The routine may be called by the names f08ksf, nagf_lapackeig_zgebrd or its LAPACK name zgebrd.
3Description
f08ksf reduces a complex matrix to real bidiagonal form by a unitary transformation: , where and are unitary matrices of order and respectively.
If , the reduction is given by:
where is a real upper bidiagonal matrix and consists of the first columns of .
If , the reduction is given by
where is a real lower bidiagonal matrix and consists of the first rows of .
The unitary matrices and 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 and in this representation (see Section 9).
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
2: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
3: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the matrix .
On exit: if , the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix , elements below the diagonal are overwritten by details of the unitary matrix and elements above the first superdiagonal are overwritten by details of the unitary matrix .
If , the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix , elements below the first subdiagonal are overwritten by details of the unitary matrix and elements above the diagonal are overwritten by details of the unitary matrix .
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08ksf is called.
Constraint:
.
5: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array d
must be at least
.
On exit: the diagonal elements of the bidiagonal matrix .
6: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array e
must be at least
.
On exit: the off-diagonal elements of the bidiagonal matrix .
7: – Complex (Kind=nag_wp) arrayOutput
Note: the dimension of the array tauq
must be at least
.
On exit: further details of the unitary matrix .
8: – Complex (Kind=nag_wp) arrayOutput
Note: the dimension of the array taup
must be at least
.
On exit: further details of the unitary matrix .
9: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
10: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08ksf is called.
If , 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, , where is the optimal block size.
Constraint:
or .
11: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7Accuracy
The computed bidiagonal form satisfies , where
is a modestly increasing function of , and is the machine precision.
The elements of themselves may be sensitive to small perturbations in or to rounding errors in the computation, but this does not affect the stability of the singular values and vectors.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08ksf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ksf 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.
9Further Comments
The total number of real floating-point operations is approximately if or if .
If , it can be more efficient to first call f08asf to perform a factorization of , and then to call f08ksf to reduce the factor to bidiagonal form. This requires approximately floating-point operations.
If , it can be more efficient to first call f08avf to perform an factorization of , and then to call f08ksf to reduce the factor to bidiagonal form. This requires approximately operations.
To form the unitary matrix f08ksf may be followed by calls to f08ktf
. For example
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.
To form the unitary matrix another call to f08kff may be made
. For example
Call zungbr('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 f08ksf.
To apply or to a complex rectangular matrix , f08ksf may be followed by a call to f08kuf.