F08NSF (ZGEHRD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08NSF (ZGEHRD) reduces a complex general matrix to Hessenberg form.

2  Specification

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:
A11 A12 A13 A22 A23 A33
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: N0.
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.
  • 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: LDAmax1,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, LWORKN×nb, where nb is the optimal block size.
Constraint: LWORKmax1,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:
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
E2 c n ε A2 ,
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):
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, &
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
A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .

9.1  Program Text

Program Text (f08nsfe.f90)

9.2  Program Data

Program Data (f08nsfe.d)

9.3  Program Results

Program Results (f08nsfe.r)

F08NSF (ZGEHRD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012