af and ipiv are not modified.
$fact = Nag_NotFactored$: The matrix $A$ will be copied to af and factorized.
$fact = Nag_EquilibrateAndFactor$: The matrix $A$ will be equilibrated if necessary, then copied to af and factorized.

Constraint:

fact = Nag_Factored

Nag_NotFactored

Nag_EquilibrateAndFactor

3: $trans$ – Nag_TransType Input

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

$trans = Nag_NoTrans$: $A X = B$ (No transpose).
$trans = Nag_Trans$ or $Nag_ConjTrans$: $A^{T} X = B$ (Transpose).

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

4: $n$ – Integer Input

On entry:

n

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

A

Constraint:

n \geq 0

5: $nrhs$ – Integer Input

On entry:

r

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

B

Constraint:

nrhs \geq 0

6: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times n)

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n \times n

matrix

A

fact = Nag_Factored

and

equed \neq Nag_NoEquilibration

, a must have been equilibrated by the scaling factors in r and/or c.

On exit: if

fact = Nag_Factored

Nag_NotFactored

, or if

fact = Nag_EquilibrateAndFactor

and

equed = Nag_NoEquilibration

, a is not modified.

fact = Nag_EquilibrateAndFactor

equed \neq Nag_NoEquilibration

A

is scaled as follows:

if $equed = Nag_RowEquilibration$ , $A = D_{R} A$ ;
if $equed = Nag_ColumnEquilibration$ , $A = A D_{C}$ ;
if $equed = Nag_RowAndColumnEquilibration$ , $A = D_{R} A D_{C}$ .

7: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, n)

8: $af [\dim]$ – double Input/Output

Note: the dimension, dim, of the array af must be at least

\max (1, pdaf \times n)

The

(i, j)

th element of the matrix is stored in

$af [(j - 1) \times pdaf + i - 1]$ when $order = Nag_ColMajor$ ;
$af [(i - 1) \times pdaf + j - 1]$ when $order = Nag_RowMajor$ .

On entry: if

fact = Nag_Factored

, af contains the factors

L

and

U

from the factorization

A = P L U

as computed by f07adc. If

equed \neq Nag_NoEquilibration

, af is the factorized form of the equilibrated matrix

A

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, af need not be set.

On exit: if

fact = Nag_NotFactored

, af returns the factors

L

and

U

from the factorization

A = P L U

of the original matrix

A

fact = Nag_EquilibrateAndFactor

, 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 = Nag_Factored

, af is unchanged from entry.

9: $pdaf$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array af.

Constraint:

pdaf \geq \max (1, n)

10: $ipiv [\dim]$ – Integer Input/Output

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

\max (1, n)

On entry: if

fact = Nag_Factored

, ipiv contains the pivot indices from the factorization

A = P L U

as computed by f07adc; at the

i

th step row

i

of the matrix was interchanged with row

ipiv [i - 1]

ipiv [i - 1] = i

indicates a row interchange was not required.

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, ipiv need not be set.

On exit: if

fact = Nag_NotFactored

, ipiv contains the pivot indices from the factorization

A = P L U

of the original matrix

A

fact = Nag_EquilibrateAndFactor

, ipiv contains the pivot indices from the factorization

A = P L U

of the equilibrated matrix

A

fact = Nag_Factored

, ipiv is unchanged from entry.

11: $equed$ – Nag_EquilibrationType * Input/Output

On entry: if

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, equed need not be set.

fact = Nag_Factored

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

if $equed = Nag_NoEquilibration$ , no equilibration;
if $equed = Nag_RowEquilibration$ , row equilibration, i.e., $A$ has been premultiplied by $D_{R}$ ;
if $equed = Nag_ColumnEquilibration$ , column equilibration, i.e., $A$ has been postmultiplied by $D_{C}$ ;
if $equed = Nag_RowAndColumnEquilibration$ , both row and column equilibration, i.e., $A$ has been replaced by $D_{R} A D_{C}$ .

