NAG Library Function Document

nag_dgbsvx (f07bbc)

+− 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

1 Purpose

nag_dgbsvx (f07bbc) uses the

L U

factorization to compute the solution to a real system of linear equations

A X = B or A^{T} X = B,

where

A

is an

n

n

band matrix with

k_{l}

subdiagonals and

k_{u}

superdiagonals, and

X

and

B

are

n

r

matrices. Error bounds on the solution and a condition estimate are also provided.

2 Specification

#include <nag.h>

#include <nagf07.h>

void

nag_dgbsvx (Nag_OrderType order, Nag_FactoredFormType fact, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, double ab[], Integer pdab, double afb[], Integer pdafb, 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)

3 Description

nag_dgbsvx (f07bbc) 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 $fact = Nag_EquilibrateAndFactor$ . 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).
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 nag_dgbsvx (f07bbc) 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 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.
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 $trans = Nag_NoTrans$ ) 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 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.

$fact = Nag_Factored$: afb 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. ab, afb and ipiv are not 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: trans – Nag_TransTypeInput

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 – IntegerInput

On entry:

n

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

A

Constraint:

n \geq 0

5: kl – IntegerInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

6: ku – IntegerInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

7: 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

8: ab[ $\dim$ ] – doubleInput/Output

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

\max (1, pdab \times n)

On entry: the

n

n

coefficient matrix

A

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

A_{i j}

, for row

i = 1, \dots, n

and column

j = \max (1, i - k_{l}), \dots, \min (n, i + k_{u})

, depends on the order argument as follows:

if $order = Nag_ColMajor$ , $A_{i j}$ is stored as $ab [(j - 1) \times pdab + ku + i - j]$ ;
if $order = Nag_RowMajor$ , $A_{i j}$ is stored as $ab [(i - 1) \times pdab + kl + j - i]$ .

See Section 8 for further details.

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

, ab is not modified.

equed \neq Nag_NoEquilibration

then, if no constraints are violated,

A

is scaled as follows:

if $equed = Nag_RowEquilibration$ , $A = D_{r} A$ ;
if $equed = Nag_ColumnEquilibration$ , $A = A D_{c}$ ;
if $equed = Nag_RowAndColumnEquilibration$ , $A = D_{r} A D_{c}$ .

9: 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 kl + ku + 1

10: 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_NotFactored

Nag_EquilibrateAndFactor

, afb need not be set.

fact = Nag_Factored

, details of the

L U

factorization of the

n

n

band matrix

A

, as computed by nag_dgbtrf (f07bdc).

The elements,

u_{i j}

, of the upper triangular band factor

U

with

k_{l} + k_{u}

super-diagonals, and the multipliers,

l_{i j}

, used to form the lower triangular factor

L

are stored. The elements

u_{i j}

, for

i = 1, \dots, n

and

j = i, \dots, \min (n, i + k_{l} + k_{u})

, and

l_{i j}

, for

i = 1, \dots, n

and

j = \max (1, i - k_{l}), \dots, i

, are stored where

A_{i j}

is stored on entry.

equed \neq Nag_NoEquilibration

, afb is the factorized form of the equilibrated matrix

A

On exit: if

fact = Nag_Factored

, afb is unchanged from entry.

Otherwise, if no constraints are violated, then if

fact = Nag_NotFactored

, afb returns details of the

L U

factorization of the band matrix

A

, and if

fact = Nag_EquilibrateAndFactor

, 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).

11: 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 2 \times kl + ku + 1

12: ipiv[ $\dim$ ] – IntegerInput/Output

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

\max (1, n)

On entry: if

fact = Nag_NotFactored

Nag_EquilibrateAndFactor

, ipiv need not be set.

fact = Nag_Factored

, ipiv contains the pivot indices from the factorization

A = L U

, as computed by nag_dgbtrf (f07bdc); row

i

of the matrix was interchanged with row

ipiv [i - 1]

On exit: if

fact = Nag_Factored

, 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 - 1]

ipiv [i - 1] = i

indicates a row interchange was not required.

fact = Nag_NotFactored

, the pivot indices are those corresponding to the factorization

A = L U

of the original matrix

A

fact = Nag_EquilibrateAndFactor

, the pivot indices are those corresponding to the factorization of

A = L U

of the equilibrated matrix

A

13: 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

14: r[ $\dim$ ] – doubleInput/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.

15: c[ $\dim$ ] – doubleInput/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

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.

16: 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.

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

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

18: 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 \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

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

20: 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})

21: 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.

22: 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).

23: 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

24: 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, $kl = 〈value〉$ .
Constraint: $kl \geq 0$ .; On entry, $ku = 〈value〉$ .
Constraint: $ku \geq 0$ .; 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, $pdb = 〈value〉$ 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_INT_3: On entry, $pdab = 〈value〉$ , $kl = 〈value〉$ and $ku = 〈value〉$ .
Constraint: $pdab \geq kl + ku + 1$ .; On entry, $pdafb = 〈value〉$ , $kl = 〈value〉$ and $ku = 〈value〉$ .
Constraint: $pdafb \geq 2 \times kl + ku + 1$ .
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_SINGULAR: $U (〈value〉, 〈value〉)$ 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 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} order = Nag_ColMajor \\ \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} \end{matrix} \begin{matrix} order = Nag_RowMajor \\ \begin{matrix} * & a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} & a_{35} \\ a_{43} & a_{44} & a_{45} & a_{46} \\ a_{54} & a_{55} & a_{56} & * \\ a_{65} & a_{66} & * & * \end{matrix} \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 nag_dgbrfs (f07bhc) 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 function is nag_zgbsvx (f07bpc).

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 Function Documentnag_dgbsvx (f07bbc)

+− 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_dgbsvx (f07bbc)