NAG Library Routine Document

F08GEF (DSPTRD)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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 = Q T Q^{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.

$UPLO ='U'$: The upper triangular part of $A$ is stored.
$UPLO ='L'$: The lower triangular part of $A$ is stored.

Constraint:

UPLO ='U'

'L'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: AP( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array AP must be at least

\max (1, N \times (N + 1) / 2)

On entry: the upper or lower triangle of the

n

n

symmetric matrix

A

, packed by columns.

More precisely,

if $UPLO ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $AP (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $UPLO ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $AP (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

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( $N - 1$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the off-diagonal elements of the tridiagonal matrix

T

6: TAU( $N - 1$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: further details of the orthogonal matrix

Q

7: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: 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

{‖E‖}_{2} \leq c (n) ε {‖A‖}_{2},

c (n)

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

\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

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

Q C

The complex analogue of this routine is F08GSF (ZHPTRD).

9 Example

This example reduces the matrix

A

to tridiagonal form, where

A = (\begin{array}{r} 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 \end{array}),

using packed storage.

NAG Library Routine DocumentF08GEF (DSPTRD)