NAG CL Interface
f04bdc (real_​posdef_​solve)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{order}$ – 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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$. 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: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2: $\mathbf{uplo}$ – Nag_UploType Input

On entry: if $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the upper triangle of the matrix $A$ is stored.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the lower triangle of the matrix $A$ is stored.

Constraint: $\mathbf{uplo}=\mathrm{Nag\_Upper}$ or $\mathrm{Nag\_Lower}$.

3: $\mathbf{n}$ – Integer Input

On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

4: $\mathbf{nrhs}$ – Integer Input

On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.

Constraint: $\mathbf{nrhs}\ge 0$.

5: $\mathbf{a}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least $\max (1,\mathbf{pda}\times \mathbf{n})$.

The $(i,j)$th element of the matrix $A$ is stored in

• $\mathbf{a}\left[(j-1)\times \mathbf{pda}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{a}\left[(i-1)\times \mathbf{pda}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: the $n\times n$ symmetric matrix $A$.

If $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the leading n by n upper triangular part of a contains the upper triangular part of the matrix $A$, and the strictly lower triangular part of a is not referenced.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the leading n by n lower triangular part of a contains the lower triangular part of the matrix $A$, and the strictly upper triangular part of a is not referenced.

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, the factor $U$ or $L$ from the Cholesky factorization $A=U^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$.

6: $\mathbf{pda}$ – Integer Input

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

Constraint: $\mathbf{pda}\ge \max (1,\mathbf{n})$.

7: $\mathbf{b}\left[\mathit{dim}\right]$ – double Input/Output

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

• $\max (1,\mathbf{pdb}\times \mathbf{nrhs})$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max (1,\mathbf{n}\times \mathbf{pdb})$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

The $(i,j)$th element of the matrix $B$ is stored in

• $\mathbf{b}\left[(j-1)\times \mathbf{pdb}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{b}\left[(i-1)\times \mathbf{pdb}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: the $n\times r$ matrix of right-hand sides $B$.

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, the $n\times r$ solution matrix $X$.

8: $\mathbf{pdb}$ – Integer Input

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

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdb}\ge \max (1,\mathbf{n})$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdb}\ge \max (1,\mathbf{nrhs})$.

9: $\mathbf{rcond}$ – double * Output

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, an estimate of the reciprocal of the condition number of the matrix $A$, computed as $\mathbf{rcond}=1/\left({\Vert A\Vert }_1{\Vert A^{-1}\Vert }_1\right)$.

10: $\mathbf{errbnd}$ – double * Output

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution $\hat{x}$, such that ${\Vert \hat{x}-x\Vert }_1/{\Vert x\Vert }_1\le \mathbf{errbnd}$, where $\hat{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, errbnd is returned as unity.

11: $\mathbf{fail}$ – 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.
The Integer allocatable memory required is n, and the double allocatable memory required is $3\times \mathbf{n}$. Allocation failed before the solution could be computed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.

NE_INT

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 0$.

On entry, $\mathbf{nrhs}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nrhs}\ge 0$.

On entry, $\mathbf{pda}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pda}>0$.

On entry, $\mathbf{pdb}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdb}>0$.

NE_INT_2

On entry, $\mathbf{pda}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pda}\ge \max (1,\mathbf{n})$.

On entry, $\mathbf{pdb}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdb}\ge \max (1,\mathbf{n})$.

On entry, $\mathbf{pdb}=⟨\mathit{\text{value}}⟩$ and $\mathbf{nrhs}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdb}\ge \max (1,\mathbf{nrhs})$.

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.

NE_POS_DEF

The principal minor of order $⟨\mathit{\text{value}}⟩$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.

NE_RCOND

A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results