The routine may be called by the names f08kpf, nagf_lapackeig_zgesvd or its LAPACK name zgesvd.
The SVD is written as
where is an matrix which is zero except for its diagonal elements, is an unitary matrix, and is an unitary matrix. The diagonal elements of are the singular values of ; they are real and non-negative, and are returned in descending order. The first columns of and are the left and right singular vectors of .
Note that the routine returns , not .
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 (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – Character(1)Input
On entry: specifies options for computing all or part of the matrix .
On entry: the first dimension of the array vt as declared in the (sub)program from which f08kpf is called.
if , ;
if , ;
12: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
13: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08kpf 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.
for optimal performance, lwork should generally be larger. Consider increasing lwork by at least , where is the optimal block size.
14: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array rwork
must be at least
On exit: if , contains the unconverged superdiagonal elements of an upper bidiagonal matrix whose diagonal is in (not necessarily sorted). satisfies , so it has the same singular values as , and singular vectors related by and .
15: – 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.
If f08kpf did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix , where
and is the machine precision. In addition, the computed singular vectors are nearly orthogonal to working precision. See Section 4.9 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08kpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kpf 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.
The total number of floating-point operations is approximately proportional to when and otherwise.
The singular values are returned in descending order.