# NAG Library Routine Document

## 1Purpose

f08gef (dsptrd) reduces a real symmetric matrix to tridiagonal form, using packed storage.

## 2Specification

Fortran Interface
 Subroutine f08gef ( uplo, n, ap, d, e, tau, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: ap(*) Real (Kind=nag_wp), Intent (Out) :: d(n), e(n-1), tau(n-1) Character (1), Intent (In) :: uplo
#include <nagmk26.h>
 void f08gef_ (const char *uplo, const Integer *n, double ap[], double d[], double e[], double tau[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dsptrd.

## 3Description

f08gef (dsptrd) reduces a real symmetric matrix $A$, held in packed storage, to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$.
The matrix $Q$ is not formed explicitly but is represented as a product of $n-1$ elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with $Q$ in this representation (see Section 9).
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the upper or lower triangle of the $n$ by $n$ symmetric matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
On exit: ap is overwritten by the tridiagonal matrix $T$ and details of the orthogonal matrix $Q$.
4:     $\mathbf{d}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the diagonal elements of the tridiagonal matrix $T$.
5:     $\mathbf{e}\left({\mathbf{n}}-1\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the off-diagonal elements of the tridiagonal matrix $T$.
6:     $\mathbf{tau}\left({\mathbf{n}}-1\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: further details of the orthogonal matrix $Q$.
7:     $\mathbf{info}$ – IntegerOutput
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.

## 7Accuracy

The computed tridiagonal matrix $T$ is exactly similar to a nearby matrix $\left(A+E\right)$, where
 $E2≤ cn ε A2 ,$
$c\left(n\right)$ is a modestly increasing function of $n$, and $\epsilon$ is the machine precision.
The elements of $T$ themselves may be sensitive to small perturbations in $A$ or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.

## 8Parallelism and Performance

f08gef (dsptrd) 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.

The total number of floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
To form the orthogonal matrix $Q$ f08gef (dsptrd) may be followed by a call to f08gff (dopgtr):
```Call dopgtr(uplo,n,ap,tau,q,ldq,work,info)
```
To apply $Q$ to an $n$ by $p$ real matrix $C$ f08gef (dsptrd) may be followed by a call to f08ggf (dopmtr). For example,
```Call dopmtr('Left',uplo,'No Transpose',n,p,ap,tau,c,ldc,work, &
info)```
forms the matrix product $QC$.
The complex analogue of this routine is f08gsf (zhptrd).

## 10Example

This example reduces the matrix $A$ to tridiagonal form, where
 $A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ,$
using packed storage.

### 10.1Program Text

Program Text (f08gefe.f90)

### 10.2Program Data

Program Data (f08gefe.d)

### 10.3Program Results

Program Results (f08gefe.r)