void	f07hac (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, double ab[], Integer pdab, double b[], Integer pdb, NagError *fail)

The function may be called by the names: f07hac, nag_lapacklin_dpbsv or nag_dpbsv.

3 Description

f07hac uses the Cholesky decomposition to factor

A

A = U^{T} U

uplo = Nag_Upper

A = L L^{T}

uplo = Nag_Lower

, where

U

is an upper triangular band matrix, and

L

is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as

A

. The factored form of

A

is then used to solve the system of equations

A X = B

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 https://www.netlib.org/lapack/lug

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: 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

3: $n$ – Integer Input

On entry:

n

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

A

Constraint:

n \geq 0

4: $kd$ – Integer Input

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

5: $nrhs$ – Integer Input

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

6: $ab [\dim]$ – double Input/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

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

fail . code =

NE_NOERROR, 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

7: $pdab$ – Integer Input

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

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: if

fail . code =

NE_NOERROR, 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_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.
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

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖ E ‖}_{1} = O (ε) {‖ A ‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖ \hat{x} - x ‖}_{1}}{{‖ x ‖}_{1}} \leq κ (A) \frac{{‖ E ‖}_{1}}{{‖ A ‖}_{1}},

where

κ (A) = {‖ A^{- 1} ‖}_{1} {‖ A ‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

f07hbc is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bfc solves

A x = b

and returns a forward error bound and condition estimate. f04bfc calls f07hac to solve the equations.

8 Parallelism and Performance

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

f07hac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07hac 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 total number of floating-point operations is approximately

n {(k + 1)}^{2} + 4 n k r

, where

k

is the number of superdiagonals and

r

is the number of right-hand sides.

The complex analogue of this function is f07hnc.

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 \\ 9.31 \\ - 5.24 \\ 11.83 \end{matrix}) .

Details of the Cholesky factorization of

A

are also output.

f07ha: FL CL CPP AD PY MB

NAG CL Interfacef07hac (dpbsv)

▸▿ 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
f07hac (dpbsv)