F08KSF (ZGEBRD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08KSF (ZGEBRD)

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

F08KSF (ZGEBRD) reduces a complex m by n matrix to bidiagonal form.

2  Specification

SUBROUTINE F08KSF ( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
INTEGER  M, N, LDA, LWORK, INFO
REAL (KIND=nag_wp)  D(*), E(*)
COMPLEX (KIND=nag_wp)  A(LDA,*), TAUQ(*), TAUP(*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name zgebrd.

3  Description

F08KSF (ZGEBRD) reduces a complex m by n matrix A to real bidiagonal form B by a unitary transformation: A=QBPH, where Q and PH are unitary matrices of order m and n respectively.
If mn, the reduction is given by:
A =Q B1 0 PH = Q1 B1 PH ,  
where B1 is a real n by n upper bidiagonal matrix and Q1 consists of the first n columns of Q.
If m<n, the reduction is given by
A =Q B1 0 PH = Q B1 P1H ,  
where B1 is a real m by m lower bidiagonal matrix and P1H consists of the first m rows of PH.
The unitary matrices Q and P are not formed explicitly but are represented as products of elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with Q and P in this representation (see Section 9).

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
2:     N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint: N0.
3:     ALDA* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the m by n matrix A.
On exit: if mn, the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix B, elements below the diagonal are overwritten by details of the unitary matrix Q and elements above the first superdiagonal are overwritten by details of the unitary matrix P.
If m<n, the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix B, elements below the first subdiagonal are overwritten by details of the unitary matrix Q and elements above the diagonal are overwritten by details of the unitary matrix P.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08KSF (ZGEBRD) is called.
Constraint: LDAmax1,M.
5:     D* – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array D must be at least max1,minM,N.
On exit: the diagonal elements of the bidiagonal matrix B.
6:     E* – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array E must be at least max1,minM,N-1.
On exit: the off-diagonal elements of the bidiagonal matrix B.
7:     TAUQ* – COMPLEX (KIND=nag_wp) arrayOutput
Note: the dimension of the array TAUQ must be at least max1,minM,N.
On exit: further details of the unitary matrix Q.
8:     TAUP* – COMPLEX (KIND=nag_wp) arrayOutput
Note: the dimension of the array TAUP must be at least max1,minM,N.
On exit: further details of the unitary matrix P.
9:     WORKmax1,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.
10:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08KSF (ZGEBRD) 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, LWORKM+N×nb, where nb is the optimal block size.
Constraint: LWORKmax1,M,N or LWORK=-1.
11:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

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 bidiagonal form B satisfies QBPH=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 B 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 singular values and vectors.

8  Parallelism and Performance

F08KSF (ZGEBRD) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08KSF (ZGEBRD) 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.

9  Further Comments

The total number of real floating-point operations is approximately 16n23m-n/3 if mn or 16m23n-m/3 if m<n.
If mn, it can be more efficient to first call F08ASF (ZGEQRF) to perform a QR factorization of A, and then to call F08KSF (ZGEBRD) to reduce the factor R to bidiagonal form. This requires approximately 8n2m+n floating-point operations.
If mn, it can be more efficient to first call F08AVF (ZGELQF) to perform an LQ factorization of A, and then to call F08KSF (ZGEBRD) to reduce the factor L to bidiagonal form. This requires approximately 8m2m+n operations.
To form the unitary matrices PH and/or Q F08KSF (ZGEBRD) may be followed by calls to F08KTF (ZUNGBR):
to form the m by m unitary matrix Q 
CALL ZUNGBR('Q',M,M,N,A,LDA,TAUQ,WORK,LWORK,INFO)
but note that the second dimension of the array A must be at least M, which may be larger than was required by F08KSF (ZGEBRD);
to form the n by n unitary matrix PH 
CALL ZUNGBR('P',N,N,M,A,LDA,TAUP,WORK,LWORK,INFO)
but note that the first dimension of the array A, specified by the argument LDA, must be at least N, which may be larger than was required by F08KSF (ZGEBRD).
To apply Q or P to a complex rectangular matrix C, F08KSF (ZGEBRD) may be followed by a call to F08KUF (ZUNMBR).
The real analogue of this routine is F08KEF (DGEBRD).

10  Example

This example reduces the matrix A to bidiagonal form, where
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .  

10.1  Program Text

Program Text (f08ksfe.f90)

10.2  Program Data

Program Data (f08ksfe.d)

10.3  Program Results

Program Results (f08ksfe.r)


F08KSF (ZGEBRD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016