F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08GEF (DSPTRD)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08GEF (DSPTRD) reduces a real symmetric matrix to tridiagonal form, using packed storage.

## 2  Specification

 SUBROUTINE F08GEF ( UPLO, N, AP, D, E, TAU, INFO)
 INTEGER N, INFO REAL (KIND=nag_wp) AP(*), D(N), E(N-1), TAU(N-1) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dsptrd.

## 3  Description

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 8).

## 4  References

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

## 5  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     AP($*$) – 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:     D(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the diagonal elements of the tridiagonal matrix $T$.
5:     E(${\mathbf{N}}-1$) – REAL (KIND=nag_wp) arrayOutput
On exit: the off-diagonal elements of the tridiagonal matrix $T$.
6:     TAU(${\mathbf{N}}-1$) – REAL (KIND=nag_wp) arrayOutput
On exit: further details of the orthogonal matrix $Q$.
7:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

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.

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).

## 9  Example

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.

### 9.1  Program Text

Program Text (f08gefe.f90)

### 9.2  Program Data

Program Data (f08gefe.d)

### 9.3  Program Results

Program Results (f08gefe.r)