# NAG CL Interfacef08jac (dstev)

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## 1Purpose

f08jac computes all the eigenvalues and, optionally, all the eigenvectors of a real $n×n$ symmetric tridiagonal matrix $A$.

## 2Specification

 #include
 void f08jac (Nag_OrderType order, Nag_JobType job, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)
The function may be called by the names: f08jac, nag_lapackeig_dstev or nag_dstev.

## 3Description

f08jac computes all the eigenvalues and, optionally, all the eigenvectors of $A$ using a combination of the $QR$ and $QL$ algorithms, with an implicit shift.

## 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix.
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 $n$ diagonal elements of the tridiagonal matrix $A$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the eigenvalues in ascending order.
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 $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.
On exit: the contents of e are destroyed.
6: $\mathbf{z}\left[\mathit{dim}\right]$double 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, z contains the orthonormal eigenvectors of the matrix $A$, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{d}}\left[i-1\right]$.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, 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 ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdz}}\ge 1$.
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 failed to converge; $⟨\mathit{\text{value}}⟩$ off-diagonal elements of e did not converge to zero.
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_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

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $‖E‖2 = O(ε) ‖A‖2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08jac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jac 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 proportional to ${n}^{2}$ if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ and is proportional to ${n}^{3}$ if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$.

## 10Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
 $A = ( 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 ) ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.

### 10.1Program Text

Program Text (f08jace.c)

### 10.2Program Data

Program Data (f08jace.d)

### 10.3Program Results

Program Results (f08jace.r)