# NAG CL Interfacef08jgc (dpteqr)

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## 1Purpose

f08jgc 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

 #include
 void f08jgc (Nag_OrderType order, Nag_ComputeZType compz, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)
The function may be called by the names: f08jgc, nag_lapackeig_dpteqr or nag_dpteqr.

## 3Description

f08jgc 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 function 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 f08jgc, which must be called with ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$. The functions which must be called to perform the reduction to tridiagonal form and form $Q$ are:
 full matrix f08fec and f08ffc full matrix, packed storage f08gec and f08gfc band matrix f08hec with ${\mathbf{vect}}=\mathrm{Nag_FormQ}$.
f08jgc 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 f08mec to compute the singular values of $B$ which are the same as the eigenvalues of $T$. The method used by the function 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: $\mathbf{order}$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: $\mathbf{compz}$Nag_ComputeZType Input
On entry: indicates whether the eigenvectors are to be computed.
${\mathbf{compz}}=\mathrm{Nag_NotZ}$
Only the eigenvalues are computed (and the array z is not referenced).
${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$
The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
${\mathbf{compz}}=\mathrm{Nag_InitZ}$
The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the function).
Constraint: ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, $\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{d}\left[\mathit{dim}\right]$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 diagonal elements of the tridiagonal matrix $T$.
On exit: the $n$ eigenvalues in descending order, unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE or NE_POS_DEF, in which case d is overwritten.
5: $\mathbf{e}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the off-diagonal elements of the tridiagonal matrix $T$.
On exit: e is overwritten.
6: $\mathbf{z}\left[\mathit{dim}\right]$double Input/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 ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$;
• $1$ when ${\mathbf{compz}}=\mathrm{Nag_NotZ}$.
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 entry: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$, z must contain the orthogonal matrix $Q$ from the reduction to tridiagonal form.
If ${\mathbf{compz}}=\mathrm{Nag_InitZ}$, z need not be set.
On exit: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE or NE_POS_DEF.
If ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, z is not referenced.
7: $\mathbf{pdz}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$.
8: $\mathbf{fail}$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
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.
NE_ENUM_INT_2
On entry, ${\mathbf{compz}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdz}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$.
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.
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.
NE_POS_DEF
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.

## 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 function) 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

f08jgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jgc 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 is typically about $30{n}^{2}$ if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$ and about $6{n}^{3}$ if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, but depends on how rapidly the algorithm converges. When ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$ can be vectorized and on some machines may be performed much faster.
The complex analogue of this function is f08juc.

## 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 (f08jgce.c)

### 10.2Program Data

Program Data (f08jgce.d)

### 10.3Program Results

Program Results (f08jgce.r)