NAG FL Interface
f08kef (dgebrd)

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1 Purpose

f08kef reduces a real m×n matrix to bidiagonal form.

2 Specification

Fortran Interface
Subroutine f08kef ( m, n, a, lda, d, e, tauq, taup, work, lwork, info)
Integer, Intent (In) :: m, n, lda, lwork
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), d(*), e(*), tauq(*), taup(*)
Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
C Header Interface
#include <nag.h>
void  f08kef_ (const Integer *m, const Integer *n, double a[], const Integer *lda, double d[], double e[], double tauq[], double taup[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08kef, nagf_lapackeig_dgebrd or its LAPACK name dgebrd.

3 Description

f08kef reduces a real m×n matrix A to bidiagonal form B by an orthogonal transformation: A=QBPT, where Q and PT are orthogonal matrices of order m and n respectively.
If mn, the reduction is given by:
A =Q ( B1 0 ) PT = Q1 B1 PT ,  
where B1 is an n×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 ) PT = Q B1 P1T ,  
where B1 is an m×m lower bidiagonal matrix and P1T consists of the first m rows of PT.
The orthogonal 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 Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
3: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the m×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 orthogonal matrix Q and elements above the first superdiagonal are overwritten by details of the orthogonal 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 orthogonal matrix Q and elements above the diagonal are overwritten by details of the orthogonal matrix P.
4: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08kef is called.
Constraint: ldamax(1,m).
5: d(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array d must be at least max(1,min(m,n)).
On exit: the diagonal elements of the bidiagonal matrix B.
6: e(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array e must be at least max(1,min(m,n)-1).
On exit: the off-diagonal elements of the bidiagonal matrix B.
7: tauq(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array tauq must be at least max(1,min(m,n)).
On exit: further details of the orthogonal matrix Q.
8: taup(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array taup must be at least max(1,min(m,n)).
On exit: further details of the orthogonal matrix P.
9: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
10: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08kef 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(m+n)×nb, where nb is the optimal block size.
Constraint: lworkmax(1,m,n) or lwork=−1.
11: info Integer Output
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 QBPT=A+E, where
E2 c (n) ε A2 ,  
c(n) 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

Background information to multithreading can be found in the Multithreading documentation.
f08kef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kef 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 floating-point operations is approximately 43n2(3m-n) if mn or 43m2(3n-m) if m<n.
If mn, it can be more efficient to first call f08aef to perform a QR factorization of A, and then to call f08kef to reduce the factor R to bidiagonal form. This requires approximately 2n2(m+n) floating-point operations.
If mn, it can be more efficient to first call f08ahf to perform an LQ factorization of A, and then to call f08kef to reduce the factor L to bidiagonal form. This requires approximately 2m2(m+n) operations.
To form the m×m orthogonal matrix Q f08kef may be followed by a call to f08kff . For example
Call dorgbr('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 f08kef.
To form the n×n orthogonal matrix PT another call to f08kff may be made . For example
Call dorgbr('P',n,n,m,a,lda,taup,work,lwork,info)
but note that the first dimension of the array a, must be at least n, which may be larger than was required by f08kef.
To apply Q or P to a real rectangular matrix C, f08kef may be followed by a call to f08kgf.
The complex analogue of this routine is f08ksf.

10 Example

This example reduces the matrix A to bidiagonal form, where
A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ) .  

10.1 Program Text

Program Text (f08kefe.f90)

10.2 Program Data

Program Data (f08kefe.d)

10.3 Program Results

Program Results (f08kefe.r)