nag_dsterf (f08jfc) (PDF version)
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NAG Library Manual

# NAG Library Function Documentnag_dsterf (f08jfc)

## 1  Purpose

nag_dsterf (f08jfc) computes all the eigenvalues of a real symmetric tridiagonal matrix.

## 2  Specification

 #include #include
 void nag_dsterf (Integer n, double d[], double e[], NagError *fail)

## 3  Description

nag_dsterf (f08jfc) computes all the eigenvalues of a real symmetric tridiagonal matrix, using a square-root-free variant of the $QR$ algorithm.
The function uses an explicit shift, and, like nag_dsteqr (f08jec), switches between the $QR$ and $QL$ variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)).

## 4  References

Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     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 $T$.
On exit: the $n$ eigenvalues in ascending order, unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE (in which case see Section 6).
3:     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 off-diagonal elements of the tridiagonal matrix $T$.
On exit: e is overwritten.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
The algorithm has failed to find all the eigenvalues after a total of $30×{\mathbf{n}}$ iterations; $⟨\mathit{\text{value}}⟩$ elements of e have not converged to zero.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
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 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 $n$.

Not applicable.

## 9  Further Comments

The total number of floating-point operations is typically about $14{n}^{2}$, but depends on how rapidly the algorithm converges. The operations are all performed in scalar mode.
There is no complex analogue of this function.

## 10  Example

This example computes all the eigenvalues of the symmetric tridiagonal matrix $T$, where
 $T = -6.99 -0.44 0.00 0.00 -0.44 7.92 -2.63 0.00 0.00 -2.63 2.34 -1.18 0.00 0.00 -1.18 0.32 .$

### 10.1  Program Text

Program Text (f08jfce.c)

### 10.2  Program Data

Program Data (f08jfce.d)

### 10.3  Program Results

Program Results (f08jfce.r)

nag_dsterf (f08jfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual