The routine may be called by its
LAPACK
name dgbbrd.
3
Description
f08lef (dgbbrd) reduces a real by band matrix to upper bidiagonal form by an orthogonal transformation: . The orthogonal matrices and , of order and respectively, are determined as a product of Givens rotation matrices, and may be formed explicitly by the routine if required. A matrix may also be updated to give .
The routine uses a vectorizable form of the reduction.
4
References
None.
5
Arguments
1: – Character(1)Input
On entry: indicates whether the matrices and/or are generated.
Neither nor is generated.
is generated.
is generated.
Both and are generated.
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: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
5: – IntegerInput
On entry: the number of subdiagonals, , within the band of .
Constraint:
.
6: – IntegerInput
On entry: the number of superdiagonals, , within the band of .
Constraint:
.
7: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
.
On entry: the original by band matrix .
The matrix is stored in rows to , more precisely, the element must be stored in
On exit: ab is overwritten by values generated during the reduction.
8: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f08lef (dgbbrd) is called.
Constraint:
.
9: – Real (Kind=nag_wp) arrayOutput
On exit: the diagonal elements of the bidiagonal matrix .
10: – Real (Kind=nag_wp) arrayOutput
On exit: the superdiagonal elements of the bidiagonal matrix .
11: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array q
must be at least
if or , and at least otherwise.
On exit: if or , contains the by orthogonal matrix .
On entry: the first dimension of the array q as declared in the (sub)program from which f08lef (dgbbrd) is called.
Constraints:
if or , ;
otherwise .
13: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array pt
must be at least
if or , and at least otherwise.
On exit: the by orthogonal matrix , if or . If or , pt is not referenced.
14: – IntegerInput
On entry: the first dimension of the array pt as declared in the (sub)program from which f08lef (dgbbrd) is called.
Constraints:
if or , ;
otherwise .
15: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array c
must be at least
.
On entry: an by matrix .
On exit: c is overwritten by . If , c is not referenced.
16: – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08lef (dgbbrd) is called.
Constraints:
if , ;
if , .
17: – Real (Kind=nag_wp) arrayWorkspace
18: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
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.
The computed matrix differs from an exactly orthogonal matrix by a matrix such that
A similar statement holds for the computed matrix .
8
Parallelism and Performance
f08lef (dgbbrd) 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 the sum of:
, if and , and
, if is updated, and
, if either or is generated (double this if both),
where , assuming . For this section we assume that .