On exit: if

fact = Nag_Factored

, 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 = Nag_Factored

equed = Nag_NoEquilibration

Nag_RowEquilibration

Nag_ColumnEquilibration

Nag_RowAndColumnEquilibration

12: $r [\dim]$ – double Input/Output

Note: the dimension, dim, of the array r must be at least

\max (1, n)

On entry: if

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, r need not be set.

fact = Nag_Factored

and

equed = Nag_RowEquilibration

Nag_RowAndColumnEquilibration

, r must contain the row scale factors for

A

D_{R}

; each element of r must be positive.

On exit: if

fact = Nag_Factored

, r is unchanged from entry.

Otherwise, if no constraints are violated and

equed = Nag_RowEquilibration

Nag_RowAndColumnEquilibration

, 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.

13: $c [\dim]$ – double Input/Output

Note: the dimension, dim, of the array c must be at least

\max (1, n)

On entry: if

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, c need not be set.

fact = Nag_Factored

and

equed = Nag_ColumnEquilibration

Nag_RowAndColumnEquilibration

, c must contain the column scale factors for

A

D_{C}

; each element of c must be positive.

On exit: if

fact = Nag_Factored

, c is unchanged from entry.

Otherwise, if no constraints are violated and

equed = Nag_ColumnEquilibration

Nag_RowAndColumnEquilibration

, c contains the row scale factors for

A

D_{C}

; each element of c is positive.

14: $b [\dim]$ – double Input/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n \times r

right-hand side matrix

B

On exit: if

equed = Nag_NoEquilibration

, b is not modified.

trans = Nag_NoTrans

and

equed = Nag_RowEquilibration

Nag_RowAndColumnEquilibration

, b is overwritten by

D_{R} B

trans = Nag_Trans

Nag_ConjTrans

and

equed = Nag_ColumnEquilibration

Nag_RowAndColumnEquilibration

, b is overwritten by

D_{C} B

15: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

16: $x [\dim]$ – double Output

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

X

is stored in

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

fail . code =

NE_NOERROR or NE_SINGULAR_WP, the

n \times r

solution matrix

X

to the original system of equations. Note that the arrays

A

and

B

are modified on exit if

equed \neq Nag_NoEquilibration

, and the solution to the equilibrated system is

D_{C}^{- 1} X

trans = Nag_NoTrans

and

equed = Nag_ColumnEquilibration

Nag_RowAndColumnEquilibration

, or

D_{R}^{- 1} X

trans = Nag_Trans

Nag_ConjTrans

and

equed = Nag_RowEquilibration

Nag_RowAndColumnEquilibration

17: $pdx$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdx \geq \max (1, nrhs)$ .

18: $rcond$ – double * 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})

19: $ferr [nrhs]$ – double Output

On exit: if

fail . code =

NE_NOERROR or NE_SINGULAR_WP, 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 - 1]

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.

20: $berr [nrhs]$ – double Output

On exit: if

fail . code =

NE_NOERROR or NE_SINGULAR_WP, 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).

21: $recip_growth_factor$ – double * Output

On exit: if

fail . code =

NE_NOERROR, the reciprocal pivot growth factor

‖ A ‖ / ‖ U ‖

, where

‖ . ‖

denotes the maximum absolute element norm. If

recip_growth_factor ≪ 1

, the stability of the

L U

factorization of (equilibrated)

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

fail . code =

NE_SINGULAR, then

recip_growth_factor

contains the reciprocal pivot growth factor for the leading

fail . errnum

columns of

A

22: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = ⟨ value ⟩$ .
Constraint: $nrhs \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdaf = ⟨ value ⟩$ .
Constraint: $pdaf > 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .

On entry, $pdx = ⟨ value ⟩$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdaf = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdaf \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $nrhs = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, nrhs)$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq \max (1, n)$ .

On entry, $pdx = ⟨ value ⟩$ and $nrhs = ⟨ value ⟩$ .
Constraint: $pdx \geq \max (1, nrhs)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR: 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.
NE_SINGULAR_WP: $U$ 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 - 1]

and a bound on

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

is returned in

ferr [j - 1]

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f07abc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07abc 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 function. 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 function is f07apc.

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.

f07ab: FL CL CPP AD PY MB

NAG CL Interfacef07abc (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

NAG CL Interface
f07abc (dgesvx)