f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dsptrd (f08gec)

## 1  Purpose

nag_dsptrd (f08gec) reduces a real symmetric matrix to tridiagonal form, using packed storage.

## 2  Specification

 #include #include
 void nag_dsptrd (Nag_OrderType order, Nag_UploType uplo, Integer n, double ap[], double d[], double e[], double tau[], NagError *fail)

## 3  Description

nag_dsptrd (f08gec) 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). Functions are provided to work with $Q$ in this representation (see Section 9).

## 4  References

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

## 5  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{ap}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, 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 rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
On exit: ap is overwritten by the tridiagonal matrix $T$ and details of the orthogonal matrix $Q$.
5:    $\mathbf{d}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the diagonal elements of the tridiagonal matrix $T$.
6:    $\mathbf{e}\left[{\mathbf{n}}-1\right]$doubleOutput
On exit: the off-diagonal elements of the tridiagonal matrix $T$.
7:    $\mathbf{tau}\left[{\mathbf{n}}-1\right]$doubleOutput
On exit: further details of the orthogonal matrix $Q$.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
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.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 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.

## 8  Parallelism and Performance

nag_dsptrd (f08gec) is not threaded by NAG in any implementation.
nag_dsptrd (f08gec) 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 approximately $\frac{4}{3}{n}^{3}$.
To form the orthogonal matrix $Q$ nag_dsptrd (f08gec) may be followed by a call to nag_dopgtr (f08gfc):
```nag_dopgtr(order,uplo,n,ap,tau,&q,pdq,&fail)
```
To apply $Q$ to an $n$ by $p$ real matrix $C$ nag_dsptrd (f08gec) may be followed by a call to nag_dopmtr (f08ggc). For example,
```nag_dopmtr(order,Nag_LeftSide,uplo,Nag_NoTrans,n,p,ap,tau,&c,
pdc,&fail)```
forms the matrix product $QC$.
The complex analogue of this function is nag_zhptrd (f08gsc).

## 10  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.

### 10.1  Program Text

Program Text (f08gece.c)

### 10.2  Program Data

Program Data (f08gece.d)

### 10.3  Program Results

Program Results (f08gece.r)