The function may be called by the names: f08msc, nag_lapackeig_zbdsqr or nag_zbdsqr.
3Description
f08msc computes the singular values and, optionally, the left or right singular vectors of a real upper or lower bidiagonal matrix . In other words, it can compute the singular value decomposition (SVD) of as
Here is a diagonal matrix with real diagonal elements (the singular values of ), such that
is an orthogonal matrix whose columns are the left singular vectors ; is an orthogonal matrix whose rows are the right singular vectors . Thus
To compute and/or , the arrays u and/or vt must be initialized to the unit matrix before f08msc is called.
The function stores the real orthogonal matrices and in complex arrays u and vt, so that it may also be used to compute the SVD of a complex general matrix which has been reduced to bidiagonal form by a unitary transformation: . If is with , then is and is ; if is with , then is and is . In this case, the matrices and/or must be formed explicitly by f08ktc and passed to f08msc in the arrays u and/or vt respectively.
f08msc also has the capability of forming , where is an arbitrary complex matrix; this is needed when using the SVD to solve linear least squares problems.
f08msc uses two different algorithms. If any singular vectors are required (i.e., if or or ), the bidiagonal algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between and variants in order to handle graded matrices effectively (see Demmel and Kahan (1990)). If only singular values are required (i.e., if ), they are computed by the differential qd algorithm (see Fernando and Parlett (1994)), which is faster and can achieve even greater accuracy.
The singular vectors are normalized so that , but are determined only to within a complex factor of absolute value .
4References
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput.11 873–912
Fernando K V and Parlett B N (1994) Accurate singular values and differential qd algorithms Numer. Math.67 191–229
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 an upper or lower bidiagonal matrix.
is an upper bidiagonal matrix.
is a lower bidiagonal matrix.
Constraint:
or .
3: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of columns of the matrix of right singular vectors. Set if no right singular vectors are required.
Constraint:
.
5: – IntegerInput
On entry: , the number of rows of the matrix of left singular vectors. Set if no left singular vectors are required.
Constraint:
.
6: – IntegerInput
On entry: , the number of columns of the matrix . Set if no matrix is supplied.
Constraint:
.
7: – 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: the singular values in decreasing order of magnitude, unless NE_CONVERGENCE (in which case see Section 6).
8: – doubleInput/Output
Note: the dimension, dim, of the array e
must be at least
.
On entry: the off-diagonal elements of the bidiagonal matrix .
Note: the dimension, dim, of the array vt
must be at least
when
and at least when
.
The th element of the matrix is stored in
when ;
when .
On entry: if , vt must contain an matrix. If the right singular vectors of are required, and vt must contain the unit matrix; if the right singular vectors of are required, vt must contain the unitary matrix returned by f08ktc with .
On exit: the matrix or of right singular vectors, stored by rows.
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
if ,
if , ;
otherwise ;
if ,
if ,
;
otherwise .
11: – ComplexInput/Output
Note: the dimension, dim, of the array u
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: if , u must contain an matrix. If the left singular vectors of are required, and u must contain the unit matrix; if the left singular vectors of are required, u must contain the unitary matrix returned by f08ktc with .
On exit: the matrix or of left singular vectors, stored as columns of the matrix.
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
if ,
;
if , .
13: – ComplexInput/Output
Note: the dimension, dim, of the array c
must be at least
when
and at least when
.
The th element of the matrix is stored in
when ;
when .
On entry: the matrix if .
On exit: c is overwritten by the matrix . If , c is not referenced.
14: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
if ,
if , ;
otherwise ;
if , .
15: – NagError *Input/Output
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_CONVERGENCE
off-diagonals did not converge. The arrays d and e contain the diagonal and off-diagonal elements, respectively, of a bidiagonal matrix orthogonally equivalent to .
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: if ,
; otherwise .
NE_INT_3
On entry, , and .
Constraint: if , ;
otherwise .
On entry, , and .
Constraint: if , ;
otherwise .
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.
7Accuracy
Each singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the function) may exclude the possibility of obtaining high relative accuracy in the small singular values of the original matrix if its singular values vary widely in magnitude.
If is an exact singular value of and is the corresponding computed value, then
where is a modestly increasing function of and , and is the machine precision. If only singular values are computed, they are computed more accurately (i.e., the function is smaller), than when some singular vectors are also computed.
If is an exact left singular vector of , and is the corresponding computed left singular vector, then the angle between them is bounded as follows:
where is the relative gap between and the other singular values, defined by
A similar error bound holds for the right singular vectors.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08msc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08msc 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
The total number of real floating-point operations is roughly proportional to if only the singular values are computed. About additional operations are required to compute the left singular vectors and about to compute the right singular vectors. The operations to compute the singular values must all be performed in scalar mode; the additional operations to compute the singular vectors can be vectorized and on some machines may be performed much faster.