The function may be called by the names: f08mdc, nag_lapackeig_dbdsdc or nag_dbdsdc.
3Description
f08mdc computes the singular value decomposition (SVD) of the (upper or lower) bidiagonal matrix as
where is a diagonal matrix with non-negative diagonal elements , such that
and and are orthogonal matrices. The diagonal elements of are the singular values of and the columns of and are respectively the corresponding left and right singular vectors of .
When only singular values are required the function uses the algorithm, but when singular vectors are required a divide and conquer method is used. The singular values can optionally be returned in compact form, although currently no function is available to apply or when stored in compact form.
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 (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_UploTypeInput
On entry: indicates whether is upper or lower bidiagonal.
is upper bidiagonal.
is lower bidiagonal.
Constraint:
or .
3: – Nag_ComputeSingularVecsTypeInput
On entry: specifies whether singular vectors are to be computed.
Compute singular values only.
Compute singular values and compute singular vectors in compact form.
Compute singular values and singular vectors.
Constraint:
, or .
4: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
5: – doubleInput/Output
Note: the dimension, dim, of the array d
must be at least
.
On entry: the diagonal elements of the bidiagonal matrix .
On exit: if NE_NOERROR, the singular values of .
6: – doubleInput/Output
Note: the dimension, dim, of the array e
must be at least
.
On entry: the off-diagonal elements of the bidiagonal matrix .
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
if , ;
otherwise .
11: – doubleOutput
Note: the dimension, dim, of the array q
must be at least
.
On exit: if , then if NE_NOERROR, q and iq contain the left and right singular vectors in a compact form, requiring space instead of . In particular, q contains all the real data in the first elements of q, where is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about ).
Note: the dimension, dim, of the array iq
must be at least
.
On exit: if , then if NE_NOERROR, q and iq contain the left and right singular vectors in a compact form, requiring space instead of . In particular, iq contains all integer data in the first elements of iq, where is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about ).
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_ENUM_INT_2
On entry, , and .
Constraint: if , ;
otherwise .
On entry, , and .
Constraint: if , ;
otherwise .
NE_INT
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
The algorithm failed to compute a singular value. The update process of divide-and-conquer failed.
7Accuracy
Each computed singular value of is accurate to nearly full relative precision, no matter how tiny the singular value. The th computed singular value, , satisfies the bound
where is the machine precision and is a modest function of .
f08mdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08mdc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
If only singular values are required, the total number of floating-point operations is approximately proportional to . When singular vectors are required the number of operations is bounded above by approximately the same number of operations as f08mec, but for large matrices f08mdc is usually much faster.
There is no complex analogue of f08mdc.
10Example
This example computes the singular value decomposition of the upper bidiagonal matrix