# NAG FL Interfacef08jgf (dpteqr)

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## 1Purpose

f08jgf computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix, or of a real symmetric positive definite matrix which has been reduced to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08jgf ( n, d, e, z, ldz, work, info)
 Integer, Intent (In) :: n, ldz Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), z(ldz,*) Real (Kind=nag_wp), Intent (Out) :: work(4*n) Character (1), Intent (In) :: compz
C Header Interface
#include <nag.h>
 void f08jgf_ (const char *compz, const Integer *n, double d[], double e[], double z[], const Integer *ldz, double work[], Integer *info, const Charlen length_compz)
The routine may be called by the names f08jgf, nagf_lapackeig_dpteqr or its LAPACK name dpteqr.

## 3Description

f08jgf computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix $T$. In other words, it can compute the spectral factorization of $T$ as
 $T=ZΛZT,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues ${\lambda }_{i}$, and $Z$ is the orthogonal matrix whose columns are the eigenvectors ${z}_{i}$. Thus
 $Tzi=λizi, i=1,2,…,n.$
The routine may also be used to compute all the eigenvalues and eigenvectors of a real symmetric positive definite matrix $A$ which has been reduced to tridiagonal form $T$:
 $A =QTQT, where ​Q​ is orthogonal =(QZ)Λ(QZ)T.$
In this case, the matrix $Q$ must be formed explicitly and passed to f08jgf, which must be called with ${\mathbf{compz}}=\text{'V'}$. The routines which must be called to perform the reduction to tridiagonal form and form $Q$ are:
 full matrix f08fef and f08fff full matrix, packed storage f08gef and f08gff band matrix f08hef with ${\mathbf{vect}}=\text{'V'}$.
f08jgf first factorizes $T$ as $LD{L}^{\mathrm{T}}$ where $L$ is unit lower bidiagonal and $D$ is diagonal. It forms the bidiagonal matrix $B=L{D}^{\frac{1}{2}}$, and then calls f08mef to compute the singular values of $B$ which are the same as the eigenvalues of $T$. The method used by the routine allows high relative accuracy to be achieved in the small eigenvalues of $T$. The eigenvectors are normalized so that ${‖{z}_{i}‖}_{2}=1$, but are determined only to within a factor $±1$.

## 4References

Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791

## 5Arguments

1: $\mathbf{compz}$Character(1) Input
On entry: indicates whether the eigenvectors are to be computed.
${\mathbf{compz}}=\text{'N'}$
Only the eigenvalues are computed (and the array z is not referenced).
${\mathbf{compz}}=\text{'V'}$
The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
${\mathbf{compz}}=\text{'I'}$
The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the routine).
Constraint: ${\mathbf{compz}}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{d}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension 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 $T$.
On exit: the $n$ eigenvalues in descending order, unless ${\mathbf{info}}>{\mathbf{0}}$, in which case d is overwritten.
4: $\mathbf{e}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the off-diagonal elements of the tridiagonal matrix $T$.
On exit: e is overwritten.
5: $\mathbf{z}\left({\mathbf{ldz}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$ and at least $1$ if ${\mathbf{compz}}=\text{'N'}$.
On entry: if ${\mathbf{compz}}=\text{'V'}$, z must contain the orthogonal matrix $Q$ from the reduction to tridiagonal form.
If ${\mathbf{compz}}=\text{'I'}$, z need not be set.
On exit: if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, the $n$ required orthonormal eigenvectors stored as columns of $Z$; the $i$th column corresponds to the $i$th eigenvalue, where $i=1,2,\dots ,n$, unless ${\mathbf{info}}>{\mathbf{0}}$.
If ${\mathbf{compz}}=\text{'N'}$, z is not referenced.
6: $\mathbf{ldz}$Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08jgf is called.
Constraints:
• if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compz}}=\text{'N'}$, ${\mathbf{ldz}}\ge 1$.
7: $\mathbf{work}\left(4×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
8: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=1,\dots ,{\mathbf{n}}$
The leading minor of order $⟨\mathit{\text{value}}⟩$ is not positive definite and the Cholesky factorization of $T$ could not be completed. Hence $T$ itself is not positive definite.
${\mathbf{info}}>{\mathbf{n}}$
The algorithm to compute the singular values of the Cholesky factor $B$ failed to converge; $⟨\mathit{\text{value}}⟩$ off-diagonal elements did not converge to zero.

## 7Accuracy

The eigenvalues and eigenvectors of $T$ are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues (and corresponding eigenvectors) will be computed more accurately than, for example, with the standard $QR$ method. However, the reduction to tridiagonal form (prior to calling the routine) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.
To be more precise, let $H$ be the tridiagonal matrix defined by $H=DTD$, where $D$ is diagonal with ${d}_{ii}={t}_{ii}^{-\frac{1}{2}}$, and ${h}_{ii}=1$ for all $i$. If ${\lambda }_{i}$ is an exact eigenvalue of $T$ and ${\stackrel{~}{\lambda }}_{i}$ is the corresponding computed value, then
 $|λ~i-λi| ≤ c (n) ε κ2 (H) λi$
where $c\left(n\right)$ is a modestly increasing function of $n$, $\epsilon$ is the machine precision, and ${\kappa }_{2}\left(H\right)$ is the condition number of $H$ with respect to inversion defined by: ${\kappa }_{2}\left(H\right)=‖H‖·‖{H}^{-1}‖$.
If ${z}_{i}$ is the corresponding exact eigenvector of $T$, 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:
 $θ (z~i,zi) ≤ c (n) ε κ2 (H) relgapi$
where ${\mathit{relgap}}_{i}$ is the relative gap between ${\lambda }_{i}$ and the other eigenvalues, defined by
 $relgapi = min i≠j |λi-λj| (λi+λj) .$

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08jgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jgf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

The total number of floating-point operations is typically about $30{n}^{2}$ if ${\mathbf{compz}}=\text{'N'}$ and about $6{n}^{3}$ if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, but depends on how rapidly the algorithm converges. When ${\mathbf{compz}}=\text{'N'}$, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$ can be vectorized and on some machines may be performed much faster.
The complex analogue of this routine is f08juf.

## 10Example

This example computes all the eigenvalues and eigenvectors of the symmetric positive definite tridiagonal matrix $T$, where
 $T = ( 4.16 3.17 0.00 0.00 3.17 5.25 -0.97 0.00 0.00 -0.97 1.09 0.55 0.00 0.00 0.55 0.62 ) .$

### 10.1Program Text

Program Text (f08jgfe.f90)

### 10.2Program Data

Program Data (f08jgfe.d)

### 10.3Program Results

Program Results (f08jgfe.r)