f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zstedc (f08jvc)

## 1  Purpose

nag_zstedc (f08jvc) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix, or of a complex full or banded Hermitian matrix which has been reduced to tridiagonal form.

## 2  Specification

 #include #include
 void nag_zstedc (Nag_OrderType order, Nag_ComputeEigVecsType compz, Integer n, double d[], double e[], Complex z[], Integer pdz, NagError *fail)

## 3  Description

nag_zstedc (f08jvc) 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
 $T = Z Λ ZT ,$
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 may also be used to compute all the eigenvalues and eigenvectors of a complex full, or banded, Hermitian matrix $A$ which has been reduced to real tridiagonal form $T$ as
 $A = QTQH ,$
where $Q$ is unitary. The spectral factorization of $A$ is then given by
 $A = QZ Λ QZH .$
In this case $Q$ must be formed explicitly and passed to nag_zstedc (f08jvc) in the array z, and the function called with ${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$. Functions which may be called to form $T$ and $Q$ are
 full matrix nag_zhetrd (f08fsc) and nag_zungtr (f08ftc) full matrix, packed storage nag_zhptrd (f08gsc) and nag_zupgtr (f08gtc) band matrix nag_zhbtrd (f08hsc), with ${\mathbf{vect}}=\mathrm{Nag_FormQ}$
When only eigenvalues are required then this function calls nag_dsterf (f08jfc) to compute the eigenvalues of the tridiagonal matrix $T$, but when eigenvectors of $T$ are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than nag_zsteqr (f08jsc), although more storage is required.

## 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
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 $\mathrm{Nag_ColMajor}$.
2:     compzNag_ComputeEigVecsTypeInput
On entry: indicates whether the eigenvectors are to be computed.
${\mathbf{compz}}=\mathrm{Nag_NotEigVecs}$
Only the eigenvalues are computed (and the array z is not referenced).
${\mathbf{compz}}=\mathrm{Nag_TridiagEigVecs}$
The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the function).
${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$
The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
Constraint: ${\mathbf{compz}}=\mathrm{Nag_NotEigVecs}$, $\mathrm{Nag_OrigEigVecs}$ or $\mathrm{Nag_TridiagEigVecs}$.
3:     nIntegerInput
On entry: $n$, the order of the symmetric tridiagonal matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     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 diagonal elements of the tridiagonal matrix.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the eigenvalues in ascending order.
5:     e[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the subdiagonal elements of the tridiagonal matrix.
On exit: e is overwritten.
6:     z[$\mathit{dim}$]ComplexInput/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 ${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$ or $\mathrm{Nag_TridiagEigVecs}$;
• $1$ otherwise.
If ${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$ then the$\left(i,j\right)$th element of the matrix $Q$ 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 ${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$, z must contain the unitary matrix $Q$ used in the reduction to tridiagonal form.
On exit: if ${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$, z contains the orthonormal eigenvectors of the original Hermitian matrix $A$, and if ${\mathbf{compz}}=\mathrm{Nag_TridiagEigVecs}$, z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix $T$.
If ${\mathbf{compz}}=\mathrm{Nag_NotEigVecs}$, z is not referenced.
7:     pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$ or $\mathrm{Nag_TridiagEigVecs}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdz}}\ge 1$.
8:     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 failed to compute an eigenvalue while working on the submatrix lying in rows and columns $⟨\mathit{\text{value}}⟩/\left({\mathbf{n}}+1\right)$ through .
NE_ENUM_INT_2
On entry, ${\mathbf{compz}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdz}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{compz}}=\mathrm{Nag_OrigEigVecs}$ or $\mathrm{Nag_TridiagEigVecs}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdz}}\ge 1$.
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

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(T+E\right)$, where
 $E2 = Oε T2 ,$
and $\epsilon$ is the machine precision.
If ${\lambda }_{i}$ is an exact eigenvalue and ${\stackrel{~}{\lambda }}_{i}$ is the corresponding computed value, then
 $λ~i - λi ≤ c n ε T2 ,$
where $c\left(n\right)$ is a modestly increasing function of $n$.
If ${z}_{i}$ is the corresponding exact eigenvector, and ${\stackrel{~}{z}}_{i}$ is the corresponding computed eigenvector, then the angle $\theta \left({\stackrel{~}{z}}_{i},{z}_{i}\right)$ between them is bounded as follows:
 $θ z~i,zi ≤ cnεT2 mini≠jλi-λj .$
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
See Section 4.7 of Anderson et al. (1999) for further details. See also nag_ddisna (f08flc).

## 8  Parallelism and Performance

nag_zstedc (f08jvc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zstedc (f08jvc) 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.

If only eigenvalues are required, the total number of floating-point operations is approximately proportional to ${n}^{2}$. When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as nag_zsteqr (f08jsc), but for large matrices nag_zstedc (f08jvc) is usually much faster.
The real analogue of this function is nag_dstedc (f08jhc).

## 10  Example

This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
 $A = -3.13i+0.00 1.94-2.10i -3.40+0.25i 0.00i+0.00 1.94+2.10i -1.91i+0.00 -0.82-0.89i -0.67+0.34i -3.40-0.25i -0.82+0.89i -2.87i+0.00 -2.10-0.16i 0.00i+0.00 -0.67-0.34i -2.10+0.16i 0.50i+0.00 .$
$A$ is first reduced to tridiagonal form by a call to nag_zhbtrd (f08hsc).

### 10.1  Program Text

Program Text (f08jvce.c)

### 10.2  Program Data

Program Data (f08jvce.d)

### 10.3  Program Results

Program Results (f08jvce.r)