The function may be called by the names: f08ksc, nag_lapackeig_zgebrd or nag_zgebrd.
3Description
f08ksc 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). Functions 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: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
4: – ComplexInput/Output
Note: the dimension, dim, of the array a
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
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 .
5: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
if ,
;
if , .
6: – doubleOutput
Note: the dimension, dim, of the array d
must be at least
.
On exit: the diagonal elements of the bidiagonal matrix .
7: – doubleOutput
Note: the dimension, dim, of the array e
must be at least
.
On exit: the off-diagonal elements of the bidiagonal matrix .
8: – ComplexOutput
Note: the dimension, dim, of the array tauq
must be at least
.
On exit: further details of the unitary matrix .
9: – ComplexOutput
Note: the dimension, dim, of the array taup
must be at least
.
On exit: further details of the unitary matrix .
10: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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.
f08ksc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ksc 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 function. 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 f08asc to perform a factorization of , and then to call f08ksc to reduce the factor to bidiagonal form. This requires approximately floating-point operations.
If , it can be more efficient to first call f08avc to perform an factorization of , and then to call f08ksc to reduce the factor to bidiagonal form. This requires approximately operations.
To form the unitary matrix f08ksc may be followed by calls to f08ktc
. For example