# NAG CL Interfacef08pxc (zhsein)

Settings help

CL Name Style:

## 1Purpose

f08pxc computes selected left and/or right eigenvectors of a complex upper Hessenberg matrix corresponding to specified eigenvalues, by inverse iteration.

## 2Specification

 #include
 void f08pxc (Nag_OrderType order, Nag_SideType side, Nag_EigValsSourceType eig_source, Nag_InitVeenumtype initv, const Nag_Boolean select[], Integer n, const Complex h[], Integer pdh, Complex w[], Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, Integer mm, Integer *m, Integer ifaill[], Integer ifailr[], NagError *fail)
The function may be called by the names: f08pxc, nag_lapackeig_zhsein or nag_zhsein.

## 3Description

f08pxc computes left and/or right eigenvectors of a complex upper Hessenberg matrix $H$, corresponding to selected eigenvalues.
The right eigenvector $x$, and the left eigenvector $y$, corresponding to an eigenvalue $\lambda$, are defined by:
The eigenvectors are computed by inverse iteration. They are scaled so that $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(|\mathrm{Re}\left({x}_{i}\right)|+|\mathrm{Im}{x}_{i}|\right)=1$.
If $H$ has been formed by reduction of a complex general matrix $A$ to upper Hessenberg form, then the eigenvectors of $H$ may be transformed to eigenvectors of $A$ by a call to f08nuc.

