NAG Library Routine Document
F08FEF (DSYTRD)
1 Purpose
F08FEF (DSYTRD) reduces a real symmetric matrix to tridiagonal form.
2 Specification
INTEGER |
N, LDA, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), D(*), E(*), TAU(*), WORK(max(1,LWORK)) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dsytrd.
3 Description
F08FEF (DSYTRD) 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). 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' or 'L'.
- 2: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 3: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the
n by
n symmetric matrix
A.
- If UPLO='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
- If UPLO='L', 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.
- 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08FEF (DSYTRD) is called.
Constraint:
LDA≥max1,N.
- 5: D(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
D
must be at least
max1,N.
On exit: the diagonal elements of the tridiagonal matrix T.
- 6: E(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
E
must be at least
max1,N-1.
On exit: the off-diagonal elements of the tridiagonal matrix T.
- 7: TAU(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAU
must be at least
max1,N-1.
On exit: further details of the orthogonal matrix Q.
- 8: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 9: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08FEF (DSYTRD) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the
WORK array, returns this value as the first entry of the
WORK array, and no error message related to
LWORK is issued.
Suggested value:
for optimal performance, LWORK≥N×nb, where nb is the optimal block size.
Constraint:
LWORK≥1 or LWORK=-1.
- 10: 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
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 F08FEF (DSYTRD) may be followed by a call to
F08FFF (DORGTR):
CALL DORGTR(UPLO,N,A,LDA,TAU,WORK,LWORK,INFO)
To apply
Q to an
n by
p real matrix
C F08FEF (DSYTRD) may be followed by a call to
F08FGF (DORMTR). For example,
CALL DORMTR('Left',UPLO,'No Transpose',N,P,A,LDA,TAU,C,LDC, &
WORK,LWORK,INFO)
forms the matrix product
QC.
The complex analogue of this routine is
F08FSF (ZHETRD).
9 Example
This example reduces the matrix
A to tridiagonal form, where
9.1 Program Text
Program Text (f08fefe.f90)
9.2 Program Data
Program Data (f08fefe.d)
9.3 Program Results
Program Results (f08fefe.r)