nag_dsytrd (f08fec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dsytrd (f08fec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsytrd (f08fec) reduces a real symmetric matrix to tridiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dsytrd (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, double d[], double e[], double tau[], NagError *fail)

3  Description

nag_dsytrd (f08fec) reduces a real symmetric matrix A 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 9).

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:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n symmetric matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, 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.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
6:     d[dim]doubleOutput
Note: the dimension, dim, of the array d must be at least max1,n.
On exit: the diagonal elements of the tridiagonal matrix T.
7:     e[dim]doubleOutput
Note: the dimension, dim, of the array e must be at least max1,n-1.
On exit: the off-diagonal elements of the tridiagonal matrix T.
8:     tau[dim]doubleOutput
Note: the dimension, dim, of the array tau must be at least max1,n-1.
On exit: further details of the orthogonal matrix Q.
9:     failNagError *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.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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 c n 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  Parallelism and Performance

nag_dsytrd (f08fec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dsytrd (f08fec) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is approximately 43 n3 .
To form the orthogonal matrix Q nag_dsytrd (f08fec) may be followed by a call to nag_dorgtr (f08ffc):
nag_dorgtr(order,uplo,n,&a,pda,tau,&fail)
To apply Q to an n by p real matrix C nag_dsytrd (f08fec) may be followed by a call to nag_dormtr (f08fgc). For example,
nag_dormtr(order,Nag_LeftSide,uplo,Nag_NoTrans,n,p,&a,pda,
  tau,&c,pdc,&fail)
forms the matrix product QC.
The complex analogue of this function is nag_zhetrd (f08fsc).

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 .

10.1  Program Text

Program Text (f08fece.c)

10.2  Program Data

Program Data (f08fece.d)

10.3  Program Results

Program Results (f08fece.r)


nag_dsytrd (f08fec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014