NAG CL Interface
f08mec (dbdsqr)

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1 Purpose

f08mec computes the singular value decomposition of a real upper or lower bidiagonal matrix, or of a real general matrix which has been reduced to bidiagonal form.

2 Specification

#include <nag.h>
void  f08mec (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer ncvt, Integer nru, Integer ncc, double d[], double e[], double vt[], Integer pdvt, double u[], Integer pdu, double c[], Integer pdc, NagError *fail)
The function may be called by the names: f08mec, nag_lapackeig_dbdsqr or nag_dbdsqr.

3 Description

f08mec computes the singular values and, optionally, the left or right singular vectors of a real upper or lower bidiagonal matrix B. In other words, it can compute the singular value decomposition (SVD) of B as
B = U Σ VT .  
Here Σ is a diagonal matrix with real diagonal elements σi (the singular values of B), such that
σ1 σ2 σn 0 ;  
U is an orthogonal matrix whose columns are the left singular vectors ui; V is an orthogonal matrix whose rows are the right singular vectors vi. Thus
Bui = σi vi   and   BT vi = σi ui ,   i = 1,2,,n .  
To compute U and/or VT, the arrays u and/or vt must be initialized to the unit matrix before f08mec is called.
The function may also be used to compute the SVD of a real general matrix A which has been reduced to bidiagonal form by an orthogonal transformation: A=QBPT. If A is m×n with mn, then Q is m×n and PT is n×n; if A is n×p with n<p, then Q is n×n and PT is n×p. In this case, the matrices Q and/or PT must be formed explicitly by f08kfc and passed to f08mec in the arrays u and/or vt respectively.
f08mec also has the capability of forming UTC, where C is an arbitrary real matrix; this is needed when using the SVD to solve linear least squares problems.
f08mec uses two different algorithms. If any singular vectors are required (i.e., if ncvt>0 or nru>0 or ncc>0), the bidiagonal QR algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between QR and QL variants in order to handle graded matrices effectively (see Demmel and Kahan (1990)). If only singular values are required (i.e., if ncvt=nru=ncc=0), 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 ui=vi=1, but are determined only to within a factor ±1.

