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: indicates whether the upper or lower triangular part of

B

is stored.

$uplo = Nag_Upper$: The upper triangular part of $B$ is stored.
$uplo = Nag_Lower$: The lower triangular part of $B$ is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

B

Constraint:

n \geq 0

4: $kb$ – Integer Input

On entry: if

uplo = Nag_Upper

, the number of superdiagonals,

k_{b}

, of the matrix

B

uplo = Nag_Lower

, the number of subdiagonals,

k_{b}

, of the matrix

B

Constraint:

kb \geq 0

5: $bb [\dim]$ – double Input/Output

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

\max (1, pdbb \times n)

On entry: the

n \times n

symmetric positive definite band matrix

B

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

B_{i j}

, depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,: $B_{i j}$ is stored in $bb [k_{b} + i - j + (j - 1) \times pdbb]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k_{b}), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,: $B_{i j}$ is stored in $bb [i - j + (j - 1) \times pdbb]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k_{b})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,: $B_{i j}$ is stored in $bb [j - i + (i - 1) \times pdbb]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k_{b})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,: $B_{i j}$ is stored in $bb [k_{b} + j - i + (i - 1) \times pdbb]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k_{b}), \dots, i$ .

On exit:

B

is overwritten by the elements of its split Cholesky factor

S

6: $pdbb$ – Integer Input

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

B

in the array bb.

Constraint:

pdbb \geq kb + 1

7: $fail$ – 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, $kb = ⟨ value ⟩$ .
Constraint: $kb \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdbb = ⟨ value ⟩$ .
Constraint: $pdbb > 0$ .
NE_INT_2: On entry, $pdbb = ⟨ value ⟩$ and $kb = ⟨ value ⟩$ .
Constraint: $pdbb \geq kb + 1$ .
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 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.
NE_POS_DEF: The factorization could not be completed, because the updated element $b (⟨ value ⟩, ⟨ value ⟩)$ would be the square root of a negative number. Hence $B$ is not positive definite. This may indicate an error in forming the matrix $B$ .

7 Accuracy

The computed factor

S

is the exact factor of a perturbed matrix

(B + E)

, where

| E | \leq c (k + 1) ε | S^{T} | | S |,

c (k + 1)

is a modest linear function of

k + 1

, and

ε

is the machine precision. It follows that

| e_{i j} | \leq c (k + 1) ε \sqrt{(b_{i i} b_{j j})}

8 Parallelism and Performance

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

f08ufc 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.