nag_dsptrd (f08gec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dsptrd (f08gec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagf08.h>
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=QTQT.
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 8).

4  References

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

5  Arguments

1:     orderNag_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 order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored.
uplo=Nag_Upper
The upper triangular part of A is stored.
uplo=Nag_Lower
The lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     ap[dim]doubleInput/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the upper or lower triangle of the n by n symmetric matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
On exit: ap is overwritten by the tridiagonal matrix T and details of the orthogonal matrix Q.
5:     d[n]doubleOutput
On exit: the diagonal elements of the tridiagonal matrix T.
6:     e[n-1]doubleOutput
On exit: the off-diagonal elements of the tridiagonal matrix T.
7:     tau[n-1]doubleOutput
On exit: further details of the orthogonal matrix Q.
8:     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 value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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 tridiagonal matrix T is exactly similar to a nearby matrix A+E, where
E2 cn ε A2 ,
cn is a modestly increasing function of n, and ε 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  Further Comments

The total number of floating point operations is approximately 43 n3 .
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).

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 (f08gece.c)

9.2  Program Data

Program Data (f08gece.d)

9.3  Program Results

Program Results (f08gece.r)


nag_dsptrd (f08gec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012