f08yef (dtgsja) computes the generalized singular value decomposition (GSVD) of two real upper trapezoidal matrices and , where is an by matrix and is a by matrix.
The routine may be called by its
LAPACK
name dtgsja.
3
Description
f08yef (dtgsja) computes the GSVD of the matrices and which are assumed to have the form as returned by f08vef (dggsvp) or f08vgf (dggsvp3)
where the by matrix and the by matrix are nonsingular upper triangular, is by upper triangular if and is by upper trapezoidal otherwise.
f08yef (dtgsja) computes orthogonal matrices , and , diagonal matrices and , and an upper triangular matrix such that
Optionally , and may or may not be computed, or they may be premultiplied by matrices , and respectively.
If then , and have the form
where .
If then , and have the form
where .
In both cases the diagonal matrix has non-negative diagonal elements, the diagonal matrix has positive diagonal elements, so that is nonsingular, and . See Section 2.3.5.3 of Anderson et al. (1999) for further information.
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References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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Arguments
1: – Character(1)Input
On entry: if , u must contain an orthogonal matrix on entry, and the product is returned.
If , u is initialized to the unit matrix, and the orthogonal matrix is returned.
If , is not computed.
Constraint:
, or .
2: – Character(1)Input
On entry: if , v must contain an orthogonal matrix on entry, and the product is returned.
If , v is initialized to the unit matrix, and the orthogonal matrix is returned.
If , is not computed.
Constraint:
, or .
3: – Character(1)Input
On entry: if , q must contain an orthogonal matrix on entry, and the product is returned.
If , q is initialized to the unit matrix, and the orthogonal matrix is returned.
If , is not computed.
Constraint:
, or .
4: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
5: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
6: – IntegerInput
On entry: , the number of columns of the matrices and .
Constraint:
.
7: – IntegerInput
8: – IntegerInput
On entry: k and l specify the sizes, and , of the subblocks of and , whose GSVD is to be computed by f08yef (dtgsja).
9: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the by matrix .
On exit: if , contains the by upper triangular matrix .
If , contains the first rows of the by upper triangular matrix , and the submatrix is returned in .
10: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08yef (dtgsja) is called.
Constraint:
.
11: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
On entry: the by matrix .
On exit: if , contains the submatrix of .
12: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08yef (dtgsja) is called.
Constraint:
.
13: – Real (Kind=nag_wp)Input
14: – Real (Kind=nag_wp)Input
On entry: tola and tolb are the convergence criteria for the Jacobi–Kogbetliantz iteration procedure. Generally, they should be the same as used in the preprocessing step performed by f08vef (dggsvp) or f08vgf (dggsvp3), say
On entry: the first dimension of the array q as declared in the (sub)program from which f08yef (dtgsja) is called.
Constraints:
if or , ;
otherwise .
23: – Real (Kind=nag_wp) arrayWorkspace
24: – IntegerOutput
On exit: the number of cycles required for convergence.
25: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
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Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The procedure does not converge after cycles.
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Accuracy
The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices and , where
and is the machine precision. See Section 4.12 of Anderson et al. (1999) for further details.
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Parallelism and Performance
f08yef (dtgsja) 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.