NAG FL Interface
f08gef (dsptrd)

1 Purpose

f08gef reduces a real symmetric matrix to tridiagonal form, using packed storage.

2 Specification

Fortran Interface
Subroutine f08gef ( uplo, n, ap, d, e, tau, info)
Integer, Intent (In) :: n
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: ap(*)
Real (Kind=nag_wp), Intent (Out) :: d(n), e(n-1), tau(n-1)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f08gef_ (const char *uplo, const Integer *n, double ap[], double d[], double e[], double tau[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f08gef, nagf_lapackeig_dsptrd or its LAPACK name dsptrd.

3 Description

f08gef 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). Routines 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: uplo Character(1) Input
On entry: indicates whether the upper or lower triangular part of A is stored.
The upper triangular part of A is stored.
The lower triangular part of A is stored.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: ap* Real (Kind=nag_wp) array Input/Output
Note: the dimension 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 columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.
On exit: ap is overwritten by the tridiagonal matrix T and details of the orthogonal matrix Q.
4: dn Real (Kind=nag_wp) array Output
On exit: the diagonal elements of the tridiagonal matrix T.
5: en-1 Real (Kind=nag_wp) array Output
On exit: the off-diagonal elements of the tridiagonal matrix T.
6: taun-1 Real (Kind=nag_wp) array Output
On exit: further details of the orthogonal matrix Q.
7: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If 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 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 Parallelism and Performance

f08gef 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.

9 Further Comments

The total number of floating-point operations is approximately 43 n3 .
To form the orthogonal matrix Q f08gef may be followed by a call to f08gff:
Call dopgtr(uplo,n,ap,tau,q,ldq,work,info)
To apply Q to an n by p real matrix C f08gef may be followed by a call to f08ggf. For example,
Call dopmtr('Left',uplo,'No Transpose',n,p,ap,tau,c,ldc,work, &
forms the matrix product QC.
The complex analogue of this routine is f08gsf.

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 (f08gefe.f90)

10.2 Program Data

Program Data (f08gefe.d)

10.3 Program Results

Program Results (f08gefe.r)