# NAG CL Interfacef08pbc (dgeesx)

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## 1Purpose

f08pbc computes the eigenvalues, the real Schur form $T$, and, optionally, the matrix of Schur vectors $Z$ for an $n×n$ real nonsymmetric matrix $A$.

## 2Specification

 #include
void  f08pbc (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)
The function may be called by the names: f08pbc, nag_lapackeig_dgeesx or nag_dgeesx.

## 3Description

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 $1×1$ and $2×2$ blocks. $2×2$ 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, 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).

## 4References

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

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 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: $\mathbf{wr}$double Input
2: $\mathbf{wi}$double Input
On entry: the real and imaginary parts of the eigenvalue.
5: $\mathbf{sense}$Nag_RCondType Input
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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
7: $\mathbf{a}\left[\mathit{dim}\right]$double Input/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 $n×n$ matrix $A$.
On exit: a is overwritten by its real Schur form $T$.
8: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{sdim}$Integer * 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: $\mathbf{wr}\left[\mathit{dim}\right]$double Output
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: $\mathbf{wi}\left[\mathit{dim}\right]$double Output
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: $\mathbf{vs}\left[\mathit{dim}\right]$double Output
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.
$i$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: $\mathbf{pdvs}$Integer Input
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: $\mathbf{rconde}$double * 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: $\mathbf{rcondv}$double * 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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.
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.
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.

## 7Accuracy

The computed Schur factorization satisfies
 $A+E = ZTZT ,$
where
 $‖E‖2 = O(ε) ‖A‖2 ,$
and $\epsilon$ is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08pbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pbc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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

## 10Example

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 positive 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.

### 10.1Program Text

Program Text (f08pbce.c)

### 10.2Program Data

Program Data (f08pbce.d)

### 10.3Program Results

Program Results (f08pbce.r)