NAG CL Interface
f08pnc (zgees)

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: $\mathbf{jobvs}$ – Nag_JobType Input

On entry: if $\mathbf{jobvs}=\mathrm{Nag\_DoNothing}$, Schur vectors are not computed.

If $\mathbf{jobvs}=\mathrm{Nag\_Schur}$, Schur vectors are computed.

Constraint: $\mathbf{jobvs}=\mathrm{Nag\_DoNothing}$ or $\mathrm{Nag\_Schur}$.

3: $\mathbf{sort}$ – Nag_SortEigValsType Input

On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.

$\mathbf{sort}=\mathrm{Nag\_NoSortEigVals}$

Eigenvalues are not ordered.

$\mathbf{sort}=\mathrm{Nag\_SortEigVals}$

Eigenvalues are ordered (see select).

Constraint: $\mathbf{sort}=\mathrm{Nag\_NoSortEigVals}$ or $\mathrm{Nag\_SortEigVals}$.

4: $\mathbf{select}$ – function, supplied by the user External Function

If $\mathbf{sort}=\mathrm{Nag\_SortEigVals}$, select is used to select eigenvalues to sort to the top left of the Schur form.

If $\mathbf{sort}=\mathrm{Nag\_NoSortEigVals}$, select is not referenced and f08pnc may be specified as NULLFN.

An eigenvalue $\mathbf{w}\left[j-1\right]$ is selected if $\mathbf{select}\left(\mathbf{w}\left[j-1\right]\right)$ is Nag_TRUE.

The specification of select is:

copy

Nag_Boolean  select (Complex w)

1: $\mathbf{w}$ – Complex Input

On entry: the real and imaginary parts of the eigenvalue.

5: $\mathbf{n}$ – Integer Input

On entry: $n$, the order of the matrix $A$.

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

6: $\mathbf{a}\left[\mathit{dim}\right]$ – Complex 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 $n\times n$ matrix $A$.

On exit: a is overwritten by its Schur form $T$.

7: $\mathbf{pda}$ – Integer Input

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

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

8: $\mathbf{sdim}$ – Integer * Output

On exit: if $\mathbf{sort}=\mathrm{Nag\_NoSortEigVals}$, $\mathbf{sdim}=0$.

If $\mathbf{sort}=\mathrm{Nag\_SortEigVals}$, $\mathbf{sdim}=$ number of eigenvalues for which select is Nag_TRUE.

9: $\mathbf{w}\left[\mathit{dim}\right]$ – Complex Output

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

On exit: contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form $T$.

10: $\mathbf{vs}\left[\mathit{dim}\right]$ – Complex Output

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

• $\max (1,\mathbf{pdvs}\times \mathbf{n})$ when $\mathbf{jobvs}=\mathrm{Nag\_Schur}$;
• $1$ otherwise.

$i$th element of the $j$th vector is stored in

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

On exit: if $\mathbf{jobvs}=\mathrm{Nag\_Schur}$, vs contains the unitary matrix $Z$ of Schur vectors.

If $\mathbf{jobvs}=\mathrm{Nag\_DoNothing}$, vs is not referenced.

11: $\mathbf{pdvs}$ – Integer Input

On entry: the stride used in the array vs.

Constraints:

• if $\mathbf{jobvs}=\mathrm{Nag\_Schur}$, $\mathbf{pdvs}\ge \max (1,\mathbf{n})$;
• otherwise $\mathbf{pdvs}\ge 1$.

12: $\mathbf{fail}$ – 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.
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_CONVERGENCE

The $QR$ algorithm failed to compute all the eigenvalues.

NE_ENUM_INT_2

On entry, $\mathbf{jobvs}=⟨\mathit{\text{value}}⟩$, $\mathbf{pdvs}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{jobvs}=\mathrm{Nag\_Schur}$, $\mathbf{pdvs}\ge \max (1,\mathbf{n})$;
otherwise $\mathbf{pdvs}\ge 1$.

NE_INT

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

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

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

NE_INT_2

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

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_SCHUR_REORDER

The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).

NE_SCHUR_REORDER_SELECT

After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy $\mathbf{select}=\mathrm{Nag\_TRUE}$. This could also be caused by underflow due to scaling.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results