# NAG Library Routine Document

## 1Purpose

f08fef (dsytrd) reduces a real symmetric matrix to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08fef ( uplo, n, a, lda, d, e, tau, work, info)
 Integer, Intent (In) :: n, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), d(*), e(*), tau(*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: uplo
C Header Interface
#include nagmk26.h
 void f08fef_ ( const char *uplo, const Integer *n, double a[], const Integer *lda, double d[], double e[], double tau[], double work[], const Integer *lwork, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dsytrd.

## 3Description

f08fef (dsytrd) reduces a real symmetric matrix $A$ 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).

## 4References

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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ symmetric matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: a is overwritten by the tridiagonal matrix $T$ and details of the orthogonal matrix $Q$ as specified by uplo.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08fef (dsytrd) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the diagonal elements of the tridiagonal matrix $T$.
6:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
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 exit: the off-diagonal elements of the tridiagonal matrix $T$.
7:     $\mathbf{tau}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On exit: further details of the orthogonal matrix $Q$.
8:     $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
9:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08fef (dsytrd) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, ${\mathbf{lwork}}\ge {\mathbf{n}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge 1$ or ${\mathbf{lwork}}=-1$.
10:   $\mathbf{info}$ – 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
 $E2≤ c n ε A2 ,$
$c\left(n\right)$ 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

f08fef (dsytrd) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fef (dsytrd) 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 approximately $\frac{4}{3}{n}^{3}$.
To form the orthogonal matrix $Q$ f08fef (dsytrd) may be followed by a call to f08fff (dorgtr):
```Call DORGTR(uplo,n,a,lda,tau,work,lwork,info)
```
To apply $Q$ to an $n$ by $p$ real matrix $C$ f08fef (dsytrd) may be followed by a call to f08fgf (dormtr). For example,
```Call DORMTR('Left',uplo,'No Transpose',n,p,a,lda,tau,c,ldc, &
work,lwork,info)```
forms the matrix product $QC$.
The complex analogue of this routine is f08fsf (zhetrd).

## 10Example

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

### 10.1Program Text

Program Text (f08fefe.f90)

### 10.2Program Data

Program Data (f08fefe.d)

### 10.3Program Results

Program Results (f08fefe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017