NAG Library Function Document

nag_zgees (f08pnc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     order – Nag_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or Nag_ColMajor.

2:     jobvs – Nag_JobTypeInput

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:     sort – Nag_SortEigValsTypeInput

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:     select – function, supplied by the userExternal 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 nag_zgees (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:

Nag_Boolean  select (Complex w)

1:     w – ComplexInput

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

5:     n – IntegerInput

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

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

6:     a[$\mathit{dim}$] – ComplexInput/Output

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

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

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

On entry: the $n$ by $n$ matrix $A$.

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

7:     pda – IntegerInput

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

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

8:     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:     w[$\mathit{dim}$] – ComplexOutput

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

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

10:   vs[$\mathit{dim}$] – ComplexOutput

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

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

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

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

On entry: the stride used in the array vs.

Constraints:

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

12:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

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 $\mathbf{jobvs}=\mathrm{Nag\_Schur}$, $\mathbf{pdvs}\ge \max \left(1,\mathbf{n}\right)$;
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: $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

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.

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  Further Comments

9  Example

9.1  Program Text

Program Text (f08pnce.c)

9.2  Program Data

Program Data (f08pnce.d)

9.3  Program Results

Program Results (f08pnce.r)