# NAG Library Function Document

## 1Purpose

nag_dhseqr (f08pec) computes all the eigenvalues and, optionally, the Schur factorization of a real Hessenberg matrix or a real general matrix which has been reduced to Hessenberg form.

## 2Specification

 #include #include
 void nag_dhseqr (Nag_OrderType order, Nag_JobType job, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, double h[], Integer pdh, double wr[], double wi[], double z[], Integer pdz, NagError *fail)

## 3Description

nag_dhseqr (f08pec) computes all the eigenvalues and, optionally, the Schur factorization of a real upper Hessenberg matrix $H$:
 $H = ZTZT ,$
where $T$ is an upper quasi-triangular matrix (the Schur form of $H$), and $Z$ is the orthogonal matrix whose columns are the Schur vectors ${z}_{i}$. See Section 9 for details of the structure of $T$.
The function may also be used to compute the Schur factorization of a real general matrix $A$ which has been reduced to upper Hessenberg form $H$:
 $A = QHQT, where ​Q​ is orthogonal, = QZTQZT.$
In this case, after nag_dgehrd (f08nec) has been called to reduce $A$ to Hessenberg form, nag_dorghr (f08nfc) must be called to form $Q$ explicitly; $Q$ is then passed to nag_dhseqr (f08pec), which must be called with ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$.
The function can also take advantage of a previous call to nag_dgebal (f08nhc) which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix $H$ has the structure:
 $H11 H12 H13 H22 H23 H33$
where ${H}_{11}$ and ${H}_{33}$ are upper triangular. If so, only the central diagonal block ${H}_{22}$ (in rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$) needs to be further reduced to Schur form (the blocks ${H}_{12}$ and ${H}_{23}$ are also affected). Therefore the values of ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ can be supplied to nag_dhseqr (f08pec) directly. Also, nag_dgebak (f08njc) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If nag_dgebal (f08nhc) has not been called however, then ${i}_{\mathrm{lo}}$ must be set to $1$ and ${i}_{\mathrm{hi}}$ to $n$. Note that if the Schur factorization of $A$ is required, nag_dgebal (f08nhc) must not be called with ${\mathbf{job}}=\mathrm{Nag_Scale}$ or $\mathrm{Nag_DoBoth}$, because the balancing transformation is not orthogonal.
nag_dhseqr (f08pec) uses a multishift form of the upper Hessenberg $QR$ algorithm, due to Bai and Demmel (1989). The Schur vectors are normalized so that ${‖{z}_{i}‖}_{2}=1$, but are determined only to within a factor $±1$.

## 4References

Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift $QR$ iteration Internat. J. High Speed Comput. 1 97–112
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

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 3.3.1.3 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:    $\mathbf{job}$Nag_JobTypeInput
On entry: indicates whether eigenvalues only or the Schur form $T$ is required.
${\mathbf{job}}=\mathrm{Nag_EigVals}$
Eigenvalues only are required.
${\mathbf{job}}=\mathrm{Nag_Schur}$
The Schur form $T$ is required.
Constraint: ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_Schur}$.
3:    $\mathbf{compz}$Nag_ComputeZTypeInput
On entry: indicates whether the Schur vectors are to be computed.
${\mathbf{compz}}=\mathrm{Nag_NotZ}$
No Schur vectors are computed (and the array z is not referenced).
${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$
The Schur vectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
${\mathbf{compz}}=\mathrm{Nag_InitZ}$
The Schur vectors of $H$ are computed (and the array z is initialized by the function).
Constraint: ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, $\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $H$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{ilo}$IntegerInput
6:    $\mathbf{ihi}$IntegerInput
On entry: if the matrix $A$ has been balanced by nag_dgebal (f08nhc), ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to $1$ and ihi to n.
Constraint: ${\mathbf{ilo}}\ge 1$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ilo}},{\mathbf{n}}\right)\le {\mathbf{ihi}}\le {\mathbf{n}}$.
7:    $\mathbf{h}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array h must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdh}}×{\mathbf{n}}\right)$.
Where ${\mathbf{H}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{h}}\left[\left(j-1\right)×{\mathbf{pdh}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{h}}\left[\left(i-1\right)×{\mathbf{pdh}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ upper Hessenberg matrix $H$, as returned by nag_dgehrd (f08nec).
On exit: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$, the array contains no useful information.
If ${\mathbf{job}}=\mathrm{Nag_Schur}$, h is overwritten by the upper quasi-triangular matrix $T$ from the Schur decomposition (the Schur form) unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE.
8:    $\mathbf{pdh}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array h.
Constraint: ${\mathbf{pdh}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:    $\mathbf{wr}\left[\mathit{dim}\right]$doubleOutput
10:  $\mathbf{wi}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the arrays wr and wi must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the real and imaginary parts, respectively, of the computed eigenvalues, unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE (in which case see Section 6). Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first. The eigenvalues are stored in the same order as on the diagonal of the Schur form $T$ (if computed); see Section 9 for details.
11:  $\mathbf{z}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array z must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdz}}×{\mathbf{n}}\right)$ when ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$;
• $1$ when ${\mathbf{compz}}=\mathrm{Nag_NotZ}$.
The $\left(i,j\right)$th element of the matrix $Z$ is stored in
• ${\mathbf{z}}\left[\left(j-1\right)×{\mathbf{pdz}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{z}}\left[\left(i-1\right)×{\mathbf{pdz}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$, z must contain the orthogonal matrix $Q$ from the reduction to Hessenberg form.
If ${\mathbf{compz}}=\mathrm{Nag_InitZ}$, z need not be set.
On exit: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, z contains the orthogonal matrix of the required Schur vectors, unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE.
If ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, z is not referenced.
12:  $\mathbf{pdz}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$.
13:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
The algorithm has failed to find all the eigenvalues after a total of $30\left({\mathbf{ihi}}-{\mathbf{ilo}}+1\right)$ iterations.
NE_ENUM_INT_2
On entry, ${\mathbf{compz}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdh}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdh}}>0$.
On entry, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdz}}>0$.
NE_INT_2
On entry, ${\mathbf{pdh}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdh}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT_3
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{ilo}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ihi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ilo}}\ge 1$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ilo}},{\mathbf{n}}\right)\le {\mathbf{ihi}}\le {\mathbf{n}}$.
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 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.

