nag_dpbsvx (Nag_OrderType order, Nag_FactoredFormType fact, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, double ab[], Integer pdab, double afb[], Integer pdafb, Nag_EquilibrationType *equed, double s[], double b[], Integer pdb, double x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)

3

Description

nag_dpbsvx (f07hbc) performs the following steps:

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

D_{S} B

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, the Cholesky decomposition is used to factor the matrix

A

(after equilibration if

fact = Nag_EquilibrateAndFactor

) as

A = U^{T} U

uplo = Nag_Upper

A = L L^{T}

uplo = Nag_Lower

, 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 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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

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.

$fact = Nag_Factored$: afb contains the factorized form of $A$ . If $equed = Nag_Equilibrated$ , the matrix $A$ has been equilibrated with scaling factors given by s. ab and afb will not be modified.
$fact = Nag_NotFactored$: The matrix $A$ will be copied to afb and factorized.
$fact = Nag_EquilibrateAndFactor$: The matrix $A$ will be equilibrated if necessary, then copied to afb 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: $kd$ – IntegerInput

On entry:

k_{d}

, the number of superdiagonals of the matrix

A

uplo = Nag_Upper

, or the number of subdiagonals if

uplo = Nag_Lower

Constraint:

kd \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: $ab [\dim]$ – doubleInput/Output

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

\max (1, pdab \times n)

On entry: the upper or lower triangle of the symmetric band matrix

A

, except if

fact = Nag_Factored

and

equed = Nag_Equilibrated

, in which case ab must contain the equilibrated matrix

D_{S} A D_{S}

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of

A_{i j}

, depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [k_{d} + i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k_{d}), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k_{d})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k_{d})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [k_{d} + j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k_{d}), \dots, i$ .

On exit: if

fact = Nag_EquilibrateAndFactor

and

equed = Nag_Equilibrated

, ab is overwritten by

D_{S} A D_{S}

8: $pdab$ – IntegerInput

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

A

in the array ab.

Constraint:

pdab \geq kd + 1

9: $afb [\dim]$ – doubleInput/Output

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

\max (1, pdafb \times n)

On entry: if

fact = Nag_Factored

, 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 = Nag_Equilibrated

, afb is the factorized form of the equilibrated matrix

A

On exit: if

fact = Nag_NotFactored

, afb returns the triangular factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

fact = Nag_EquilibrateAndFactor

, 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).

10: $pdafb$ – IntegerInput

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

A

in the array afb.

Constraint:

pdafb \geq kd + 1

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 $equed = Nag_Equilibrated$ , 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

12: $s [\dim]$ – 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.

13: $b [\dim]$ – 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

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

15: $x [\dim]$ – 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

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

17: $rcond$ – 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})

18: $ferr [nrhs]$ – 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.

19: $berr [nrhs]$ – 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).

20: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $kd = 〈value〉$ .
Constraint: $kd \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = 〈value〉$ .
Constraint: $nrhs \geq 0$ .

On entry, $pdab = 〈value〉$ .
Constraint: $pdab > 0$ .

On entry, $pdafb = 〈value〉$ .
Constraint: $pdafb > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .

On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $pdab = 〈value〉$ and $kd = 〈value〉$ .
Constraint: $pdab \geq kd + 1$ .

On entry, $pdafb = 〈value〉$ and $kd = 〈value〉$ .
Constraint: $pdafb \geq kd + 1$ .

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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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

Parallelism and Performance

nag_dpbsvx (f07hbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dpbsvx (f07hbc) 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

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 function is nag_zpbsvx (f07hpc).

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 Function Document

nag_dpbsvx (f07hbc)

▸▿ 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