The routine may be called by the names f08vqf, nagf_lapackeig_zggsvd3 or its LAPACK name zggsvd3.
3Description
Given an complex matrix and a complex matrix , the generalized singular value decomposition is given by
where , and are unitary matrices. Let be the effective numerical rank of and be the effective numerical rank of the matrix , then the first generalized singular values are infinite and the remaining are finite. is a nonsingular upper triangular matrix, and are and ‘diagonal’ matrices structured as follows:
if ,
where
and
is stored as a submatrix of with elements stored as on exit.
If ,
where
and
is stored as a submatrix of with stored as , and is stored as a submatrix of with stored as .
The routine computes , , and, optionally, the unitary transformation matrices , and .
In particular, if is an nonsingular matrix, then the GSVD of and implicitly gives the SVD of :
If has orthonormal columns, then the GSVD of and is also equal to the CS decomposition of and . Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
In some literature, the GSVD of and is presented in the form
where and are orthogonal and is nonsingular, and and are ‘diagonal’. The former GSVD form can be converted to the latter form by setting
A two stage process is used to compute the GSVD of the matrix pair . The pair is first reduced to upper triangular form by unitary transformations using f08vuf. The GSVD of the resulting upper triangular matrix pair is then performed by f08ysf which uses a variant of the Kogbetliantz algorithm (a cyclic Jacobi method).
4References
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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – Character(1)Input
On entry: if , the unitary matrix is computed.
If , is not computed.
Constraint:
or .
2: – Character(1)Input
On entry: if , the unitary matrix is computed.
If , is not computed.
Constraint:
or .
3: – Character(1)Input
On entry: if , the unitary matrix is computed.
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 columns of the matrices and .
Constraint:
.
6: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
7: – IntegerOutput
8: – IntegerOutput
On exit: k and l specify the dimension of the subblocks and as described in Section 3; is the effective numerical rank of .
9: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the matrix .
On exit: contains the triangular matrix , or part of . See Section 3 for details.
10: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08vqf is called.
Constraint:
.
11: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
On entry: the matrix .
On exit: contains the triangular matrix if . See Section 3 for details.
12: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08vqf is called.
On entry: the first dimension of the array q as declared in the (sub)program from which f08vqf is called.
Constraints:
if , ;
otherwise .
21: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
22: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08vqf 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, lwork must generally be larger than the minimum; increase workspace by, say, , where is the optimal block size.
Constraints:
if ,
if , ;
if , .
23: – Real (Kind=nag_wp) arrayWorkspace
24: – Integer arrayOutput
On exit: stores the sorting information. More precisely, if is the ordered set of indices of alpha containing (denote as , see beta), then the corresponding elements contain the swap pivots, , that sorts such that is in descending numerical order.
The following pseudocode sorts the set :
25: – 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.
The Jacobi-type procedure failed to converge.
7Accuracy
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08vqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08vqf 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
This routine replaces the deprecated routine f08vnf which used an unblocked algorithm and, therefore, did not make best use of Level 3 BLAS routines.