f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dgees (f08pac)

## 1  Purpose

nag_dgees (f08pac) computes the eigenvalues, the real Schur form $T$, and, optionally, the matrix of Schur vectors $Z$ for an $n$ by $n$ real nonsymmetric matrix $A$.

## 2  Specification

 #include #include
void  nag_dgees (Nag_OrderType order, Nag_JobType jobvs, Nag_SortEigValsType sort,
 Nag_Boolean (*select)(double wr, double wi),
Integer n, double a[], Integer pda, Integer *sdim, double wr[], double wi[], double vs[], Integer pdvs, NagError *fail)

## 3  Description

The real Schur factorization of $A$ is given by
 $A = Z T ZT ,$
where $Z$, the matrix of Schur vectors, is orthogonal and $T$ is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with $1$ by $1$ and $2$ by $2$ blocks. $2$ by $2$ blocks will be standardized in the form
 $a b c a$
where $bc<0$. The eigenvalues of such a block are $a±\sqrt{bc}$.
Optionally, nag_dgees (f08pac) also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left. The leading columns of $Z$ form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Arguments

1:     orderNag_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:     jobvsNag_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:     sortNag_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:     selectfunction, 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 ${\mathbf{wr}}\left[j-1\right]+\sqrt{-1}×{\mathbf{wi}}\left[j-1\right]$ 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:     wrdoubleInput
2:     widoubleInput
On entry: the real and imaginary parts of the eigenvalue.
5:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     a[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ 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:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     sdimInteger *Output
On exit: if ${\mathbf{sort}}=\mathrm{Nag_NoSortEigVals}$, ${\mathbf{sdim}}=0$.
If ${\mathbf{sort}}=\mathrm{Nag_SortEigVals}$, ${\mathbf{sdim}}=\text{}$ 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:     wr[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array wr must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: see the description of wi.
10:   wi[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array wi must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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:   vs[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array vs must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvs}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvs}}=\mathrm{Nag_Schur}$;
• $1$ otherwise.
The $i$th element of the $j$th vector is stored in
• ${\mathbf{vs}}\left[\left(j-1\right)×{\mathbf{pdvs}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vs}}\left[\left(i-1\right)×{\mathbf{pdvs}}+j-1\right]$ 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:   pdvsIntegerInput
On entry: the stride used in the array vs.
Constraints:
• if ${\mathbf{jobvs}}=\mathrm{Nag_Schur}$, ${\mathbf{pdvs}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvs}}\ge 1$.
13:   failNagError *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.
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 ${\mathbf{jobvs}}=\mathrm{Nag_Schur}$, ${\mathbf{pdvs}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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

The computed Schur factorization satisfies
 $A+E=ZT ZT ,$
where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

The total number of floating point operations is proportional to ${n}^{3}$.
The complex analogue of this function is nag_zgees (f08pnc).

## 9  Example

This example finds the Schur factorization of the matrix
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ,$
such that the real eigenvalues of $A$ are the top left diagonal elements of the Schur form, $T$.

### 9.1  Program Text

Program Text (f08pace.c)

### 9.2  Program Data

Program Data (f08pace.d)

### 9.3  Program Results

Program Results (f08pace.r)