# NAG CL Interfacef08jyc (zstegr)

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## 1Purpose

f08jyc computes selected eigenvalues and, optionally, the corresponding eigenvectors of a real $n×n$ symmetric tridiagonal matrix.

## 2Specification

 #include
 void f08jyc (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Integer n, double d[], double e[], double vl, double vu, Integer il, Integer iu, Integer *m, double w[], Complex z[], Integer pdz, Integer isuppz[], NagError *fail)
The function may be called by the names: f08jyc, nag_lapackeig_zstegr or nag_zstegr.

## 3Description

f08jyc computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix $T$. That is, the function computes the (partial) spectral factorization of $T$ given by
 $ZΛZT ,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the selected eigenvalues, ${\lambda }_{i}$, of $T$ and $Z$ is an orthogonal matrix whose columns are the corresponding eigenvectors, ${z}_{i}$, of $T$. Thus
 $Tzi= λi zi , i = 1,2,…,m$
where $m$ is the number of selected eigenvectors computed.
The function stores the real orthogonal matrix $Z$ in a complex array, so that it may also be used to compute selected eigenvalues and the corresponding eigenvectors of a complex Hermitian matrix $A$ which has been reduced to tridiagonal form $T$:
 $(QZ)Λ(QZ)H, where ​Q​ is unitary.$
In this case, the matrix $Q$ must be explicitly applied to the output matrix $Z$. The functions which must be called to perform the reduction to tridiagonal form and apply $Q$ are:
 full matrix f08fsc and f08fuc full matrix, packed storage f08gsc and f08guc band matrix f08hsc with ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ and f16zac.
This function uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. f08jyc can usually compute all the eigenvalues and eigenvectors in $O\left({n}^{2}\right)$ floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal functions in this chapter when all the eigenvectors are required, particularly so compared to those functions that are based on the $QR$ algorithm.

## 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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps SIAM J. Appl. Math. 25 858–899
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

## 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: $\mathbf{job}$Nag_JobType Input
On entry: indicates whether eigenvectors are computed.
${\mathbf{job}}=\mathrm{Nag_EigVals}$
Only eigenvalues are computed.
${\mathbf{job}}=\mathrm{Nag_DoBoth}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$.
3: $\mathbf{range}$Nag_RangeType Input
On entry: indicates which eigenvalues should be returned.
${\mathbf{range}}=\mathrm{Nag_AllValues}$
All eigenvalues will be found.
${\mathbf{range}}=\mathrm{Nag_Interval}$
All eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
${\mathbf{range}}=\mathrm{Nag_Indices}$
The ilth through iuth eigenvectors will be found.
Constraint: ${\mathbf{range}}=\mathrm{Nag_AllValues}$, $\mathrm{Nag_Interval}$ or $\mathrm{Nag_Indices}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{d}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ diagonal elements of the tridiagonal matrix $T$.
On exit: d is overwritten.
6: $\mathbf{e}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: ${\mathbf{e}}\left[0\right]$ to ${\mathbf{e}}\left[{\mathbf{n}}-2\right]$ are the subdiagonal elements of the tridiagonal matrix $T$. ${\mathbf{e}}\left[{\mathbf{n}}-1\right]$ need not be set.
On exit: e is overwritten.
7: $\mathbf{vl}$double Input
8: $\mathbf{vu}$double Input
On entry: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, vl and vu contain the lower and upper bounds respectively of the interval to be searched for eigenvalues.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Indices}$, vl and vu are not referenced.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
9: $\mathbf{il}$Integer Input
10: $\mathbf{iu}$Integer Input
On entry: if ${\mathbf{range}}=\mathrm{Nag_Indices}$, il and iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Interval}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$;
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$.
11: $\mathbf{m}$Integer * Output
On exit: the total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$, ${\mathbf{m}}={\mathbf{n}}$.
If ${\mathbf{range}}=\mathrm{Nag_Indices}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
12: $\mathbf{w}\left[\mathit{dim}\right]$double 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 exit: the eigenvalues in ascending order.
13: $\mathbf{z}\left[\mathit{dim}\right]$Complex 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 ${\mathbf{job}}=\mathrm{Nag_DoBoth}$;
• $1$ otherwise.
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 exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, then if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the columns of z contain the orthonormal eigenvectors of the matrix $T$, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{w}}\left[i-1\right]$.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, z is not referenced.
14: $\mathbf{pdz}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdz}}\ge 1$.
15: $\mathbf{isuppz}\left[\mathit{dim}\right]$Integer Output
Note: the dimension, dim, of the array isuppz must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{m}}\right)$.
On exit: the support of the eigenvectors in $Z$, i.e., the indices indicating the nonzero elements in $Z$. The $i$th eigenvector is nonzero only in elements ${\mathbf{isuppz}}\left[2×i-2\right]$ through ${\mathbf{isuppz}}\left[2×i-1\right]$.
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
Inverse iteration failed to converge.
The $\mathrm{dqds}$ algorithm failed to converge.
NE_ENUM_INT_2
On entry, ${\mathbf{job}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdz}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdz}}\ge 1$.
NE_ENUM_INT_3
On entry, ${\mathbf{range}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{il}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{iu}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$;
if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$.
NE_ENUM_REAL_2
On entry, ${\mathbf{range}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{vl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{vu}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdz}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdz}}>0$.
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

See Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.

## 8Parallelism and Performance

f08jyc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jyc 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 required to compute all the eigenvalues and eigenvectors is approximately proportional to ${n}^{2}$.
The real analogue of this function is f08jlc.

## 10Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
 $T = ( 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 ) .$

### 10.1Program Text

Program Text (f08jyce.c)

### 10.2Program Data

Program Data (f08jyce.d)

### 10.3Program Results

Program Results (f08jyce.r)