NAG Library Function Document

nag_dposvx (f07fbc)

nag_dposvx (Nag_OrderType order, Nag_FactoredFormType fact, Nag_UploType uplo, Integer n, Integer nrhs, double a[], Integer pda, double af[], Integer pdaf, Nag_EquilibrationType *equed, double s[], double b[], Integer pdb, double x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)

3 Description

nag_dposvx (f07fbc) performs the following steps:

If $fact = Nag_EquilibrateAndFactor$ , 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 $B$ by $D_{S} B$ .
If $fact = Nag_NotFactored$ or $Nag_EquilibrateAndFactor$ , the Cholesky decomposition is used to factor the matrix $A$ (after equilibration if $fact = Nag_EquilibrateAndFactor$ ) as $A = U^{T} U$ if $uplo = Nag_Upper$ or $A = L L^{T}$ if $uplo = Nag_Lower$ , where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix.
If the leading $i$ by $i$ principal minor of $A$ is not positive definite, then the function returns with $fail . errnum = i$ and $fail . code =$ NE_MAT_NOT_POS_DEF. 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, $fail . code =$ NE_SINGULAR_WP is returned as a warning, but the function 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 Arguments

1: order – Nag_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: fact – Nag_FactoredFormTypeInput

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 contains the factorized form of $A$ . If $equed = Nag_Equilibrated$ , the matrix $A$ has been equilibrated with scaling factors given by s. a and af will not be 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: uplo – Nag_UploTypeInput

On entry: if

uplo = Nag_Upper

, the upper triangle of

A

is stored.

uplo = Nag_Lower

, the lower triangle of

A

is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

4: n – IntegerInput

On entry:

n

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

A

Constraint:

n \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: a[ $\dim$ ] – doubleInput/Output

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

\max (1, pda \times n)

On entry: the

n

n

symmetric matrix

A

fact = Nag_Factored

and

equed = Nag_Equilibrated

, a must have been equilibrated by the scaling factor in s as

D_{S} A D_{S}

order = Nag_ColMajor

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

A_{i j}

is stored in

a [(i - 1) \times pda + j - 1]

uplo = Nag_Upper

, the upper triangular part of

A

must be stored and the elements of the array below the diagonal are not referenced.

uplo = Nag_Lower

, the lower triangular part of

A

must be stored and the elements of the array above the diagonal are not referenced.

On exit: if

fact = Nag_Factored

Nag_NotFactored

, or if

fact = Nag_EquilibrateAndFactor

and

equed = Nag_NoEquilibration

, a is not modified.

fact = Nag_EquilibrateAndFactor

and

equed = Nag_Equilibrated

, a is overwritten by

D_{S} A D_{S}

7: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array a.

Constraint:

pda \geq \max (1, n)

8: af[ $\dim$ ] – doubleInput/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 triangular factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

, in the same storage format as a. If

equed \neq Nag_NoEquilibration

, af is the factorized form of the equilibrated matrix

D_{S} A D_{S}

On exit: if

fact = Nag_NotFactored

, af returns the triangular factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

of the original matrix

A

fact = Nag_EquilibrateAndFactor

, af 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 a for the form of the equilibrated matrix).

9: pdaf – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array af.

Constraint:

pdaf \geq \max (1, n)

10: 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 $equed = Nag_Equilibrated$ , equilibration was performed, i.e., $A$ has been replaced by $D_{S} A D_{S}$ .

On exit: if

fact = Nag_Factored

, 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 = Nag_Factored

equed = Nag_NoEquilibration

Nag_Equilibrated

11: s[ $\dim$ ] – doubleInput/Output

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

\max (1, n)

On entry: if

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, s need not be set.

fact = Nag_Factored

and

equed = Nag_Equilibrated

, s must contain the scale factors,

D_{S}

, for

A

; each element of s must be positive.

On exit: if

fact = Nag_Factored

, s is unchanged from entry.

Otherwise, if no constraints are violated and

equed = Nag_Equilibrated

, s contains the scale factors,

D_{S}

, for

A

; each element of s is positive.

12: b[ $\dim$ ] – doubleInput/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

r

right-hand side matrix

B

On exit: if

equed = Nag_NoEquilibration

, b is not modified.

equed = Nag_Equilibrated

, b is overwritten by

D_{S} B

13: pdb – IntegerInput

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 $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

14: x[ $\dim$ ] – doubleOutput

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

r

solution matrix

X

to the original system of equations. Note that the arrays

A

and

B

are modified on exit if

equed = Nag_Equilibrated

, and the solution to the equilibrated system is

D_{S}^{- 1} X

15: pdx – IntegerInput

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 $order = Nag_RowMajor$ , $pdx \geq \max (1, nrhs)$ .

16: 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})

17: ferr[nrhs] – doubleOutput

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.

18: berr[nrhs] – doubleOutput

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).

19: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
NE_MAT_NOT_POS_DEF: 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.
NE_SINGULAR_WP: $U$ (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 $uplo = Nag_Upper$ , $|E| \leq c (n) ε |U^{T}| |U|$ ;
if $uplo = Nag_Lower$ , $|E| \leq c (n) ε |L| |L^{T}|$ ,

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 - 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 Further Comments

The factorization of

A

requires approximately

\frac{1}{3} n^{3}

floating point operations.

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

4 n^{2}

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

6 n^{2}

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

2 n^{2}

operations.

The complex analogue of this function is nag_zposvx (f07fpc).

9 Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite matrix

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix})

and

B = (\begin{matrix} 8.70 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \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 Function Documentnag_dposvx (f07fbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_dposvx (f07fbc)