void	f07hec (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, const double ab[], Integer pdab, double b[], Integer pdb, NagError *fail)

The function may be called by the names: f07hec, nag_lapacklin_dpbtrs or nag_dpbtrs.

3 Description

f07hec is used to solve a real symmetric positive definite band system of linear equations

A X = B

, the function must be preceded by a call to f07hdc which computes the Cholesky factorization of

A

. The solution

X

is computed by forward and backward substitution.

uplo = Nag_Upper

A = U^{T} U

, where

U

is upper triangular; the solution

X

is computed by solving

U^{T} Y = B

and then

U X = Y

uplo = Nag_Lower

A = L L^{T}

, where

L

is lower triangular; the solution

X

is computed by solving

L Y = B

and then

L^{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_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: $uplo$ – Nag_UploType Input

On entry: specifies how

A

has been factorized.

$uplo = Nag_Upper$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $kd$ – Integer Input

On entry:

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

5: $nrhs$ – Integer Input

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

6: $ab [\dim]$ – const double Input

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

\max (1, pdab \times n)

On entry: the Cholesky factor of

A

, as returned by f07hdc.

7: $pdab$ – Integer Input

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 kd + 1

8: $b [\dim]$ – 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: the

n \times r

solution matrix

X

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

10: $fail$ – 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, $kd = ⟨ value ⟩$ .
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, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pdab = ⟨ value ⟩$ and $kd = ⟨ value ⟩$ .
Constraint: $pdab \geq kd + 1$ .

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

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

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 $uplo = Nag_Upper$ , $| E | \leq c (k + 1) ε | U^{T} | | U |$ ;
if $uplo = Nag_Lower$ , $| E | \leq c (k + 1) ε | L | | L^{T} |$ ,

c (k + 1)

is a modest linear function of

k + 1

, and

ε

is the machine precision.

\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 + 1) 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)

Forward and backward error bounds can be computed by calling f07hhc, and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling f07hgc.

8 Parallelism and Performance

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

f07hec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07hec 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

4 n k r

, assuming

n ≫ k

This function may be followed by a call to f07hhc to refine the solution and return an error estimate.

The complex analogue of this function is f07hsc.

10 Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 5.49 & 2.68 & 0.00 & 0.00 \\ 2.68 & 5.63 & - 2.39 & 0.00 \\ 0.00 & - 2.39 & 2.60 & - 2.22 \\ 0.00 & 0.00 & - 2.22 & 5.17 \end{matrix}) and B = (\begin{matrix} 22.09 & 5.10 \\ 9.31 & 30.81 \\ - 5.24 & - 25.82 \\ 11.83 & 22.90 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by f07hdc.

f07he: FL CL CPP AD PY MB

NAG CL Interfacef07hec (dpbtrs)

▸▿ 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
f07hec (dpbtrs)