nag_dgeesx (f08pbc) (PDF version)
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NAG C Library Manual

# NAG Library Function Documentnag_dgeesx (f08pbc)

## 1  Purpose

nag_dgeesx (f08pbc) 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_dgeesx (Nag_OrderType order, Nag_JobType jobvs, Nag_SortEigValsType sort,
 Nag_Boolean (*select)(double wr, double wi),
Nag_RCondType sense, Integer n, double a[], Integer pda, Integer *sdim, double wr[], double wi[], double vs[], Integer pdvs, double *rconde, double *rcondv, 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_dgeesx (f08pbc) also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of $Z$ form an orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.8 of Anderson et al. (1999) (where these quantities are called $s$ and $\mathrm{sep}$ respectively).

## 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_dgeesx (f08pbc) 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:     senseNag_RCondTypeInput
On entry: determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\mathrm{Nag_NotRCond}$
None are computed.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$
Computed for average of selected eigenvalues only.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$
Computed for selected right invariant subspace only.
${\mathbf{sense}}=\mathrm{Nag_RCondBoth}$
Computed for both.
If ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$, $\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$, ${\mathbf{sort}}=\mathrm{Nag_SortEigVals}$.
Constraint: ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$, $\mathrm{Nag_RCondEigVals}$, $\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$.
6:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
7:     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 $T$.
8:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     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 $2$.)
10:   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.
11:   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.
12:   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.
13:   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 ${\mathbf{pdvs}}\ge 1$.
14:   rcondedouble *Output
On exit: if ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$ or $\mathrm{Nag_RCondBoth}$, contains the reciprocal condition number for the average of the selected eigenvalues.
If ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$ or $\mathrm{Nag_RCondEigVecs}$, rconde is not referenced.
15:   rcondvdouble *Output
On exit: if ${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$, rcondv contains the reciprocal condition number for the selected right invariant subspace.
If ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$ or $\mathrm{Nag_RCondEigVals}$, rcondv is not referenced.
16:   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.
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 ${\mathbf{jobvs}}=\mathrm{Nag_Schur}$, ${\mathbf{pdvs}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
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
 $A+E = ZTZT ,$
where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

## 8  Further Comments

The total number of floating point operations is proportional to ${n}^{3}$.
The complex analogue of this function is nag_zgeesx (f08ppc).

## 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, $T$. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding invariant subspace are also returned.

### 9.1  Program Text

Program Text (f08pbce.c)

### 9.2  Program Data

Program Data (f08pbce.d)

### 9.3  Program Results

Program Results (f08pbce.r)

nag_dgeesx (f08pbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual