NAG Library Routine Document
F08NSF (ZGEHRD)
1 Purpose
F08NSF (ZGEHRD) reduces a complex general matrix to Hessenberg form.
2 Specification
INTEGER |
N, ILO, IHI, LDA, LWORK, INFO |
COMPLEX (KIND=nag_wp) |
A(LDA,*), TAU(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name zgehrd.
3 Description
F08NSF (ZGEHRD) reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: A=QHQH. H has real subdiagonal elements.
The matrix
Q 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
Q in this representation (see
Section 8).
The routine can take advantage of a previous call to
F08NVF (ZGEBAL), which may produce a matrix with the structure:
where
A11 and
A33 are upper triangular. If so, only the central diagonal block
A22, in rows and columns
ilo to
ihi, needs to be reduced to Hessenberg form (the blocks
A12 and
A23 will also be affected by the reduction). Therefore the values of
ilo and
ihi determined by
F08NVF (ZGEBAL) can be supplied to the routine directly. If
F08NVF (ZGEBAL) has not previously been called however, then
ilo must be set to
1 and
ihi to
n.
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 2: ILO – INTEGERInput
- 3: IHI – INTEGERInput
On entry: if
A has been output by
F08NVF (ZGEBAL), then
ILO and
IHI must contain the values returned by that routine. Otherwise,
ILO must be set to
1 and
IHI to
N.
Constraints:
- if N>0, 1≤ ILO≤ IHI≤ N ;
- if N=0, ILO=1 and IHI=0.
- 4: A(LDA,*) – COMPLEX (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 general matrix A.
On exit:
A is overwritten by the upper Hessenberg matrix
H and details of the unitary matrix
Q. The subdiagonal elements of
H are real.
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08NSF (ZGEHRD) is called.
Constraint:
LDA≥max1,N.
- 6: TAU(*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAU
must be at least
max1,N-1.
On exit: further details of the unitary matrix Q.
- 7: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 8: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08NSF (ZGEHRD) 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≥max1,N or LWORK=-1.
- 9: 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 Hessenberg matrix
H 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 H 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, eigenvectors or Schur factorization.
8 Further Comments
The total number of real floating point operations is approximately 83q22q+3n, where q=ihi-ilo; if ilo=1 and ihi=n, the number is approximately 403n3.
To form the unitary matrix
Q F08NSF (ZGEHRD) may be followed by a call to
F08NTF (ZUNGHR):
CALL ZUNGHR(N,ILO,IHI,A,LDA,TAU,WORK,LWORK,INFO)
To apply
Q to an
m by
n complex matrix
C F08NSF (ZGEHRD) may be followed by a call to
F08NUF (ZUNMHR). For example,
CALL ZUNMHR('Left','No Transpose',M,N,ILO,IHI,A,LDA,TAU,C,LDC, &
WORK,LWORK,INFO)
forms the matrix product QC.
The real analogue of this routine is
F08NEF (DGEHRD).
9 Example
This example computes the upper Hessenberg form of the matrix
A, where
9.1 Program Text
Program Text (f08nsfe.f90)
9.2 Program Data
Program Data (f08nsfe.d)
9.3 Program Results
Program Results (f08nsfe.r)