4 References

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

5 Arguments

1: order Nag_OrderType Input
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 order=Nag_RowMajor. 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: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: indicates whether B is an upper or lower bidiagonal matrix.
uplo=Nag_Upper
B is an upper bidiagonal matrix.
uplo=Nag_Lower
B is a lower bidiagonal matrix.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix B.
Constraint: n0.
4: ncvt Integer Input
On entry: ncvt, the number of columns of the matrix VT of right singular vectors. Set ncvt=0 if no right singular vectors are required.
Constraint: ncvt0.
5: nru Integer Input
On entry: nru, the number of rows of the matrix U of left singular vectors. Set nru=0 if no left singular vectors are required.
Constraint: nru0.
6: ncc Integer Input
On entry: ncc, the number of columns of the matrix C. Set ncc=0 if no matrix C is supplied.
Constraint: ncc0.
7: d[dim] double Input/Output
Note: the dimension, dim, of the array d must be at least max(1,n).
On entry: the diagonal elements of the bidiagonal matrix B.
On exit: the singular values in decreasing order of magnitude, unless fail.code= NE_CONVERGENCE (in which case see Section 6).
8: e[dim] double Input/Output
Note: the dimension, dim, of the array e must be at least max(1,n-1).
On entry: the off-diagonal elements of the bidiagonal matrix B.
On exit: e is overwritten, but if fail.code= NE_CONVERGENCE see Section 6.
9: vt[dim] double Input/Output
Note: the dimension, dim, of the array vt must be at least max(1,pdvt×ncvt) when order=Nag_ColMajor and at least max(1,pdvt×n) when order=Nag_RowMajor.
The (i,j)th element of the matrix is stored in
  • vt[(j-1)×pdvt+i-1] when order=Nag_ColMajor;
  • vt[(i-1)×pdvt+j-1] when order=Nag_RowMajor.
On entry: if ncvt>0, vt must contain an n×ncvt matrix. If the right singular vectors of B are required, ncvt=n and vt must contain the unit matrix; if the right singular vectors of A are required, vt must contain the orthogonal matrix PT returned by f08kfc with vect=Nag_ApplyP.
On exit: the n×ncvt matrix VT or VTPT of right singular vectors, stored by rows.
If ncvt=0, vt is not referenced.
10: pdvt Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
  • if order=Nag_ColMajor,
    • if ncvt>0, pdvt max(1,n) ;
    • otherwise pdvt1;
  • if order=Nag_RowMajor,
    • if ncvt>0, pdvtncvt;
    • otherwise pdvt1.
11: u[dim] double Input/Output
Note: the dimension, dim, of the array u must be at least
  • max(1,pdu×n) when order=Nag_ColMajor;
  • max(1,nru×pdu) when order=Nag_RowMajor.
The (i,j)th element of the matrix U is stored in
  • u[(j-1)×pdu+i-1] when order=Nag_ColMajor;
  • u[(i-1)×pdu+j-1] when order=Nag_RowMajor.
On entry: if nru>0, u must contain an nru×n matrix. If the left singular vectors of B are required, nru=n and u must contain the unit matrix; if the left singular vectors of A are required, u must contain the orthogonal matrix Q returned by f08kfc with vect=Nag_ApplyQ.
On exit: the nru×n matrix U or QU of left singular vectors, stored as columns of the matrix.
If nru=0, u is not referenced.
12: pdu Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
  • if order=Nag_ColMajor, pdu max(1,nru) ;
  • if order=Nag_RowMajor, pdumax(1,n).
13: c[dim] double Input/Output
Note: the dimension, dim, of the array c must be at least max(1,pdc×ncc) when order=Nag_ColMajor and at least max(1,pdc×n) when order=Nag_RowMajor.
The (i,j)th element of the matrix C is stored in
  • c[(j-1)×pdc+i-1] when order=Nag_ColMajor;
  • c[(i-1)×pdc+j-1] when order=Nag_RowMajor.
On entry: the n×ncc matrix C if ncc>0.
On exit: c is overwritten by the matrix UTC. If ncc=0, c is not referenced.
14: pdc Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
  • if order=Nag_ColMajor,
    • if ncc>0, pdc max(1,n) ;
    • otherwise pdc1;
  • if order=Nag_RowMajor, pdcmax(1,ncc).
15: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 value had an illegal value.
NE_CONVERGENCE
value 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 B.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, ncc=value.
Constraint: ncc0.
On entry, ncvt=value.
Constraint: ncvt>0.
On entry, ncvt=value.
Constraint: ncvt0.
On entry, nru=value.
Constraint: nru0.
On entry, pdc=value.
Constraint: pdc>0.
On entry, pdu=value.
Constraint: pdu>0.
On entry, pdvt=value.
Constraint: pdvt>0.
NE_INT_2
On entry, pdc=value and ncc=value.
Constraint: pdcmax(1,ncc).
On entry, pdu=value and n=value.
Constraint: pdumax(1,n).
On entry, pdu=value and nru=value.
Constraint: pdu max(1,nru) .
On entry, pdvt=value and ncvt=value.
Constraint: if ncvt>0, pdvtncvt;
otherwise pdvt1.
NE_INT_3
On entry, ncc=value, pdc=value and n=value.
Constraint: if ncc>0, pdc max(1,n) ;
otherwise pdc1.
On entry, pdvt=value, ncvt=value and n=value.
Constraint: if ncvt>0, pdvt max(1,n) ;
otherwise pdvt1.
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.

7 Accuracy

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 σi is an exact singular value of B and σ~i is the corresponding computed value, then
|σ~i-σi| p (m,n) ε σi  
where p(m,n) is a modestly increasing function of m and n, and ε is the machine precision. If only singular values are computed, they are computed more accurately (i.e., the function p(m,n) is smaller), than when some singular vectors are also computed.
If ui is the corresponding exact left singular vector of B, and u~i is the corresponding computed left singular vector, then the angle θ(u~i,ui) between them is bounded as follows:
θ (u~i,ui) p (m,n) ε relgapi  
where relgapi is the relative gap between σi and the other singular values, defined by
relgapi = min ij |σi-σj| (σi+σj) .  
A similar error bound holds for the right singular vectors.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08mec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08mec 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.

9 Further Comments

The total number of floating-point operations is roughly proportional to n2 if only the singular values are computed. About 6n2×nru additional operations are required to compute the left singular vectors and about 6n2×ncvt 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.
The complex analogue of this function is f08msc.

10 Example

This example computes the singular value decomposition of the upper bidiagonal matrix B, where
B = ( 3.62 1.26 0.00 0.00 0.00 -2.41 -1.53 0.00 0.00 0.00 1.92 1.19 0.00 0.00 0.00 -1.43 ) .  
See also the example for f08kfc, which illustrates the use of the function to compute the singular value decomposition of a general matrix.

10.1 Program Text

Program Text (f08mece.c)

10.2 Program Data

Program Data (f08mece.d)

10.3 Program Results

Program Results (f08mece.r)