## 7Accuracy

The computed Schur factorization is the exact factorization of a nearby matrix $\left(H+E\right)$, where
 $E2 = Oε H2 ,$
and $\epsilon$ is the machine precision.
If ${\lambda }_{i}$ is an exact eigenvalue, and ${\stackrel{~}{\lambda }}_{i}$ is the corresponding computed value, then
 $λ~i - λi ≤ c n ε H2 si ,$
where $c\left(n\right)$ is a modestly increasing function of $n$, and ${s}_{i}$ is the reciprocal condition number of ${\lambda }_{i}$. The condition numbers ${s}_{i}$ may be computed by calling nag_dtrsna (f08qlc).

## 8Parallelism and Performance

nag_dhseqr (f08pec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dhseqr (f08pec) 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 depends on how rapidly the algorithm converges, but is typically about:
• $7{n}^{3}$ if only eigenvalues are computed;
• $10{n}^{3}$ if the Schur form is computed;
• $20{n}^{3}$ if the full Schur factorization is computed.
The Schur form $T$ has the following structure (referred to as canonical Schur form).
If all the computed eigenvalues are real, $T$ is upper triangular, and the diagonal elements of $T$ are the eigenvalues; ${\mathbf{wr}}\left[\mathit{i}-1\right]={t}_{\mathit{i}\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, and ${\mathbf{wi}}\left[i-1\right]=0.0$.
If some of the computed eigenvalues form complex conjugate pairs, then $T$ has $2$ by $2$ diagonal blocks. Each diagonal block has the form
 $tii ti,i+1 ti+1,i ti+1,i+1 = α β γ α$
where $\beta \gamma <0$. The corresponding eigenvalues are $\alpha ±\sqrt{\beta \gamma }$; ${\mathbf{wr}}\left[i-1\right]={\mathbf{wr}}\left[i\right]=\alpha$; ${\mathbf{wi}}\left[i-1\right]=+\sqrt{\left|\beta \gamma \right|}$; ${\mathbf{wi}}\left[i\right]=-{\mathbf{wi}}\left[i-1\right]$.
The complex analogue of this function is nag_zhseqr (f08psc).

## 10Example

This example computes all the eigenvalues and the Schur factorization of the upper Hessenberg matrix $H$, where
 $H = 0.3500 -0.1160 -0.3886 -0.2942 -0.5140 0.1225 0.1004 0.1126 0.0000 0.6443 -0.1357 -0.0977 0.0000 0.0000 0.4262 0.1632 .$
See also Section 10 in nag_dorghr (f08nfc), which illustrates the use of this function to compute the Schur factorization of a general matrix.

### 10.1Program Text

Program Text (f08pece.c)

### 10.2Program Data

Program Data (f08pece.d)

### 10.3Program Results

Program Results (f08pece.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017