f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dhseqr (f08pec)

## 1  Purpose

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.

## 2  Specification

 #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)

## 3  Description

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 8 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 ${\mathbf{job}}=\mathrm{Nag_Schur}$ 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 $±1$.

## 4  References

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

## 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:     jobNag_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:     compzNag_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).
${\mathbf{compz}}=\mathrm{Nag_InitZ}$
The Schur vectors of $H$ are computed (and the array z is initialized by the function).
${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$
The Schur vectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
Constraint: ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, $\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$.
4:     nIntegerInput
On entry: $n$, the order of the matrix $H$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     iloIntegerInput
6:     ihiIntegerInput
On entry: if the matrix $A$ has been balanced by nag_dgebal (f08nhc), then 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:     h[$\mathit{dim}$]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 NE_CONVERGENCE.
8:     pdhIntegerInput
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:     wr[$\mathit{dim}$]doubleOutput
10:   wi[$\mathit{dim}$]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 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 8 for details.
11:   z[$\mathit{dim}$]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 $\mathrm{Nag_InitZ}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,×{\mathbf{pdz}}\right)$ when ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ 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 NE_CONVERGENCE.
If ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, z is not referenced.
12:   pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{compz}}=\mathrm{Nag_InitZ}$ or $\mathrm{Nag_UpdateZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• 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:   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 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 ${\mathbf{compz}}=\mathrm{Nag_InitZ}$ or $\mathrm{Nag_UpdateZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$.
On entry, ${\mathbf{compz}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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.

## 7  Accuracy

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).

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).

## 9  Example

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 9 in nag_dorghr (f08nfc), which illustrates the use of this function to compute the Schur factorization of a general matrix.

### 9.1  Program Text

Program Text (f08pece.c)

### 9.2  Program Data

Program Data (f08pece.d)

### 9.3  Program Results

Program Results (f08pece.r)