f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_zstegr (f08jyc)

1  Purpose

nag_zstegr (f08jyc) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix.

2  Specification

 #include #include
 void nag_zstegr (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)

3  Description

nag_zstegr (f08jyc) computes all the eigenvalues and, optionally, the eigenvectors, of a real symmetric tridiagonal matrix $T$. That is, the function computes the spectral factorization of $T$ given by
 $T = ZΛZT ,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues, ${\lambda }_{i}$, of $T$ and $Z$ is an orthogonal matrix whose columns are the eigenvectors, ${z}_{i}$, of $T$. Thus
 $Tzi= λi zi , i = 1,2,…,n .$
The function stores the real orthogonal matrix $Z$ in a complex array, so that it may also be used to compute all the eigenvalues and eigenvectors of a complex Hermitian matrix $A$ which has been reduced to tridiagonal form $T$:
 $A =QTQH, where ​Q​ is unitary =QZΛQZH.$
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 nag_zhetrd (f08fsc) and nag_zunmtr (f08fuc) full matrix, packed storage nag_zhptrd (f08gsc) and nag_zupmtr (f08guc) band matrix nag_zhbtrd (f08hsc) with ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ and nag_zgemm (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. nag_zstegr (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.

4  References

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 http://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

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 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:     rangeNag_RangeTypeInput
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:     nIntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     d[$\mathit{dim}$]doubleInput/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:     e[$\mathit{dim}$]doubleInput/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:     vldoubleInput
8:     vudoubleInput
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:     ilIntegerInput
10:   iuIntegerInput
On entry: if ${\mathbf{range}}=\mathrm{Nag_Indices}$, il and iu contains 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:   mInteger *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:   w[$\mathit{dim}$]doubleOutput
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:   z[$\mathit{dim}$]ComplexOutput
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 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:   pdzIntegerInput
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:   isuppz[$\mathit{dim}$]IntegerOutput
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:   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
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.

7  Accuracy

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

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 nag_dstegr (f08jlc).

9  Example

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

9.1  Program Text

Program Text (f08jyce.c)

9.2  Program Data

Program Data (f08jyce.d)

9.3  Program Results

Program Results (f08jyce.r)