void	nag_dgbtrs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, const double ab[], Integer pdab, const Integer ipiv[], double b[], Integer pdb, NagError *fail)

3

Description

nag_dgbtrs (f07bec) is used to solve a real band system of linear equations

A X = B

A^{T} X = B

, the function must be preceded by a call to nag_dgbtrf (f07bdc) which computes the

L U

factorization of

A

A = P L U

. The solution is computed by forward and backward substitution.

trans = Nag_NoTrans

, the solution is computed by solving

P L Y = B

and then

U X = Y

trans = Nag_Trans

Nag_ConjTrans

, the solution is computed by solving

U^{T} Y = B

and then

L^{T} P^{T} X = Y

4

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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: $trans$ – Nag_TransTypeInput

On entry: indicates the form of the equations.

$trans = Nag_NoTrans$: $A X = B$ is solved for $X$ .
$trans = Nag_Trans$ or $Nag_ConjTrans$: $A^{T} X = B$ is solved for $X$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

3: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $kl$ – IntegerInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

5: $ku$ – IntegerInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

6: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

7: $ab [\dim]$ – const doubleInput

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

\max (1, pdab \times n)

On entry: the

L U

factorization of

A

, as returned by nag_dgbtrf (f07bdc).

8: $pdab$ – IntegerInput

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

Constraint:

pdab \geq 2 \times kl + ku + 1

9: $ipiv [\dim]$ – const IntegerInput

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

\max (1, n)

On entry: the pivot indices, as returned by nag_dgbtrf (f07bdc).

10: $b [\dim]$ – 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: the

n

r

solution matrix

X

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

12: $fail$ – 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, $kl = 〈value〉$ .
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, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
NE_INT_3: On entry, $pdab = 〈value〉$ , $kl = 〈value〉$ and $ku = 〈value〉$ .
Constraint: $pdab \geq 2 \times kl + ku + 1$ .
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.

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

|E| \leq c (k) ε P |L| |U|,

c (k)

is a modest linear function of

k = k_{l} + k_{u} + 1

, and

ε

is the machine precision. This assumes

k ≪ n

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (k) cond (A, x) ε

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

Note that

cond (A, x)

can be much smaller than

cond (A)

, and

cond (A^{T})

can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_dgbrfs (f07bhc), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_dgbcon (f07bgc) with

norm = Nag_InfNorm

8

Parallelism and Performance

nag_dgbtrs (f07bec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgbtrs (f07bec) 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 total number of floating-point operations is approximately

2 n (2 k_{l} + k_{u}) r

, assuming

n ≫ k_{l}

and

n ≫ k_{u}

This function may be followed by a call to nag_dgbrfs (f07bhc) to refine the solution and return an error estimate.

The complex analogue of this function is nag_zgbtrs (f07bsc).

10

Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0.00 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0.00 & 2.56 & 2.46 & 4.07 \\ 0.00 & 0.00 & - 4.78 & - 3.82 \end{matrix}) and B = (\begin{matrix} 4.42 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_dgbtrf (f07bdc).

NAG Library Function Document

nag_dgbtrs (f07bec)

▸▿ 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