NAG Library Routine Document
F08GEF (DSPTRD) reduces a real symmetric matrix to tridiagonal form, using packed storage.
||AP(*), D(N), E(N-1), TAU(N-1)
The routine may be called by its
F08GEF (DSPTRD) reduces a real symmetric matrix , held in packed storage, to symmetric tridiagonal form by an orthogonal similarity transformation: .
is not formed explicitly but is represented as a product of
elementary reflectors (see the F08 Chapter Introduction
for details). Routines are provided to work with
in this representation (see Section 9
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
- 1: – CHARACTER(1)Input
: indicates whether the upper or lower triangular part of
- The upper triangular part of is stored.
- The lower triangular part of is stored.
- 2: – INTEGERInput
On entry: , the order of the matrix .
- 3: – REAL (KIND=nag_wp) arrayInput/Output
the dimension of the array AP
must be at least
: the upper or lower triangle of the
, packed by columns.
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
is overwritten by the tridiagonal matrix
and details of the orthogonal matrix
- 4: – REAL (KIND=nag_wp) arrayOutput
On exit: the diagonal elements of the tridiagonal matrix .
- 5: – REAL (KIND=nag_wp) arrayOutput
On exit: the off-diagonal elements of the tridiagonal matrix .
- 6: – REAL (KIND=nag_wp) arrayOutput
On exit: further details of the orthogonal matrix .
- 7: – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed tridiagonal matrix
is exactly similar to a nearby matrix
is a modestly increasing function of
is the machine precision
The elements of themselves may be sensitive to small perturbations in or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.
8 Parallelism and Performance
F08GEF (DSPTRD) 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.
The total number of floating-point operations is approximately .
To form the orthogonal matrix
F08GEF (DSPTRD) may be followed by a call to F08GFF (DOPGTR)
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, &
forms the matrix product
The complex analogue of this routine is F08GSF (ZHPTRD)
This example reduces the matrix
to tridiagonal form, where
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)