The routine may be called by the names f08kdf, nagf_lapackeig_dgesdd or its LAPACK name dgesdd.
The SVD is written as
where is an matrix which is zero except for its diagonal elements, is an orthogonal matrix, and is an orthogonal 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 .
All columns of and all rows of are returned in the arrays u and vt.
The first columns of and the first rows of are returned in the arrays u and vt.
If , the first columns of are overwritten on the array a and all rows of are returned in the array vt. Otherwise, all columns of are returned in the array u and the first rows of are overwritten in the array vt.
No columns of or rows of are computed.
, , or .
2: – IntegerInput
On entry: , the number of rows of the matrix .
3: – IntegerInput
On entry: , the number of columns of the matrix .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the matrix .
On exit: if , a is overwritten with the first columns of (the left singular vectors, stored column-wise) if ; a is overwritten with the first rows of (the right singular vectors, stored row-wise) otherwise.
On entry: the first dimension of the array vt as declared in the (sub)program from which f08kdf is called.
if or ( and ), ;
if , ;
11: – Real (Kind=nag_wp) arrayWorkspace
12: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08kdf 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.
if , ;
if , ;
if or , ;
13: – Integer arrayWorkspace
14: – 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.
f08kdf did not converge, the updating process failed.
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
f08kdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kdf 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.