## 4References

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: $\mathbf{side}$Nag_SideType Input
On entry: indicates whether left and/or right eigenvectors are to be computed.
${\mathbf{side}}=\mathrm{Nag_RightSide}$
Only right eigenvectors are computed.
${\mathbf{side}}=\mathrm{Nag_LeftSide}$
Only left eigenvectors are computed.
${\mathbf{side}}=\mathrm{Nag_BothSides}$
Both left and right eigenvectors are computed.
Constraint: ${\mathbf{side}}=\mathrm{Nag_RightSide}$, $\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$.
3: $\mathbf{eig_source}$Nag_EigValsSourceType Input
On entry: indicates whether the eigenvalues of $H$ (stored in w) were found using f08psc.
${\mathbf{eig_source}}=\mathrm{Nag_HSEQRSource}$
The eigenvalues of $H$ were found using f08psc; thus if $H$ has any zero subdiagonal elements (and so is block triangular), then the $j$th eigenvalue can be assumed to be an eigenvalue of the block containing the $j$th row/column. This property allows the function to perform inverse iteration on just one diagonal block.
${\mathbf{eig_source}}=\mathrm{Nag_NotKnown}$
No such assumption is made and the function performs inverse iteration using the whole matrix.
Constraint: ${\mathbf{eig_source}}=\mathrm{Nag_HSEQRSource}$ or $\mathrm{Nag_NotKnown}$.
4: $\mathbf{initv}$Nag_InitVeenumtype Input
On entry: indicates whether you are supplying initial estimates for the selected eigenvectors.
${\mathbf{initv}}=\mathrm{Nag_NoVec}$
No initial estimates are supplied.
${\mathbf{initv}}=\mathrm{Nag_UserVec}$
Initial estimates are supplied in vl and/or vr.
Constraint: ${\mathbf{initv}}=\mathrm{Nag_NoVec}$ or $\mathrm{Nag_UserVec}$.
5: $\mathbf{select}\left[\mathit{dim}\right]$const Nag_Boolean Input
Note: the dimension, dim, of the array select must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: specifies which eigenvectors are to be computed. To select the eigenvector corresponding to the eigenvalue ${\mathbf{w}}\left[j-1\right]$, ${\mathbf{select}}\left[j-1\right]$ must be set to Nag_TRUE.
6: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $H$.
Constraint: ${\mathbf{n}}\ge 0$.
7: $\mathbf{h}\left[\mathit{dim}\right]$const Complex Input
Note: the dimension, dim, of the array h must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdh}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $H$ is stored in
• ${\mathbf{h}}\left[\left(j-1\right)×{\mathbf{pdh}}+i-1\right]$ 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 $n×n$ upper Hessenberg matrix $H$. If a NaN is detected in h, the function will return with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_BAD_PARAM.
Constraint: No element of h is equal to NaN.
8: $\mathbf{pdh}$Integer Input
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: $\mathbf{w}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array w must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the eigenvalues of the matrix $H$. If ${\mathbf{eig_source}}=\mathrm{Nag_HSEQRSource}$, the array must be exactly as returned by f08psc.
On exit: the real parts of some elements of w may be modified, as close eigenvalues are perturbed slightly in searching for independent eigenvectors.
10: $\mathbf{vl}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array vl must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvl}}×{\mathbf{mm}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdvl}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vl}}\left[\left(j-1\right)×{\mathbf{pdvl}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vl}}\left[\left(i-1\right)×{\mathbf{pdvl}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{initv}}=\mathrm{Nag_UserVec}$ and ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, vl must contain starting vectors for inverse iteration for the left eigenvectors. Each starting vector must be stored in the same row or column as will be used to store the corresponding eigenvector (see below).
If ${\mathbf{initv}}=\mathrm{Nag_NoVec}$, vl need not be set.
On exit: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, vl contains the computed left eigenvectors (as specified by select). The eigenvectors are stored consecutively in the rows or columns of the array (depending on the value of order), in the same order as their eigenvalues.
If ${\mathbf{side}}=\mathrm{Nag_RightSide}$, vl is not referenced.
11: $\mathbf{pdvl}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvl}}\ge {\mathbf{n}}$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pdvl}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pdvl}}\ge 1$.
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pdvl}}\ge 1$.
12: $\mathbf{vr}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array vr must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvr}}×{\mathbf{mm}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdvr}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vr}}\left[\left(j-1\right)×{\mathbf{pdvr}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vr}}\left[\left(i-1\right)×{\mathbf{pdvr}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{initv}}=\mathrm{Nag_UserVec}$ and ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, vr must contain starting vectors for inverse iteration for the right eigenvectors. Each starting vector must be stored in the same row or column as will be used to store the corresponding eigenvector (see below).
If ${\mathbf{initv}}=\mathrm{Nag_NoVec}$, vr need not be set.
On exit: if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, vr contains the computed right eigenvectors (as specified by select). The eigenvectors are stored consecutively in the rows or columns of the array (depending on the value of order), in the same order as their eigenvalues.
If ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, vr is not referenced.
13: $\mathbf{pdvr}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvr}}\ge {\mathbf{n}}$;
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pdvr}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pdvr}}\ge 1$.
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pdvr}}\ge 1$.
14: $\mathbf{mm}$Integer Input
On entry: the number of columns in the arrays vl and/or vr if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ or the number of rows in the arrays if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. This must be an upper bound on the actual number of rows or columns required, that is, the number of elements of select, in the first n, that are set to Nag_TRUE.
Constraint: ${\mathbf{mm}}\ge \mathit{m}$.
15: $\mathbf{m}$Integer * Output
On exit: $\mathit{m}$, the number of selected eigenvectors.
16: $\mathbf{ifaill}\left[\mathit{dim}\right]$Integer Output
Note: the dimension, dim, of the array ifaill must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$;
• $1$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
On exit: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, then ${\mathbf{ifaill}}\left[i-1\right]=0$ if the selected left eigenvector converged and ${\mathbf{ifaill}}\left[i-1\right]=j\ge 0$ if the eigenvector stored in the $i$th row or column of vl (corresponding to the $j$th eigenvalue) failed to converge.
If ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ifaill is not referenced.
17: $\mathbf{ifailr}\left[\mathit{dim}\right]$Integer Output
Note: the dimension, dim, of the array ifailr must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$;
• $1$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$.
On exit: if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, then ${\mathbf{ifailr}}\left[i-1\right]=0$ if the selected right eigenvector converged and ${\mathbf{ifailr}}\left[i-1\right]=j\ge 0$ if the eigenvector stored in the $i$th column of vr (corresponding to the $j$th eigenvalue) failed to converge.
If ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ifailr is not referenced.
18: $\mathbf{fail}$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.
Constraint: No element of h is equal to NaN.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
$⟨\mathit{\text{value}}⟩$ eigenvectors (as indicated by arguments ifaill and/or ifailr) failed to converge. The corresponding columns of vl and/or vr contain no useful information.
NE_ENUM_INT_2
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pdvl}}\ge 1$.
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvl}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{mm}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$;
if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pdvl}}\ge 1$.
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvl}}\ge {\mathbf{n}}$;
if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{pdvl}}\ge 1$.
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvr}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pdvr}}\ge 1$.
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvr}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{mm}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$;
if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pdvr}}\ge 1$.
On entry, ${\mathbf{side}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvr}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_RightSide}$ or $\mathrm{Nag_BothSides}$, ${\mathbf{pdvr}}\ge {\mathbf{n}}$;
if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{pdvr}}\ge 1$.
NE_INT
On entry, ${\mathbf{mm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mm}}\ge \mathit{m}$.
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, ${\mathbf{pdvl}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdvl}}>0$.
On entry, ${\mathbf{pdvr}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdvr}}>0$.
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_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.

## 7Accuracy

Each computed right eigenvector ${x}_{i}$ is the exact eigenvector of a nearby matrix $A+{E}_{i}$, such that $‖{E}_{i}‖=\mathit{O}\left(\epsilon \right)‖A‖$. Hence the residual is small:
 $‖Axi-λixi‖ = O(ε) ‖A‖ .$
However, eigenvectors corresponding to close or coincident eigenvalues may not accurately span the relevant subspaces.
Similar remarks apply to computed left eigenvectors.

## 8Parallelism and Performance

f08pxc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pxc 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.