NAG Library Function Document

nag_dgees (f08pac)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{order}$ – 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 2.3.1.3 in How to Use the NAG Library and its Documentation 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_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:    $\mathbf{sort}$ – 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:    $\mathbf{select}$ – 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_dgees (f08pac) may be specified as NULLFN.

An eigenvalue $\mathbf{wr}\left[j-1\right]+\sqrt{-1}\times \mathbf{wi}\left[j-1\right]$ is selected if $\mathbf{select}\left(\mathbf{wr}\left[j-1\right],\mathbf{wi}\left[j-1\right]\right)$ is Nag_TRUE. If either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy $\mathbf{select}\left(\mathbf{wr}\left[j-1\right],\mathbf{wi}\left[j-1\right]\right)=\mathrm{Nag\_TRUE}$ after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case $\mathbf{fail}\mathbf{.}\mathbf{errnum}$ is set to $\mathbf{n}+2$.

The specification of select is:

Nag_Boolean  select (double wr, double wi)

1:    $\mathbf{wr}$ – doubleInput

2:    $\mathbf{wi}$ – doubleInput

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

5:    $\mathbf{n}$ – IntegerInput

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

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

6:    $\mathbf{a}\left[\mathit{dim}\right]$ – doubleInput/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 real Schur form $T$.

7:    $\mathbf{pda}$ – 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:    $\mathbf{sdim}$ – Integer *Output

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

If $\mathbf{sort}=\mathrm{Nag\_SortEigVals}$, $\mathbf{sdim}=$ number of eigenvalues (after sorting) for which select is Nag_TRUE. (Complex conjugate pairs for which select is Nag_TRUE for either eigenvalue count as $2$.)

9:    $\mathbf{wr}\left[\mathit{dim}\right]$ – doubleOutput

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

On exit: see the description of wi.

10:  $\mathbf{wi}\left[\mathit{dim}\right]$ – doubleOutput

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

On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form $T$. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

11:  $\mathbf{vs}\left[\mathit{dim}\right]$ – doubleOutput

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 orthogonal matrix $Z$ of Schur vectors.

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

12:  $\mathbf{pdvs}$ – 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 $\mathbf{pdvs}\ge 1$.

13:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 $\mathbf{jobvs}=\mathrm{Nag\_Schur}$, $\mathbf{pdvs}\ge \max \left(1,\mathbf{n}\right)$;
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

Program Text (f08pace.c)

10.2  Program Data

Program Data (f08pace.d)

10.3  Program Results

Program Results (f08pace.r)